3DLDF User and Reference Manual

Copyright © 2003, 2004 Laurence D. Finston.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Please note that the figures cannot be shown in the Info output format.

• Introduction:
• Contributing to 3DLDF:
• Using 3DLDF:
• Points:
• Transforming Points:
• Transforms:
• Drawing and Labeling Points:
• Paths:
• Plane Figures:
• Solid Figures:
• Pictures:
• Intersections:
• Installing and Running 3DLDF:
• Typedefs and Utility Structures:
• Global Constants and Variables:
• Dynamic Allocation of Shapes:
• System Information:
• Color Reference:
• Input and Output:
• Shape Reference:
• Transform Reference:
• Label Reference:
• Picture Reference:
• Point Reference:
• Focus Reference:
• Line Reference:
• Plane Reference:
• Path Reference:
• Polygon Reference:
• Regular Polygon Reference:
• Rectangle Reference:
• Regular Closed Plane Curve Reference:
• Ellipse Reference:
• Circle Reference:
• Pattern Reference:
• Solid Reference:
• Faced Solid Reference:
• Cuboid Reference:
• Polyhedron Reference:
• Utility Functions:
• Adding a File:
• Future Plans:
• Changes:
• Bibliography:
• GNU Free Documentation License:
• Data Type and Variable Index:
• Function Index:
• Concept Index:

  Introduction

• Sources of Information:
• About This Manual:
• CWEB Documentation:
• Metafont and MetaPost:
• Caveats:
• Ports:

  About This Manual

• Manual Conventions:
• Illustrations:

  Caveats

• Accuracy:
• No Input Routine:

  Points

• Declaring and Initializing Points:
• Setting and Assigning to Points:
• Shifting Points:
• Scaling Points:
• Shearing Points:
• Rotating Points:

  Transforms

• Applying Transforms to Points Intro:
• Inverting Transforms:

  Drawing and Labeling Points

• Drawing Points Intro:
• Labeling Points Intro:

  Paths

• Declaring and Initializing Paths:
• Drawing and Filling Paths Intro:

  Plane Figures

• Regular Polygons Getstart:
• Rectangles Getstart:
• Ellipses Getstart:
• Circles Getstart:

  Solid Figures

• Cuboid Getstart:
• Polyhedron Getstart:

  Polyhedron

• Tetrahedron Getstart:
• Dodecahedron Getstart:
• Icosahedron Getstart:

  Pictures

• Projections:
• Focuses Getstart:
• Surface Hiding:

  Projections

• Parallel Projections:
• The Perspective Projection:

  Installing and Running 3DLDF

• Installing 3DLDF:
• Running 3DLDF:

  Installing 3DLDF

• Template Functions:

  Running 3DLDF

• Converting EPS Files:
• Command Line Arguments:

  Converting EPS Files

• Converting EPS Files ELISP:

  System Information

• Endianness:
• Register Width:
• Get Second Largest Real:

  Color Reference

• Color Data Members:
• Color Constructors and Setting Functions:
• Color Operators:
• Modifying Colors:
• Showing Colors:
• Querying Colors:
• Defining and Initializing Colors:
• Namespace Colors:

  Input and Output

• I/O Global Variables:
• I/O Functions:

  Shape Reference

• Shape Data Members:
• Shape Operators:
• Copying Shapes:
• Modifying Shapes:
• Affine Transformations for Shapes:
• Applying Transformations to Shapes:
• Clearing Shapes:
• Querying Shapes:
• Showing Shapes:
• Outputting Shapes:

  Transform Reference

• Transform Data Members:
• Transform Global Variables and Constants:
• Transform Constructors :
• Transform Operators:
• Matrix Inversion:
• Setting Values Transforms:
• Querying Transforms:
• Returning Information for Transforms:
• Showing Transforms:
• Affine Transformations for Transforms:
• Alignment with an Axis for Transforms:
• Resetting Transforms:
• Cleaning Transforms:

  Label Reference

• Label Data Members:
• Copying Labels:
• Outputting Labels:

  Picture Reference

• Picture Data Members:
• Picture Global Variables:
• Picture Constructors:
• Picture Operators:
• Affine Transformations for Pictures:
• Modifying Pictures:
• Showing Pictures:
• Outputting Pictures:

  Outputting

• Picture Output Namespaces:
• Picture Output Functions:

  Namespaces

• Namespace Projections:
• Namespace Sorting:

  Point Reference

• Point Data Members:
• Point Typedefs and Utility Structures:
• Point Global Constants and Variables:
• Point Constructors and Setting Functions:
• Point Destructor:
• Point Operators:
• Copying Points:
• Querying Points:
• Returning Coordinates:
• Returning Information for Points:
• Modifying Points:
• Affine Transformations for Points:
• Applying Transformations to Points:
• Projecting Points:
• Vector Operations:
• Points and Lines:
• Point Intersections:
• Point Drawing Functions:
• Labelling Points:
• Showing Points:
• Outputting Points:

  Focus Reference

• Focus Data Members:
• Focus Global Variables:
• Focus Constructors and Setting Functions:
• Focus Operators:
• Modifying Focuses:
• Querying Focuses:
• Showing Focuses:

  Line Reference

• Line Data Members:
• Line Global Constants:
• Line Constructors:
• Line Operators:
• Get Path:
• Showing:

  Plane Reference

• Planes Data Members:
• Planes Global Constants:
• Planes Constructors:
• Planes Operators:
• Planes Returning Information:
• Plane Intersections:
• Planes Showing:

  Path Reference

• Path Data Members:
• Path Constructors and Setting Functions:
• Path Destructor:
• Path Operators:
• Appending to Paths :
• Copying Paths:
• Clearing Paths :
• Modifying Paths :
• Affine Transformations for Paths:
• Aligning Paths with an Axis:
• Applying Transformations to Paths:
• Drawing and Filling Paths:
• Labelling Paths:
• Showing Paths:
• Querying Paths:
• Outputting Paths:
• Path Intersections:

  Polygon Reference

• Polygon Data Members:
• Polygon Operators:
• Querying Polygons:
• Affine Transformations for Polygons:
• Polygon Intersections:

  Regular Polygon Reference

• Regular Polygon Data Members:
• Regular Polygon Constructors and Setting Functions:
• Regular Polygon Operators:
• Querying Regular Polygons:
• Circles for Regular Polygons:

  Rectangle Reference

• Rectangle Data Members:
• Rectangle Constructors and Setting Functions:
• Rectangle Operators:
• Returning Points for Rectangles:
• Querying Rectangles:
• Ellipses for Rectangles:

  Regular Closed Plane Curve Reference

• Regular Closed Plane Curve Data Members:
• Querying Regular Closed Plane Curves:
• Regular Closed Plane Curve Intersections:
• Regular Closed Plane Curve Segments:

  Ellipse Reference

• Ellipse Data Members:
• Ellipse Constructors and Setting Functions:
• Performing Transformations on Ellipses:
• Ellipse Operators:
• Labeling Ellipses:
• Affine Transformations for Ellipses:
• Querying Ellipses:
• Returning Elements and Information for Ellipses:
• Ellipse Intersections:
• Solving Ellipses:
• Rectangles for Ellipses:

  Circle Reference

• Circle Data Members:
• Circle Constructors and Setting Functions:
• Circle Operators:
• Querying Circles:
• Circle Intersections:

  Pattern Reference

• Plane Tesselations:
• Roulettes and Involutes:

  Roulettes and Involutes

• Epicycloids:

  Solid Reference

• Solid Data Members:
• Solid Constructors and Setting Functions:
• Solid Destructor:
• Solid Operators:
• Copying Solids:
• Setting Solid Members:
• Querying Solids:
• Returning Elements and Information Solids:
• Showing Solids:
• Affine Transformations for Solids:
• Applying Transformations to Solids:
• Outputting Solids:
• Drawing and Filling Solids:
• Clearing Solids:

  Returning Elements and Information

• Getting Shape Centers Solids:
• Getting Shapes Solids:

  Faced Solid Reference

• Solid_Faced Data Members:

  Cuboid Reference

• Cuboid Data Members:
• Cuboid Constructors and Setting Functions:
• Cuboid Operators:

  Polyhedron Reference

• Polyhedron Data Members:
• Regular Platonic Polyhedra:
• Semi-Regular Archimedean Polyhedra:

  Regular Platonic Polyhedra

• Tetrahedron:
• Dodecahedron:
• Icosahedron:

  Tetrahedron

• Tetrahedron Data Members:
• Tetrahedron Constructors and Setting Functions:
• Tetrahedron Net:

  Dodecahedron

• Dodecahedron Data Members:
• Dodecahedron Constructors and Setting Functions:
• Dodecahedron Net:

  Icosahedron

• Icosahedron Data Members:
• Icosahedron Constructors and Setting Functions:
• Icosahedron Net:

  Semi-Regular Archimedean Polyhedra

• Truncated Octahedron:

  Truncated Octahedron

• Truncated Octahedron Data Members:
• Truncated Octahedron Constructors and Setting Functions:
• Truncated Octahedron Net:

  Utility Functions

• Perspective Functions:

  Future Plans

• Geometry:
• Curves and Surfaces:
• Shadows:
• Multi-Threading:

  Changes

• Changes in 3DLDF 1.1.5.1:
• Changes in 3DLDF 1.1.5:
• Changes in 3DLDF 1.1.4.2:
• Changes in 3DLDF 1.1.4.1:
• Changes in 3DLDF 1.1.4:
• Initial version:

Introduction

3DLDF is a free software package for three-dimensional drawing written by Laurence D. Finston, who is also the author of this manual. It is written in C++ using CWEB and it outputs MetaPost code.

3DLDF is a GNU package. It is part of the GNU Project of the Free Software Foundation and is published under the GNU General Public License. See the website http://www.gnu.org for more information. 3DLDF is available for downloading from http://ftp.gnu.org/gnu/3dldf. The official 3DLDF website is http://www.gnu.org/software/3dldf. More information about 3DLDF can be found at the author's website: http://wwwuser.gwdg.de/~lfinsto1.

Please send bug reports to:

     bug-3DLDF@gnu.org and
     

Two other mailing lists may be of interest to users of 3DLDF: help-3DLDF@gnu.org is for people to ask other users for help and info-3DLDF@gnu.org is for sending announcements to users. To subscribe, send an email to the appropriate mailing list or lists with the word "subscribe" as the subject. The author's website is http://wwwuser.gwdg.de/~lfinsto1.

My primary purpose in writing 3DLDF was to make it possible to use MetaPost for three-dimensional drawing. I've always enjoyed using MetaPost, and thought it was a shame that I could only use it for making two-dimensional drawings. 3DLDF is a front-end that operates on three-dimensional data, performs the necessary calculations for the projection onto two dimensions, and writes its output in the form of MetaPost code.

While 3DLDF's data types and operations are modelled on those of Metafont and MetaPost, and while the only form of output 3DLDF currently produces is MetaPost code, it is nonetheless not in principle tied to MetaPost. It could be modified to produce PostScript code directly, or output in other formats. It would also be possible to modify 3DLDF so that it could be used for creating graphics interactively on a terminal, by means of an appropriate interface to the computer's graphics hardware.

The name "3DLDF" ("3D" plus the author's initials) was chosen because, while not pretty, it's unlikely to conflict with any of the other programs called "3D"-something.

• Sources of Information:
• About This Manual:
• CWEB Documentation:
• Metafont and MetaPost:
• Caveats:
• Ports:

Sources of Information

This handbook, and the use of 3DLDF itself, presuppose at least some familiarity on the part of the reader with Metafont, MetaPost, CWEB, and C++ . If you are not familiar with any or all of them, I recommend the following sources of information:

Knuth, Donald Ervin. The METAFONTbook. Computers and Typesetting; C. Addison Wesley Publishing Company, Inc. Reading, Massachusetts 1986.

Hobby, John D. A User's Manual for MetaPost. AT & T Bell Laboratories. Murray Hill, NJ. No date.

Knuth, Donald E. and Silvio Levy. The CWEB System of Structured Documentation. Version 3.64--February 2002.

Stroustrup, Bjarne. The C++ Programming Language. Special Edition. Reading, Massachusetts 2000. Addison-Wesley. ISBN 0-201-70073-5.

The manuals for MetaPost and CWEB are available from the Comprehensive TeX Archive Network (CTAN). See one of the following web sites for more information:

Germany

http://dante.ctan.org, http://ftp.dante.de
http://www.dante.de.

United Kingdom

http://www.cam.ctan.org
http://ftp.tex.ac.uk.

USA

http://www.tug.ctan.org
http://www.ctan.tug.org.

About This Manual

This manual has been created using Texinfo, a documentation system which is part of the GNU Project, whose main sponsor is the Free Software Foundation. Texinfo can be used to generate online and printed documentation from the same input files.

For more information about Texinfo, see:

Stallmann, Richard M. and Robert J. Chassell. Texinfo. The GNU Documentation Format. The Free Software Foundation. Boston 1999.

For more information about the GNU Project and the Free Software Foundation, see the following web site: http://www.gnu.org.

The edition of this manual is 1.1.5.1 and it documents version 1.1.5.1 of 3DLDF. The edition number of the manual and the version number of the program are the same (as of 16 January 2004), but may diverge at a later date.

Note that "I", "me", etc., in this manual refers to Laurence D. Finston, so far the sole author of both 3DLDF and this manual. "Currently" and similar formulations refer to version 1.1.5.1 of 3DLDF as of 16 January 2004.

This manual is intended for both beginning and advanced users of 3DLDF. So, if there's something you don't understand, it's probably best to skip it and come back to it later. Some of the more difficult points, or ones that presuppose familiarity with features not yet described, are in the footnotes.

I firmly believe that an adequate program with good documentation is more useful than a great program with poor or no documentation. The ideal case, of course, is a great program with great documentation. I'm sorry to say, that this manual is not yet as good as I'd like it to be. I apologize for the number of typos and other errors. I hope they don't detract too much from its usefulness. I would have liked to have proofread and corrected it again before publication, but for reasons external to 3DLDF, it is necessary for me to publish now. I plan to set up an errata list on the official 3DLDF website, and/or my own website.

Unless I've left anything out by mistake, this manual documents all of the data types, constants and variables, namespaces, and functions defined in 3DLDF. However, some of the descriptions are terser than I would like, and I'd like to have more examples and illustrations. There is also more to be said on a number of topics touched on in this manual, and some topics I haven't touched on at all. In general, while I've tried to give complete information on the "what and how", the "why and wherefore" has sometimes gotten short shrift. I hope to correct these defects in future editions.

• Manual Conventions:
• Illustrations:

Conventions

Data types are formatted like this: int, Point, Path. Plurals are formatted in the same way: ints, Points, Paths. It is poor typographical practice to typeset a single word using more than one font, e.g., ints, Points, Paths. This applies to data types whose plurals do not end in "s" as well, e.g., the plural of the C++ class Polyhedron is Polyhedra.

When C++ functions are discussed in this manual, I always include a pair of parentheses to make it clear that the item in question is a function and not a variable, but I generally do not include the arguments. For example, if I mention the function foo(), this doesn't imply that foo() takes no arguments. If it were appropriate, I would include the argument type:

     foo(int)
     

or the argument type and a placeholder name:

     foo(int arg)
     

or I would write

     foo(void)
     

to indicate that foo() takes no arguments. Also, I generally don't indicate the return type, unless it is relevant. If it is a member function of a class, I may indicate this, e.g.,, bar_class::foo(), or not, depending on whether this information is relevant. This convention differs from that used in the Function Index, which is generated automatically by Texinfo. There, only the name of the function appears, without parentheses, parameters, or return values. The class type of member functions may appear in the Function Index, (e.g., bar_class::foo), but only in index entries that have been entered explicitly by the author; such entries are not generated by Texinfo automatically.

Examples are formatted as follows:

     Point p0(1, 2, 3);
     Point p1(5, 6, 7.9);
     Path pa(p0, p1);
     p0.show("p0:");
     -| p0: (1, 2, 3)
     

The beautiful mathematical typesetting produced by TeX unfortunately does not appear in the Info and HTML versions of this manual. In these, the following symbols are used to replace the proper mathematical symbols.

^

Precedes a superscript. For example, read a^2 as "a squared".

_

Precedes a subscript. For example, read x_1 as "x sub one".

*

Multiplication. For example, read x * y as "x times y".

sqrt()

The square root function. For example, read sqrt(x) as "the square root of x".

In addition, examples can contain the following symbols:

-|

Indicates output to the terminal when 3DLDF is run.

=>

Indicates a result of some sort. It may precede a illustration generated by the code in the example.

error-->

Indicates that the following text is an error message.

This manual does not use all of the symbols provided by Texinfo. If you find a symbol you don't understand in this manual (which shouldn't happen), see page 103 of the Texinfo manual.

Symbols:

N

The set of the natural numbers {0, 1, 2, 3, 4, ...}

I

The set of the integers {..., -3, -2, -1, 0, 1, 2, 3, 4, ...}

R

The set of the real numbers.

Illustrations

The illustrations in this manual have been created using 3DLDF. The code that generates them is in the Texinfo files themselves, that contain the text of the manual. Texinfo is based on TeX, so it's possible to make use of the latter's facility for writing ASCII text to files using TeX's \write command.

The file 3DLDF-1.1.5.1/CWEB/exampman.web contains the C++ code, and the file 3DLDF-1.1.5.1/CWEB/examples.mp contains the MetaPost code for generating the illustrations. 3DLDF was built using GCC 2.95 when the illustrations were generated. For some reason, GCC 3.3 has difficulty with them. It works to generate them in batches of about 50 with GCC 3.3.

MetaPost outputs Encapsulated PostScript files. These can be included in TeX files, as explained below. However, in order to display the illustrations in the HTML version of this manual, I had to convert them to PNG ("Portable Network Graphics") format. See Converting EPS Files, for instructions on how to do this.

Please note that the illustrations cannot be shown in the Info output format!

If you have problems including the illustrations in the printed version, for example, if your installation doesn't have dvips, look for the following lines in 3DLDF.texi:

     \doepsftrue    %% One of these two lines should be commented-out.
     %\doepsffalse
     

Now, remove the % from in front of \doepsffalse and put one in front of \doepsftrue. This will prevent the illustrations from being included. This should only be done as a last resort, however, because it will make it difficult if not impossible to understand this manual.

The C++ code in an example is not always the complete code used to create the illustration that follows it, since the latter may be cluttered with commands that would detract from the clarity of the example. The actual code used always follows the example in the Texinfo source file, so the latter may be referred to, if the reader wishes to see exactly what code was used to generate the illustration.

You may want to skip the following paragraphs in this section, if you're reading this manual for the first time. Don't worry if you don't understand it, it's meaning should become clear after reading the manual and some experience with using 3DLDF.

The file 3DLDF.texi in the directory 3DLDF-1.1.5.1/DOC/TEXINFO, the driver file for this manual, contains the following TeX code:

     \newif\ifmakeexamples
     \makeexamplestrue     %% One of these two lines should be commented-out.
     %\makeexamplesfalse
     

When texi2dvi is run on 3DLDF.texi, \makeexamplestrue is not commented-out, and \makeexamplesfalse is, the C++ code for the illustrations is written to the file examples.web. If the EPS files don't already exist (in the directory 3DLDF-1.1.5.1/DOC/TEXINFO/EPS), the TeX macro \PEX, which includes them in the Texinfo files, will signal an error each time it can't find one. Just type s at the command line to tell TeX to keep going. If you want to be sure that these are indeed the only errors, you can type <RETURN> after each one instead.

texi2dvi 3DLDF.texi also generates the file extext.tex, which contains TeX code for including the illustrations by themselves.

examples.web must now be moved to 3DLDF-1.1.5.1/CWEB/ and ctangled, examples.c must compiled, and 3DLDF must be relinked. ctangle examples also generates the header file example.h, which is included in main.web. Therefore, if the contents of examples.h have changed since the last time main.web was ctangled, main.web will have to be ctangled, and main.c recompiled, before 3dldf is relinked.¹

Running 3dldf and MetaPost now generates the EPS (Encapsulated PostScript) files 3DLDFmp.1 through (currently) 3DLDFmp.199 for the illustrations. They must be moved to 3DLDF-1.1.5.1/DOC/TEXINFO/EPS. Now, when texi2dvi 3DLDF.texi is run again, the dvips command \epsffile includes the EPS files for the illustrations in the manual. 3DLDF.texi includes the line \input epsf, so that \epsffile works. Of course, dvips (or some other program that does the job) must be used to convert 3DLDF.dvi to a PostScript file. To see exactly how this is done, take a look at the .texi source files of this manual.²

In the 3DLDF.texi belonging to the 3DLDF distribution, \makeexamplestrue will be commented-out, and makeexamplesfalse won't be, because the EPS files for the illustrations are included in the distribution.

The version of examples.web in 3DLDF-1.1.5.1/CWEB merely includes the files subex1.web and subex2.web. If you rename 3DLDF-1.1.5.1/CWEB/exampman.web to examples.web, you can generate the illustrations.

CWEB Documentation

As mentioned above, 3DLDF has been programmed using CWEB, which is a "literate programming" tool developed by Donald E. Knuth and Silvio Levy. See Sources of Information, for a reference to the CWEB manual. Knuth's TeX--The Program and Metafont--The Program both include a section "How to read a WEB" (pp. x-xv, in both volumes).

CWEB files combine source code and documentation. Running ctangle on a CWEB file, for example, main.web, produces the file main.c containing C or C++ code. Running cweave main.web creates a TeX file with pretty-printed source code and nicely formatted documentation. I find that using CWEB makes it more natural to document my code as I write it, and makes the source files easier to read when editing them. It does have certain consequences with regard to compilation, but these are taken care of by make. See Adding a File, and Changes, for more information.

The CWEB files in the directory 3DLDF-1.1.5.1/CWEB/ contain the source code for 3DLDF. The file 3DLDFprg.web in this directory is only ever used for cweaving; it is never ctangled and contains no C++ code for compilation. It does, however, include all of the other CWEB files, so that cweave 3DLDFprg.web generates the TeX file containing the complete documentation of the source code of 3DLDF.

The files 3DLDF-1.1.5.1/CWEB/3DLDFprg.tex, 3DLDF-1.1.5.1/CWEB/3DLDFprg.dvi, and 3DLDF-1.1.5.1/CWEB/3DLDFprg.ps are included in the distribution of 3DLDF as a convenience. However, users may generate them themselves, should there be some reason for doing so, by entering make ps from the command line of a shell from the working directory 3DLDF-1.1.5.1/ or 3DLDF-1.1.5.1/CWEB. Alternatively, the user may generate them by hand from the working directory 3DLDF-1.1.5.1/CWEB/ in the following way:

1. cweave 3DLDFprg.web generates 3DLDFprg.tex.
2. tex 3DLDFprg or tex 3DLDFprg.tex generates 3DLDFprg.dvi.
3. dvips -o 3DLDFprg.ps 3DLDFprg (possibly with additional options) generates 3DLDFprg.ps.
4. lpr -P<print queue> 3DLDFprg.ps sends 3DLDFprg.ps to a printer, on a UNIX or UNIX-like system.

The individual commands may differ, depending on the system you're using.

Metafont and MetaPost

Metafont is a system created by Donald E. Knuth for generating fonts, in particular for use with TeX, his well-known typsetting system.³ Expressed in a somewhat simplified way, Metafont is a system for programming curves, which are then digitized and output in the form of run-time encoded bitmaps. (See Knuth's The Metafontbook for more information).

John D. Hobby modified Metafont's source code to create MetaPost, which functions in much the same way, but outputs encapsulated PostScript (EPS) files instead of bitmaps. MetaPost is very useful for creating graphics and is a convenient interface to PostScript. It is also easy both to imbed TeX code in MetaPost programs, for instance, for typesetting labels, and to include MetaPost graphics in ordinary TeX files, e.g., by using dvips.⁴ Apart from simply printing the PostScript file output by dvips, there are many programs that can process ordinary or encapsulated PostScript files and convert them to other formats. Just two of the many possibilities are ImageMagick and GIMP, both of which can be used to create animations from MetaPost graphics.

However, MetaPost inherited a significant limitation from Metafont: it's not possible to use it for making three-dimensional graphics, except in a very limited way. One insuperable problem is the severe limitation on the magnitude of user-defined numerical variables in Metafont and MetaPost.⁵ This made sense for Metafont's and MetaPost's original purposes, but they make it impossible to perform the calculations needed for 3D graphics.

Another problem is the data types defined in Metafont: Points are represented as pairs of real values and affine transformations as sets of 6 real values. This corresponds to the representation of points and affine transformations in the plane as a two-element vector on the one hand and a six element matrix on the other. While it is possible to work around the limitation imposed by having points be represented by only two values, it is impracticable in the case of the transformations.

For these reasons, I decided to write a program that would behave more or less like Metafont, but with suitable extensions, and the ability to handle three dimensional data; namely 3DLDF. It stores the data and performs the transformations and other necessary calculations and is not subject to the limitations of MetaPost and its data types. Upon output, it performs a perspective transformation, converting the 3D image into a 2D one. The latter can now be expressed as an ordinary MetaPost program, so 3DLDF writes its output as MetaPost code to a file.

In the following, it may be a little unclear why I sometimes refer to Metafont and sometimes to MetaPost. The reason is that Metafont inherited much of its functionality from Metafont. Certain operations in Metafont have no meaning in MetaPost and so have been removed, while MetaPost's function of interfacing with PostScript has caused other operations to be added. For example, in MetaPost, color is a data type, but not in Metafont. Unless otherwise stated, when I refer to Metafont, it can be assumed that what I say applies to MetaPost as well. However, when I refer to MetaPost, it will generally be in connection with features specific to MetaPost.

Caveats

• Accuracy:
• No Input Routine:

Accuracy

When 3DLDF is run, it uses the three-dimensional data contained in the user code to create a two-dimensional projection. Currently, this can be a perspective projection, or a parallel projection onto one of the major planes. MetaPost code representing this projection is then written to the output file. 3DLDF does no scan conversion,⁶ so all of the curves in the projection are generated by means of the algorithms MetaPost inherited from Metafont. These algorithms, however, are designed to find the "most pleasing curve"⁷ given one or more two-dimensional points and connectors; they do not account for the the fact that the two-dimensional points are projections of three-dimensional ones. This can lead to unsatisfactory results, especially where extreme foreshortening occurs. In particular, curl, dir, tension, and control points should be used cautiously, or avoided altogether, when specifying connectors.

3DLDF operates on the assumption that, given an adequate number of points, MetaPost will produce an adequate approximation to the desired curve in perspective, since the greater the number of points given for a curve, the less "choice" MetaPost has for the path through them. My experience with 3DLDF bears this out. Generally, the curves look quite good. Where problems arise, it usually helps to increase the number of points in a curve.

A more serious problem is the imprecision resulting from the operation of rotation. Rotations use the trigonometric functions, which return approximate values. This has the result that points that should have identical coordinate values, sometimes do not. This has consequences for the functions that compare points. The more rotations are applied to points, the greater the divergence between their actual coordinate values, and the values they should have. So far, I haven't found a solution for this problem. On the other hand, it hasn't yet affected the usability of 3DLDF.

No Input Routine

3DLDF does not yet include a routine for reading input files. This means that user code must be written in C++ , compiled, and linked with the rest of the program. I admit, this is not ideal, and writing an input routine for user code is one of the next things I plan to add to 3DLDF.

I plan to use Flex and Bison to write the input routine.⁸ The syntax of the input code should be as close as possible to that of MetaPost, while taking account of the differences between MetaPost and 3DLDF.

For the present, however, the use of 3DLDF is limited to those who feel comfortable using C++ and compiling and relinking programs. Please don't be put off by this! It's not so difficult, and make does most of the work of recompiling and running 3DLDF. See Installing and Running 3DLDF, for more information.

Ports

I originally developed 3DLDF on a DECalpha Personal Workstation with two processors running under the operating system Tru64 Unix 5.1, using the DEC C++ compiler. I then ported it to a PC Pentium 4 running under Linux 2.4, using the GNU C++ compiler GCC 2.95.3, and a PC Pentium II XEON under Linux 2.4, using GCC 3.3. I am currently only maintaining the last version. I do not believe that it's worthwhile to maintain a version for GCC 2.95. While I would like 3DLDF to run on as many platforms as possible, I would rather spend my time developing it than porting it. This is something where I would be grateful for help from other programmers.

Although I am no longer supporting ports to other systems, I have left some conditionally compiled code for managing platform dependencies in the CWEB sources of 3DLDF. This may make it easier for other people who want to port 3DLDF to other platforms.

Currently, the files io.web, loader.web, main.web, points.web, and pspglb.web contain conditionally compiled code, depending on which compiler, or in the case of GCC, which version of the compiler, is used. The DEC C++ compiler defines the preprocessor macro __DECCXX and GCC defines __GNUC__. In order to distinguish between GCC 2.95.3 and GCC 3.3, I've added the macros LDF_GCC_2_95 and LDF_GCC_3_3 in loader.web, which should be defined or undefined, depending on which compiler you're using. In the distribution, LDF_GCC_3_3 is defined and LDF_GCC_2_95 is undefined, so if you want to try using GCC 2.95, you'll have to change this (it's not guaranteed to work).

3DLDF 1.1.5.1 now uses Autoconf and Automake, and the configure script generates a config.h file, which is now included in loader.web. Some of the preprocessor macros defined in config.h are used to conditionally include library header files, but so far, there is no error handling code for the case that a file can't be included. I hope to improve the way 3DLDF works together with Autoconf and Automake in the near future.

3DLDF 1.1.5 is the first release that contains template functions. Template instantiation differs from compiler to compiler, so using template functions will tend to make 3DLDF less portable. See Template Functions, for more information. I am no longer able to build 3DLDF on the DECalpha Personal Workstation. I'm fairly sure that it would be possible to port it, but I don't plan to do this, since Tru64 Unix 5.1 and the DEC C++

compiler are non-free software.

Contributing to 3DLDF

So far, I've been the sole author and user of 3DLDF. I would be very interested in having other programmers contribute to it. I would be particularly interested in help in making 3DLDF conform as closely as possible to the GNU Coding Standards. I would be grateful if someone would write proper Automake and Autoconf files, since I haven't yet learned how to do so (I'm working on it).

See Introduction, for information on how to contact the author.

Using 3DLDF

Since 3DLDF does not yet have an input routine, user code must be written in C++ (in main.web, or some other file) and compiled. Then, 3DLDF must be relinked, together with the new file of object code resulting from the compilation. For now, the important point is that the text of the examples in this manual represent C++ code. See Installing and Running 3DLDF, for more information.

Points

• Declaring and Initializing Points:
• Setting and Assigning to Points:

Declaring and Initializing Points

The most basic drawable object in 3DLDF is class Point. It is analogous to pair in Metafont. For example, in Metafont one can define a pair using the "z" syntax as follows:

     z0 = (1cm, 1cm);
     

There are other ways of defining pairs in Metafont (and MetaPost), but this is the usual way.

In 3DLDF, a Point is declared and initialized as follows:

     Point pt0(1, 2, 3);
     

This simple example demonstrates several differences between Metafont and 3DLDF. First of all, there is no analog in 3DLDF to Metafont's "z" syntax. If I want to have Points called "pt0", "pt1", "pt2", etc., then I must declare each of them to be a Point:

     Point pt0(10, 15, 2);
     Point pt1(13, 41, 5.5);
     Point pt2(62.9, 7.02, 8);
     

Alternatively, I could declare an array of Points:

     Point pt[3];
     

Now I can refer to pt[0], pt[1], and pt[2].

In the Metafont example, the x and y-coordinates of the pair z0 are specified using the unit of measurement, in this case, centimeters. This is currently not possible in 3DLDF. The current unit of measurement is stored in the static variable Point::measurement_units, which is a string. Its default value is "cm" for "centimeters". At present, it is best to stick with one unit of measurement for a drawing. After I've defined an input routine, 3DLDF should handle units of measurement in the same way that Metafont does.

Another difference is that the Points pt0, pt1, and pt2 have three coordinates, x, y, and z, whereas z0 has only two, x and y. Actually, the difference goes deeper than this. In Metafont, a pair has two parts, xpart and ypart, which can be examined by the user. In 3DLDF, a Point contains the following sets of coordinates:

     world_coordinates
     user_coordinates
     view_coordinates
     projective_coordinates
     

These are sets of 3-dimensional homogeneous coordinates, which means that they contain four coordinates: x, y, z, and w. Homogeneous coordinates are used in the affine and perspective transformations (see Transforms).

Currently, only world_coordinates and projective_coordinates are used in 3DLDF. The world_coordinates refer to the position of a Point in 3DLDF's basic, unchanging coordinate system. The projective_coordinates are the coordinates of the two-dimensional projection of the Point onto a plane. This projection is what is ultimately printed out or displayed on the computer screen. Please note, that when the coordinates of a Point are referred to in this manual, the world_coordinates are meant, unless otherwise stated.

Points can be declared and their values can be set in different ways.

     Point pt0;
     Point pt1(1);
     Point pt2(2.3, 52);
     Point pt3(4.5, 7, 13.205);
     

pt0 is declared without any arguments, i.e., using the default constructor, so the values of its x, y, and z-coordinates are all 0.

pt1 is declared and initialized with one argument for the x-coordinate, so its y and z-coordinates are initialized with the values of CURR_Y and CURR_Z respectively. The latter are static constant data members of class Point, whose values are 0 by default. They can be reset by the user, who should make sure that they have sensible values.

pt2 is declared and initialized with two arguments for its x and y-coordinates, so its z-coordinate is initialized to the value of CURR_Z. Finally, pt3 has an argument for each of its coordinates.

Please note that pt0 is constructed using a the default constructor, whereas the other Points are constructed using a constructor with one required argument (for the x-coordinate), and two optional arguments (for the y and z-coordinates). The default constructor always sets all the coordinates to 0, irrespective of the values of CURR_Y and CURR_Z.

Setting and Assigning to Points

It is possible to change the value of the coordinates of Points by using the assignment operator = (Point::operator=()) or the function Point::set() (with appropriate arguments):

     Point pt0(2, 3.3, 7);
     Point pt1;
     pt1 = pt0;
     pt0.set(34, 99, 107.5);
     pt0.show("pt0:");
     -| pt0: (34, 99, 107.5)
     pt1.show("pt1:");
     -| pt1: (2, 3.3, 7)
     

In this example, pt0 is initialized with the coordinates (2, 3.3, 7), and pt1 with the coordinates (0, 0, 0). pt1 = pt0 causes pt1 to have the same coordinates as pt0, then the coordinates of pt0 are changed to (34, 99, 107.5). This doesn't affect pt1, whose coordinates remain (2, 3.3, 7).

Another way of declaring and initializing Points is by using the copy constructor:

     Point pt0(1, 3.5, 19);
     Point pt1(pt0);
     Point pt2 = pt0;
     Point pt3;
     pt3 = pt0;
     

In this example, pt1 and pt2 are both declared and initialized using the copy constructor; Point pt2 = pt0 does not invoke the assignment operator. pt3, on the other hand, is declared using the default constructor, and not initialized. In the following line, pt3 = pt0 does invoke the assignment operator, thus resetting the coordinate values of pt3 to those of pt0.

Transforming Points

Points don't always have to remain in the same place. There are various ways of moving or transforming them:

• Shifting. This is often called "translating", but the operation in Metafont that performs translation is called shift, so I call it "shifting".
• Scaling.
• Shearing.
• Rotating about an axis.

class Point has several member functions for applying these affine transformations⁹ to a Point. Most of the arguments to these functions are of type real. As you may know, there is no such data type in C++ . I have defined real using typedef to be either float or double, depending on the value of a preprocessor switch for conditional compilation.¹⁰ 3DLDF uses many real values and I wanted to be able to change the precision used by making one change (in the file pspglb.web) rather than having to examine all the places in the program where float or double are used. Unfortunately, setting real to double currently doesn't work.

• Shifting Points:
• Scaling Points:
• Shearing Points:
• Rotating Points:

Shifting

The function shift() adds its arguments to the corresponding world_coordinates of a Point. In the following example, the function show() is used to print the world_coordinates of p0 to standard output.

     Point p0(0, 0, 0);
     p0.shift(1, 2, 3);
     p0.show("p0:");
     -| p0: (1, 2, 3)
     p0.shift(10);
     p0.show("p0:");
     -| p0: (11, 2, 3)
     p0.shift(0, 20);
     p0.show("p0:");
     -| p0: (11, 22, 3)
     p0.shift(0, 0, 30);
     p0.show("p0:");
     -| p0: (11, 22, 33)
     

shift takes three real arguments, whereby the second and third are optional. To shift a Point in the direction of the positive or negative y-axis, and/or the positive or negative z-axis only, then a 0 argument for the x direction, and possibly one for the y direction must be used as placeholders, as in the example above.

shift() can be invoked with a Point argument instead of real arguments. In this case, the x, y, and z-coordinates of the argument are used for shifting the Point:

     Point a(10, 10, 10);
     Point b(1, 2, 3);
     a.shift(b);
     a.show("a:")
     -| a: (11, 12, 13)
     

Another way of shifting Points is to use the binary += operator (Point::operator+=()) with a Point argument.

     Point a0(1, 1, 1);
     Point a1(2, 2, 2);
     a0 += a1;
     a0.show("a0:");
     -| a0: (3, 3, 3)
     

Scaling

The function scale() takes three real arguments. The x, y, and z-coordinates of the Point are multiplied by the first, second, and third arguments respectively. Only the first argument is required; the default for the others is 1.

If one wants to perform scaling in either the y-dimension only, or the y and z-dimensions only, a dummy argument of 1 must be passed for scaling in the x-dimension. Similarly, if one wants to perform scaling in the z-dimension only, dummy arguments of 1 must be passed for scaling in the x and y-dimensions.

     Point p0(1, 2, 3);
     p0.scale(2, 3, 4);
     p0.show("p0:");
     -| p0: (2, 6, 12)
     p0.scale(2);
     p0.show("p0:");
     -| p0: (4, 6, 12)
     p0.scale(1, 3);
     p0.show("p0:");
     -| p0: (4, 18, 12)
     p0.scale(1, 1, 3);
     p0.show("p0:");
     -| p0: (4, 18, 36)
     

Shearing

Shearing is more complicated than shifting or scaling. The function shear() takes six real arguments. If p is a Point, then p.shear(a, b, c, d, e, f) sets x_p to x_p + ay_p + bz_p, y_p to y_p + cx_p + dz_p, and z_p to z_p + ex_p + fy_p. In this way, each coordinate of a Point is modified based on the values of the other two coordinates, whereby the influence of the other coordinates on the new value is weighted according to the arguments.

     Point p(1, 1, 1);
     p.shear(1);
     p.show("p:");
     -| p: (2, 1, 1)
     p.set(1, 1, 1);
     p.shear(1, 1);
     p.show("p:");
     -| p: (3, 1, 1)
     p.set(1, 1, 1);
     p.shear(1, 1, 2, 2, 3, 3);
     p.show("p:");
     -| p: (3, 5, 7)
     

[next figure] demonstrates the effect of shearing the points of a rectangle in the x-y plane.

     Point P0;
     Point P1(3);
     Point P2(3, 3);
     Point P3(0, 3);
     Rectangle r(p0, p1, p2, p3);
     r.draw();
     Rectangle q(r);
     q.shear(1.5);
     q.draw(black, "evenly");
     

Fig. 1.

Rotating

The function rotate() rotates a Point about one or more of the main axes. It takes three real arguments, specifying the angles of rotation in degrees about the x, y, and z-axes respectively. Only the first argument is required, the other two are 0 by default. If rotation about the y-axis, or the y and z-axes only are required, then 0 must be used as a placeholder for the first and possibly the second argument.

     Point p(0, 1);
     p.rotate(90);
     p.show("p:");
     -| p: (0, 0, -1)
     p.rotate(0, 90);
     p.show("p:");
     -| p: (1, 0, 0)
     p.rotate(0, 0, 90);
     p.show("p:");
     -| p: (0, 1, 0)
     

The rotations are performed successively about the x, y, and z-axes. However, rotation is not a commutative operation, so if rotation about the main axes in a different order is required, then rotate() must be invoked more than once:

     Point A(2, 3, 4);
     Point B(A);
     A.rotate(30, 60, 90);
     A.show("A:");
     -| A: (-4.59808, -0.700962, 2.7141)
     B.rotate(0, 0, 90);
     B.rotate(0, 60);
     B.rotate(30);
     B.show("B:");
     -| B: (-4.9641, 1.43301, -1.51795)
     

Rotation need not be about the main axes; it can also be performed about a line defined by two Points. The function rotate() with two Point arguments and a real argument for the angle of rotation (in degrees) about the axis. The real argument is optional, with 180 degrees

as the default.

     Point p0 (-1.06066, 0, 1.06066);
     Point p1 (1.06066, 0, -1.06066);
     p1 *= p0.rotate(0, 30, 30);
     p0.show("p0:");
     -| p0: (-1.25477, -0.724444, 0.388228)
     p1.show("p1:");
     -| p1: (1.25477, 0.724444, -0.388228)
     p0.draw(p1);
     Point p2(1.06066, 0, 1.06066);
     p2.show("p2:");
     -| p2: (1.06066, 0, 1.06066)
     Point p3(p2);
     p3.rotate(p1, p0, 45);
     p3.show("p3:");
     -| p3 (1.09721, 1.15036, 1.17879)
     Point p4(p2);
     p4.rotate(p1, p0, 90);
     p4.show("p4:");
     -| p4: (0.882625, 2.05122, 0.485242)
     Point p5(p2);
     p5.rotate(p1, p0, 135);
     p5.show("p5:");
     -| p5: (0.542606, 2.17488, -0.613716)
     Point p6(p2);
     p6.rotate(p1, p0);
     p6.show("p6:");
     -| p6: (0.276332, 1.44889, -1.47433)
     

Fig. 2.

I have sometimes gotten erroneous results using rotate() for rotation about two Points. It's usually worked to reverse the order of the Point arguments, or to change sign of the angle argument. I think I've fixed the problem, though.

Transforms

When Points are transformed using shift(), shear(), or one of the other transformation functions, the world_coordinates are not modified directly. Instead, another data member of class Point is used to store the information about the transformation, namely transform of type class Transform. A Transform object has a single data element of type Matrix and a number of member functions. A Matrix is simply a 4 X 4 array¹¹ of reals defined using typedef real Matrix[4][4]. Such a matrix suffices for performing all of the transformations (affine and perspective) possible in three-dimensional space.¹² Any combination of transformations can be represented by a single transformation matrix. This means that consecutive transformations of a Point can be "saved up" and applied to its coordinates all at once when needed, rather than updating them for each transformation.

Transforms work by performing matrix multiplication of Matrix with the homogeneous world_coordinates of Points. If a set of homogeneous coordinates \alpha = (x, y, z, w) and

     Matrix M =
     a e i m
     b f j n
     c g k o
     d h l p
     

     \beta = \alpha\times M = ((xa + yb + zc + wd),  (xe + yf + zg + wh),
     (xi + yj + zk + wl), (xm + yn + zo + wp))
     

Operations on matrices are very important in computer graphics applications and are described in many books about computer graphics and geometry. For 3DLDF, I've mostly used Huw Jones' Computer Graphics through Key Mathematics and David Salomon's Computer Graphics and Geometric Modeling.

It is often useful to declare and use Transform objects in 3DLDF, just as it is for transforms in Metafont. Transformations can be stored in Transforms and then be used to transform Points by means of Point::operator*=(const Transform&).

     1. Transform t;
     2. t.shift(0, 1);
     3. Point p(1, 0, 0);
     4. p *= t;
     5. p.show("p:");
     -| p: (1, 1, 0)
     

When a Transform is declared (line 1), it is initialized to an identity matrix. All identity matrices are square, all of the elements of the main diagonal (upper left to lower right) are 1, and all of the other elements are 0. So a 4 X 4 identity matrix, as used in 3DLDF, looks like this:

     1 0 0 0
     0 1 0 0
     0 0 1 0
     0 0 0 1
     

The same affine transformations are applied in the same way to Transforms as they are to Points, i.e., the functions scale(), shift(), shear(), and rotate() correspond to the Point versions of these functions, and they take the same arguments:

     Point p;
     Transform t;
     p.shift(3, 4, 5);
     t.shift(3, 4, 5);
     => p.transform == t
     p.show_transform("p:");
     -| p:
        Transform:
              0   0.707   0.707       0
         -0.866   0.354  -0.354       0
           -0.5  -0.612   0.612       0
              0       0       0       1
     t.show("t:");
     -| t:
              0   0.707   0.707       0
         -0.866   0.354  -0.354       0
           -0.5  -0.612   0.612       0
              0       0       0       1
     
     

• Applying Transforms to Points Intro:
• Inverting Transforms:

Applying Transforms to Points

A Transform t is applied to a Point P using the binary *= operation (Point::operator*=(const Transform&)) which performs matrix multiplication of P.transform by t. See Point Reference; Operators.

     Point P(0, 1);
     Transform t;
     t.rotate(90);
     t.show("t:");
     -| t:
        1       0       0       0
        0       0      -1       0
        0       1       0       0
        0       0       0       1
     P *= t;
     P.show_transform("P:");
     -| P:
     Transform:
        1       0       0       0
        0       0      -1       0
        0       1       0       0
        0       0       0       1
     P.show("P:");
     -| P: (0, 0, -1)
     

In the example above, there is no real need to use a Transform, since P.rotate(90) could have been called directly. As constructions become more complex, the power of Transforms becomes clear:

     1. Point p0(0, 0, 0);
     2. Point p1(10, 5, 10);
     3. Point p2(16, 14, 32);
     4. Point p3(25, 50, 99);
     5. Point p4(12, 6, 88);
     6. Transform a;
     7. a.shift(2, 3, 4);
     8. a.scale(1, 3, 1);
     9. p2 *= p3 *= a;
     10. a.rotate(p0, p1, 75);
     11. p4 *= a;
     12. p2.show("p2:");
        -| p2: (18, 51, 36)
     13. p3.show("p3:");
        -| p3: (27, 159, 103)
     14. p4.show("p4:");
        -| p4: (24.4647, -46.2869, 81.5353)
     

In this example, a is shifted and scaled, and a is applied to both in line 9. This works, because the binary operation operator*=(const Transform& t) returns t, making it possible to chain invocations of *=. Following this, a is rotated 75 degrees

about the line through p_0 and p_1. Finally, all three transformations, which are stored in a, are applied to p_4.

Inverting Transforms

Inversion is another operation that can be performed on Transforms. This makes it possible to reverse the effect of a Transform, which may represent multiple transformations.

     Point p;
     Transform t;
     t.shift(1, 2, 3);
     t.scale(2, 3, 4);
     t.rotate(45, 45, 30);
     t.show("t:");
     -| t:
        1.22   0.707    1.41       0
       0.238    2.59    -1.5       0
       -3.15    1.45       2       0
       -7.74    10.2    4.41       1
     p *= t;
     p.show("p:");
     -| p: (-7.74, 10.2, 4.41)
     Transform u;
     u = t.inverse();
     u.show("u:");
     -| u:
       0.306  0.0265  -0.197 2.85e-09
       0.177   0.287  0.0906 -1.12e-09
       0.354  -0.167   0.125       0
          -1      -2      -3       1
     p *= u;
     p.show("p:");
     -| p: (0, 0, 0)
     u *= t;
     u.show("u:");
     -| u:
           1       0       0       0
           0       1       0       0
           0       0       1       0
           0       0       0       1
     

If inverse() is called with no argument, or with the argument false, it returns a Transform representing its inverse, and remains unchanged. If it is called with the argument true, it is set to its inverse.

Complete reversal of the transformations applied to a Point, as in the previous example, probably won't make much sense. However, partial reversal is a valuable technique. For example, it is used in rotate() for rotation about a line defined by two Points. The following example merely demonstrates the basic principle; an example that does something useful would be too complicated.

     Transform t;
     t.shift(3, 4, 5);
     t.rotate(45);
     t.scale(2, 2, 2);
     Point p;
     p *= t;
     p.show("p:");
     -| p: (6, 12.7279, 1.41421)
     t.inverse(true);
     p.rotate(90, 90);
     p *= t;
     p.show("p:");
     -| p: (3.36396, -5.62132, -2.37868)
     

Drawing and Labeling Points

• Drawing Points Intro:
• Labeling Points Intro:

Drawing Points

It's all very well to declare Points, place them at particular locations, print their locations to standard output, and transform them, but none of these operations produce any MetaPost output. In order to do this, the first step is to use drawing and filling commands. The drawing and filling commands in 3DLDF are modelled on those in Metafont.

The following example demonstrates how to draw a dot specifying a Color (see Color Reference) and a pen¹³.

     Point P(0, 1);
     P.drawdot(Colors::black, "pencircle scaled 3mm");
     

Fig. 3.

In drawdot(), a Color argument precedes the string argument for the pen, so "Colors::black" must be specified as a placeholder in the call to drawdot().¹⁴

The following example "undraws" a dot at the same location using a smaller pen. undraw() does not take a Color argument.

     p.undrawdot("pencircle scaled 2mm");
     

Fig. 4.

For complete descriptions of drawdot() and undrawdot(), see Point Reference; Drawing.

Drawing and undrawing dots is not very exciting. In order to make a proper drawing it is necessary to connect the Points. The most basic way of doing this is to use the Point member function draw() with a Point argument:

     Point p0;
     Point p1(2, 2);
     p0.draw(p1);
     

Fig. 5.

p0.draw(p1) is equivalent in its effect to p1.draw(p0).

The function Point::draw() takes a required Point& argument (a reference¹⁵ to a Point) an optional Color argument, and optional string arguments for the dash pattern and the pen. The string arguments, if present, are passed unchanged to the output file. The empty string following the argument p1 is a placeholder for the dash pattern argument, which isn't used here.

     p0.draw(p1, Colors::gray, "", "pensquare scaled .5cm rotated 45");
     

Fig. 6.

The function Point::undraw() takes a required Point& argument and optional string arguments for the dash pattern and the pen. Unlike Point::draw(), a Color argument would have no meaning for Point::undraw(). The string arguments are passed unchanged to the output file.

undraw() can be used to "hollow out" the region drawn in [the previous figure] . Since a dash pattern is used, portions of the middle of the region are not undrawn.

     p0.undraw(p1, "evenly scaled 6", "pencircle scaled .2cm");
     

Fig. 7.

For complete descriptions of draw() and undraw(), see Point Reference; Drawing.

Labeling Points

The labels in the previous examples were made by using the functions Point::label() and Point::dotlabel(), which make it possible to include TeX text in a drawing.

label() and dotlabel() take string arguments for the text of the label and the position of the label with respect to the Point. The label text is formatted using TeX, so it can contain math mode material between dollar signs. Please note that double backslashes must be used, where a single backslash would suffice in a file of MetaPost code, for example, for TeX control sequences. Alternatively, a short argument can be used for the label.

The position argument is optional, with "top" as the default. If the empty string "" is used, the label will centered about the Point itself. This will usually only make sense for label(), because it would otherwise interfere with the dot. Valid arguments for the position are the same as in MetaPost: "top", "bot" (bottom), "lft" (left), "rt" (right), "ulft" (upper left), "urt" (upper right), "llft" (lower left), and "lrt" (lower right).

     Point p0;
     Point p1(1);
     Point p2(2);
     Point p3(p0);
     Point p4(p1);
     Point p5(p2);
     p3 *= p4 *= p5.shift(0, 1);
     p0.draw(p1);
     p1.draw(p2);
     p2.draw(p5);
     p5.draw(p4);
     p4.draw(p3);
     p3.draw(p0);
     p0.label($p_0$, "");
     p1.dotlabel(1);
     p2.dotlabel("p2", "bot");
     p3.dotlabel("This is $p_3$", "lft");
     p4.label(4);
     p5.label("$\\leftarrow p_5$", "rt");
     

Fig. 8.

For complete descriptions of Point::label() and Point::dotlabel(), see Points; Labelling.

Paths

Points alone are not enough for making useful drawings. The next step is to combine them into Paths, which are similar to Metafont's paths, except that they are three-dimensional. A Path consists of a number of Points and strings representing the connectors. The latter are not processed by 3DLDF, but are passed unchanged to the output file. They must be valid connectors for MetaPost, e.g.:

     ..
     ...
     --
     ---
     &
     curl{2}..
     {dir 60}..
     {z1 - z2}..
     .. tension 1 and 1.5..
     ..controls z1 and z2..
     

Usually, it will only make sense to use .. or -, and not ..., --, tension, curl, controls, or any of the other possibilities, in Paths, unless you are sure that they will only be viewed with no foreshortening due to the perspective projection. This can be the case, when a Path lies in a plane parallel to one of the major planes, and is projected using parallel projection onto that plane. Otherwise, the result of using these connectors is likely to be unsatisfactory, because MetaPost performs its calculations based purely on the two-dimensional values of the points in the perspective projection. While the Points on the Path will be projected correctly, the course of the Path between these Points is likely to differ, depending on the values of the Focus used (see Focuses), so that different views of the same Path may well be mutually inconsistent. This problem doesn't arise with "-", since the perspective projection does not "unstraighten" straight lines, but it does with "..", even without tension, curl, or controls. The solution is to use enough Points, since a greater number of Points on a Path tends to reduce the number of possible courses through the Points.¹⁶

• Declaring and Initializing Paths:
• Drawing and Filling Paths Intro:

Declaring and Initializing Paths

There are various ways of declaring and initializing Paths. The simplest is to use the constructor taking two Point arguments:

     Point A;
     Point B(2, 2);
     Path p(A, B);
     p.draw();
     

Fig. 9.

Paths created in this way are important, because they are guaranteed to be linear, as long as no operations are performed on them that cause them to become non-linear. Linear Paths can be used to find intersections. See Path Intersections.

Paths can be declared and initialized using a single connector and an arbitrary number of Points. The first argument is a string specifying the connector. It is followed by a bool, indicating whether the Path is cyclical or not. Then, an arbitrary number of pointers to Point follow. The last argument must be 0.¹⁷

     Point p[3];
     p[0].shift(1);
     p[1].set(1, 2, 2);
     p[2].set(1, 0, 2);
     Path pa("--", true, &p[0], &p[1], &p[2], 0);
     pa.draw();
     

Fig. 10.

Another constructor must be used for Paths with more than one connector and an arbitrary number of Points. The argument list starts with a pointer to Point, followed by string for the first connector. Then, pointer to Point arguments alternate with string arguments for the connectors. Again, the list of arguments ends in 0. There is no need for a bool to indicate whether the Path is cyclical or not; if it is, the last non-zero argument will be a connector, otherwise, it will be a pointer to Point.

     Point p[8];
     p[0].set(-2);
     p[1].set(2);
     p[2].set(0, 0, -2);
     p[3].set(0, 0, 2);
     p[4] = p[0].mediate(p[2]);
     p[5] = p[2].mediate(p[1]);
     p[6] = p[1].mediate(p[3]);
     p[7] = p[3].mediate(p[0]);
     p[4] *= p[5] *= p[6] *= p[7].shift(0, 1);
     Path pa(&p[0], "..", &p[4], "...", &p[2],
             "..", &p[5], "...", &p[1], "..", &p[6],
             "...", &p[3], "..", &p[7], "...", 0);
     pa.draw();
     

Fig. 11.

As mentioned above (see Accuracy), specifying connectors is problematic for three-dimensional Paths, because MetaPost ultimately calculates the "most pleasing curve" based on the two-dimensional points in the MetaPost code written by 3DLDF.¹⁸ For this reason, it's advisable to avoid specifying curl, dir, tension or control points in connectors. The more Points a (3DLDF) Path or other object contains, the less freedom MetaPost has to determine the (MetaPost) path through them. So a three-dimensional Path or other object in 3DLDF should have enough Points to ensure satisfactory results. The Path in [the previous figure] does not really have enough Points. It may require some trial and error to determine what a sufficient number of Points is in a given case.

Paths are very flexible, but not always convenient. 3DLDF provides a number of classes representing common geometric Shapes, which will be described in subsequent sections, and I intend to add more in the course of time.

Drawing and Filling Paths

The easiest way to draw a Path is with no arguments.

     Point pt[5];
     pt[0].set(-1, -2);
     pt[1].set(0, -3);
     pt[2].set(1, 0);
     pt[3].set(2, 1);
     pt[4].set(-1, 2);
     Path pa("..", true, &pt[0], &pt[1], &pt[2], &pt[3], &pt[4], 0);
     pa.draw();
     

Fig. 12.

Since pa is closed, it can be filled as well as drawn. The following example uses fill() with a Color argument, in order to avoid having a large splotch of black on the page. Common Colors are declared in the namespace Colors. See Color Reference.

     pa.fill(Colors::gray);
     

Fig. 13.

Closed Paths can be filled and drawn, using the function filldraw(). This function draws the Path using the pen specified, or MetaPost's currentpen by default. A Color for drawing the Path can also be specified, otherwise, the default color (currently Colors::black) is used. In addition, the Path is filled using a second Color, which can be specified, or the background_color (Colors::background_color), by default. Filling a Path using the background color causes it to hide objects that lie behind it. See Surface Hiding, for a description of the surface hiding algorithm, and examples. Currently, this algorithm is quite primitive and only works for simple cases.

     Point p0(-3, 0, 1);
     Point p1(3, 1, 1);
     p0.draw(p1);
     pa.filldraw();
     

Fig. 14.

The following example uses arguments for the Colors used for drawing and filling, and the pen. The empty string argument before the pen argument is a placeholder for the dash pattern argument.

     pa.filldraw(black, gray, "",
        "pensquare xscaled 3mm yscaled 1mm rotated 60");
     

Fig. 15.

Paths can also be "undrawn", "unfilled", and "unfilldrawn", using the corresponding functions:

     pa.fill(gray);
     p0.undraw(p1, "", "pencircle scaled 3mm");
     

Fig. 16.

     pa.fill(gray);
     Path q;
     q = pa;
     q.scale(.5, .5);
     q.unfill();
     

Fig. 17.

The function unfilldraw() takes a Color argument for drawing the Path, which is *Colors::background_color by default. This makes it possible to unfill the Path while drawing the outline with a visible Color. On the other hand, it also makes it necessary to specify *Colors::background_color or Colors::white, if the user wants to use the dash pattern and/or pen arguments, without drawing the Path.

     pa.fill(gray);
     q.unfilldraw(white, "", "pensquare xscaled 3mm yscaled 1mm");
     

Fig. 18.

The following example demonstrates the use of unfilldraw() with black as its Color argument. Unfortunately, it also demonstrates one of the limitations of the surface hiding algorith: The line from p0 to p1 is hidden by the filled Path pa. Since the portion of pa covered by Path q has been unfilled, the line from p_0 to p_1 should be visible as it passes through q. However, from the point of view of 3DLDF, there is no relationship between pa and q; nor does it "know" whether a Path has been filled or unfilled. If it's on a Picture, it will hide objects lying behind it, unless the surface hiding algorithm fails for another reason. See Surface Hiding, for more information.

     p0.draw(p1);
     pa.fill(gray);
     q.unfilldraw(black, "", "pensquare xscaled 3mm yscaled 1mm");
     

Fig. 19.

See Paths; Drawing and Filling, for more information, and complete descriptions of the functions.

Plane Figures

3DLDF currently includes the following classes representing plane geometric figures: Polygon, Reg_Cl_Plane_Curve ("Regular Closed Plane Curve"), Reg_Polygon ("Regular Polygon"), Rectangle, Ellipse and Circle. Polygon and Reg_Cl_Plane_Curve are derived from Path, Reg_Polygon and Rectangle are derived from Polygon, and Ellipse and Circle are derived from Reg_Cl_Plane_Curve. Polygon and Reg_Cl_Plane_Curve are meant to be used as base classes only, so objects of these types should normally never be declared.

Since Reg_Polygon, Rectangle, Ellipse, and Circle all ultimately derive from Path, they are really just special kinds of Path. In particular, they inherit their drawing and filling functions from Path, and their transformation functions take the same arguments as the Path versions. They also have constructors and setting functions that work in a similar way, with a few minor differences, to account for their different natures. See Polygon Reference, Rectangle Reference, Ellipse Reference, and Circle Reference, for complete information on these classes.

• Regular Polygons Getstart:
• Rectangles Getstart:
• Ellipses Getstart:
• Circles Getstart:

Regular Polygons

The following example creates a pentagon in the x-z plane, centered about the origin, whose enclosing circle has a radius equal to 3cm.

     default_focus.set(2, 3, -10, 2, 3, 10, 10);
     Reg_Polygon p(origin, 5, 3);
     p.draw();
     

Fig. 20.

Three additional arguments cause the pentagon to be rotated about the x, y, and z axes by the amount indicated. In this example, it's rotated 90 degrees

about the x-axis, so that it comes to lie in the x-y plane:

     Reg_Polygon p(origin, 5, 3, 90);
     p.draw();
     

Fig. 21.

In this example, it's rotated 36 degrees

about the y-axis, so that it appears to point in the opposite direction from the first example:

     Reg_Polygon p(origin, 5, 3, 0, 36);
     p.draw();
     

Fig. 22.

In this example, it's rotated 90 degrees

about the z-axis, so that it lies in the z-y plane:

     Reg_Polygon p(origin, 5, 3, 0, 0, 90);
     p.draw();
     

Fig. 23.

In this example, it's rotated 45 degrees

about the x, y, and z-axes in that order:

     Reg_Polygon p(origin, 5, 3, 45, 45, 45);
     p.draw();
     

Fig. 24.

Reg_Polygons need not be centered about the origin. If another Point pt is used as the first argument, the Reg_Polygon is first created with its center at the origin, then the specified rotations, if any, are performed. Finally, the Reg_Polygon is shifted such that its center comes to lie on pt:

     Point P(-2, 1, 1);
     Reg_Polygon hex(P, 6, 4, 60, 30, 30);
     hex.draw();
     

Fig. 25.

In the following example, the Reg_Polygon polygon is first declared using the default constructor, which creates an empty Reg_Polygon. Then, the polygon is repeatedly changed using the setting function corresponding to the constructor used in the previous examples. [next figure] demonstrates that a given Reg_Polygon need not always have the same number of sides.

     Point p(0, -3);
     Reg_Polygon polygon;
     for (int i = 3; i < 9; ++i)
       {
         polygon.set(p, i, 3);
         polygon.draw();
         p.shift(0, 1);
       }
     

Fig. 26.

Rectangles

A Rectangle can be constructed in the x-z plane by specifying a center Point, the width, and the height:

     Rectangle r(origin, 2, 3);
     r.draw();
     

Fig. 27.

Three additional arguments can be used to specify rotation about the x, y, and z-axes respectively:

     Rectangle r(origin, 2, 3, 30, 45, 15);
     r.draw();
     

Fig. 28.

If a Point p other than the origin is specified as the center of the Rectangle, the latter is first created in the x-z plane, centered about the origin, as above. Then, any rotations specified are performed. Finally, the Rectangle is shifted such that its center comes to lie at p:

     Point p0(.5, 1, 3);
     Rectangle r(p0, 4, 2, 30, 30, 30);
     r.draw();
     

Fig. 29.

This constructor has a corresponding setting function:

     Rectangle r;
     for (int i = 0; i < 180; i += 30)
       {
         r.set(origin, 4, 2, i);
         r.draw();
       }
     

Fig. 30.

Rectangles can also be specified using four Points as arguments, whereby they must be ordered so that they are contiguous in the resulting Rectangle:

     Point pt[4];
     pt[0].shift(-1, -2);
     pt[2] = pt[1] = pt[0];
     pt[1].rotate(180);
     pt[3] = pt[1];
     pt[2] *= pt[3].rotate(0, 180);
     Rectangle r(pt[0], pt[2], pt[3], pt[1]);
     r.draw();
     

Fig. 31.

This constructor checks whether the Point arguments are coplanar, however, it does not check whether they are really the corners of a valid rectangle; the user, or the code that calls this function, must ensure that they are. In the following example, r, although not rectangular, is a Rectangle, as far as 3DLDF is concerned:

     pt[0].shift(0, -1);
     pt[3].shift(0, 1);
     Rectangle q(pt[0], pt[2], pt[3], pt[1]);
     q.draw();
     

Fig. 32.

This constructor is not really intended to be used directly, but should mostly be called from within other functions, that should ensure that the arguments produce a rectangular Rectangle. There is also no guarantee that transformations or other functions called on Rectangle, Circle, or other classes representing geometric figures won't cause them to become non-rectangular, non-circular, or otherwise irregular. Sometimes, this might even be desirable. I plan to add the function Rectangle::is_rectangular() soon, so that users can test Rectangles for rectangularity.

Ellipses

Ellipse has a constructor similar to those for Reg_Polygon and Rectangle. The first argument is the center of the Ellipse, and the following two specify the lengths of the horizontal and vertical axes respectively. The Ellipse is first created in the x-z plane, centered about the origin. The horizontal axis lies along the x-axis and the vertical axis lies along the z-axis. The three subsequent arguments specify the amounts of rotation about the x, y, and z-axes respectively and default to 0. Finally, Ellipse is shifted such that its center comes to lie at the Point specified in the first argument.

     Point pt(-1, 1, 1);
     Ellipse e(pt, 3, 6, 90);
     e.draw();
     

Fig. 33.

As you may expect, this constructor has a corresponding setting function:

     Ellipse e;
     real h_save = 1.5;
     real v_save = 2;
     real h = h_save;
     real v = v_save;
     Point p(-1);
     for (int i = 0; i < 5; ++i)
       {
           e.set(p, h, v, 90);
           e.draw();
           h_save += .25;
           v_save += .25;
           h *= sqrt(h_save);
           v *= sqrt(v_save);
           p.shift(0, 0, 2);
       }
     

Fig. 34.

Circles

Circles are constructed just like Ellipses, except that the vertical and horizontal axes are per definition the same, so there's only one argument for the diameter, instead of two for the horizontal and vertical axes:

     Point P(0, 2, 1);
     Circle c(P, 3.5, 90, 90);
     c.draw();
     

Fig. 35.

This constructor, too, has a corresponding setting function:

     Circle c;
     Point p(-1, 0, 5);
     for (int i = 0; i < 16; ++i)
       {
           c.set(p, 5, i * 22.5, 0, 0, 64);
           c.draw();
       }
     

Fig. 36.

In the preceding example, the last argument to set(), namely "64", is for the number of Points used for constructing the perimeter of the Circle. The default value is 16, however, if it is used, foreshortening distorts the most nearly horizontal Circle. Increasing the number of points used improves its appearance. However, there may be a limit to how much improvement is possible. See Accuracy.

Solid Figures

• Cuboid Getstart:
• Polyhedron Getstart:

Cuboids

A cuboid is a solid figure consisting of six rectangular faces that meet at right angles. A cube is a special form of cuboid, whose faces are all squares. The constructor for the class Cuboid follows the pattern familiar from the constructors for the plane figures: The first argument is the center of the Cuboid, followed by three real arguments for the height, width, and depth, and then three more real arguments for the angles of rotation about the x, y, and z-axes. The Cuboid is first constructed with its center at the origin. Its width, height, and depth are measured along the x, y, and z-axes respectively. If rotations are specified, it is rotated about the x, y, z-axes in that order. Finally, it is shifted such that its center comes to lie on its Point argument, if the latter is not the origin.

If the width, height, and depth arguments are equal, the Cuboid is a cube:

     Cuboid c0(origin, 3, 3, 3, 0, 30);
     c0.draw();
     

Fig. 37.

In the following example, the Cuboid is "filldrawn", so that the lines dilineating the hidden surfaces of the Cuboid are covered.

     Cuboid c1(origin, 3, 4, 5, 0, 30);
     c1.filldraw();
     

Fig. 38.

Polyhedron

The class Polyhedron is meant for use only as a base class; no objects of type Polyhedron should be declared. Instead, there is a class for each of the different drawable polyhedra. Currently, 3DLDF defines only three: Tetrahedron, Dodecahedron, and Icosahedron. There's no need for a Cube class, because cubes can be drawn using Cuboid (see Cuboid Getstart).

Polyhedra have a high priority in my plans for 3DLDF. I intend to add Octahedron soon, which will complete the set of regular Platonic polyhedra. Then I will begin adding the semi-regular Archimedean polyhedra, and their duals.

The constructors for the classes derived from Polyhedron follow the pattern familiar from the classes already described. The constructors for the classes described below have identical arguments: First, a Point specifying the center, then a real for the diameter of the surrounding circle (Umkreis, in German) of one of its polygonal faces, followed by three real arguments for the angles of rotation about the main axes.

• Tetrahedron Getstart:
• Dodecahedron Getstart:
• Icosahedron Getstart:

Tetrahedron

The center of a tetrahedron is the intersection of the lines from a vertex to the center of the opposite side. At least, in 3DLDF, this is the center of a Tetrahedron. I'm not 100 degrees certain that this is mathematically correct.

     Tetrahedron t(origin, 4);
     t.draw();
     t.get_center().dotlabel("$c$");
     

Fig. 39.

Dodecahedron

A dodecahedron has 12 similar regular pentagonal faces. The following examples show the same Dodecahedron using different projections:

     default_focus.set(2, 5, -10, 2, 5, 10, 10);
     Dodecahedron d(origin, 3);
     d.draw();
     

Fig. 40.

Fig. 41.

Please note that the Dodecahedron in [next figure] is drawn, and not filldrawn!

Fig. 42.

Fig. 43.

In [next figure] , d is filldrawn. In this case, the surface hiding algorithm has worked properly. See Surface Hiding.

Fig. 44.

Icosahedron

An icosahedron has 20 similar regular triangular faces. The following examples show the same Icosahedron using different projections:

     default_focus.set(3, 0, -10, 2, 0, 10, 10);
     Icosahedron i(origin, 3);
     i.draw();
     

Fig. 45.

Fig. 46.

Fig. 47.

Fig. 48.

In [next figure] , i is filldrawn. In this case, the surface hiding algorithm has worked properly. See Surface Hiding.

Fig. 49.

Pictures

Applying drawing and filling operations to the drawable objects described in the previous chapters isn't enough to produce output. These operations merely modify the Picture object that was passed to them as an argument (current_picture, by default).

Pictures in 3DLDF are quite different from pictures in MetaPost. When a drawing or filling operation is applied to an object O, a copy of O, C, is allocated on the free store, a pointer to Shape S is pointed at C, and S is pushed onto the vector<Shape*> shapes on the Picture P, which was passed as an argument to the drawing or filling command. The arguments for the pen, dash pattern, Color, and any others, are used to set the corresponding data members of C (not O).

In order to actually cause MetaPost code to be written to the output file, it is necessary to invoke P.output(). Now, the appropriate version of output() is applied to each of the objects pointed to by a pointer on P.shapes. output() is a pure virtual function in Shape, so all classes derived from Shape must have an output() function. So, if shapes[0] points to a Path, Path::output() is called, if shapes[1] points to a Point, Point::output() is called, and if shapes[2] points to an object of a type derived from Solid, Solid::output() is called. Point, Path, and Solid are namely the only classes derived from Shape for which a version of output() is defined. All other Shapes are derived from one of these classes. These output() functions then write the MetaPost code to the output file through the output file stream out_stream.

     beginfig(1);
     default_focus.set(0, 0, -10, 0, 0, 10, 10);
     Circle c(origin, 3, 90);
     c.draw();
     c.shift(1.5);
     c.draw();
     current_picture.output();
     endfig(1);
     

Fig. 50.

The C++ code for [the previous figure] starts with the command beginfig(1) and ends with the command endfig(1). They simply write "beginfig(<arg> )" and "endfig()" to out_stream, The optional unsigned int argument to endfig() is not written to out_stream, it's merely "syntactic sugar" for the user.

In MetaPost, the endfig command causes output and then clears currentpicture. This is not the case in 3DLDF, where Picture::output() and Picture::clear() must be invoked explicitly:

     beginfig(1);
     Point p0;
     Point p1(1, 2, 3);
     p0.draw(p1);
     current_picture.output();
     endfig(1);
     
     beginfig(2);
     current_picture.clear();
     Circle C(origin, 3);
     C.fill();
     current_picture.output();
     endfig(2);
     

In [next figure] , two Pictures are used within a single figure.

     beginfig(1);
     Picture my_picture;
     default_focus.set(0, 0, -10, 0, 0, 10, 10);
     Circle c(origin, 3, 90);
     c.draw(my_picture);
     my_picture.output();
     c.shift(1.5);
     c.fill(light_gray);
     current_picture.output();
     endfig(1);
     

Fig. 51.

Multiple objects, or complex objects made up of sub-objects, can be stored in a Picture, so that operations can be applied to them as a group:

     default_focus.set(7, 5, -10, 7, 5, 10, 10);
     Cuboid c0(origin, 5, 5, 5);
     c0.shift(0, 0, 3);
     c0.draw();
     Circle z0(c0.get_rectangle_center(0), 2.5, 90, 0, 0, 64);
     z0.draw();
     Circle z1(z0);
     z1.shift(0, 0, -1);
     z1.draw();
     int i;
     int j = z0.get_size();
     for (i = 0; i < 8; ++i)
         z0.get_point(i * j/8).draw(z1.get_point(i * j/8));
     Cuboid c1(c0.get_rectangle_center(4), 5, 3, 3);
     c1.shift(0, 2.5);
     c1.draw();
     Rectangle r0 = *c1.get_rectangle_ptr(3);
     Point p[10];
     for (i = 0; i < 4; ++i)
       p[i] = r0.get_point(i);
     p[4] = r0.get_mid_point(0);
     p[5] = r0.get_mid_point(2);
     p[6] = p[4].mediate(p[5], 2/3.0);
     Circle z2(p[6], 2, 90, 90, 0, 16);
     z2.draw();
     Circle z3 = z2;
     z3.shift(3);
     z3.draw();
     j = z2.get_size();
     for (i = 0; i < 8; ++i)
         z2.get_point(i * j/8).draw(z3.get_point(i * j/8));
     p[7] = c0.get_rectangle_center(2);
     p[7].shift(-4);
     p[8] = c0.get_rectangle_center(3);
     p[8].shift(4);
     current_picture.output();
     current_picture.rotate(45, 45);
     current_picture.shift(10, 0, 3);
     current_picture.output();
     

Fig. 52.

Let's say the complex object in [the previous figure] represents a furnace. From the point of view of 3DLDF, however, it's not an object at all, and the drawing consists of a collection of unrelated Cuboids, Circles, Rectangles, and Paths. If we hadn't put it into a Picture, we could still have rotated and shifted it, but only by applying the operations to each of the sub-objects individually.

One consequence of the way Pictures are output in 3DLDF is, that the following code will not work:

     beginfig(1);
     Point p(1, 2);
     Point q(1, 3);
     out_stream << "pickup pencircle scaled .5mm;" << endl;
     origin.draw(p);
     out_stream << "pickup pensquare xscaled .3mm rotated 30;" << endl;
     origin.draw(q);
     current_picture.output();
     endfig();
     

This is the MetaPost code that results:

     beginfig(1);
     pickup pencircle scaled .5mm;
     pickup pensquare xscaled .3mm rotated 30;
     draw (0.000000cm, -3.000000cm) -- (1.000000cm, -1.000000cm);
     draw (0.000000cm, -3.000000cm) -- (1.000000cm, 0.000000cm);
     endfig;
     

It's perfectly legitimate to write raw MetaPost code to out_stream, as in lines 4 and 6 of this example. However, the draw() commands do not cause any output to out_stream. The MetaPost drawing commands are written to out_stream when current_picture.output() is called. Therefore, the pickup commands are "bunched up" before the drawing commands. In this example, setting currentpen to pencircle scaled .5mm has no effect, because it is immediately reset to pensquare xscaled .3mm rotated 30 in the MetaPost code, before the draw commands. It is not possible to change currentpen in this way within a Picture. Since the draw() commands in the 3DLDF code didn't specify a pen argument, currentpen with its final value is used for both of the MetaPost draw commands. For any given invocation of Picture::output(), there can only be one value of currentpen. All other pens must be passed as arguments to the drawing commands.

• Projections:
• Focuses Getstart:
• Surface Hiding:

Projections

In order for a 3D graphic program to be useful, it must be able to make two-dimensional projections of its three-dimensional constructions so that they can be displayed on computer screens and printed out. These are some of the possible projections:

• Parallel projection onto one of the major planes

These projections are trivial, and can be performed by 3DLDF. They are discussed in the following section.

• Parallel projection onto another plane

I haven't programmed these projections yet, but they might be useful, so I probably will, when I get around to it.

• The perspective projection

This is the projection most people think of, when they think of 3D-graphics. It is discussed in detail in The Perspective Projection.

• The isometric and axonometric projections

These projections are important for engineering and drafting. I have not yet implemented them in 3DLDF, but they are on my list of "Things To Do".

The function Picture::output() takes a const unsigned short argument specifying the projection to be used. The user should probably avoid using explicit unsigned shorts, but should use the constants defined for this purpose in the namespace Projections.¹⁹ The constants are PERSP, PARALLEL_X_Y, PARALLEL_X_Z, PARALLEL_Z_Y, AXON, and ISO. The latter two should not be used, because the axonometric and isometric projections have not yet been implemented.

• Parallel Projections:
• The Perspective Projection:

Parallel Projections

When a Picture is projected onto the x-y plane, the x and y-values from the world_coordinates of the Points belonging to the objects on the Picture are copied to their projective_coordinates, which are used in the MetaPost code written to out_stream. If a Picture p contains an object in the x-y plane, or in a plane parallel to the x-y plane, then the result of p.output(Projections::PARALLEL_X_Y) is more-or-less equivalent to just using MetaPost without 3DLDF.

     Rectangle r(origin, 3, 3, 90);
     Circle c(origin, 3, 90);
     c *= r.shift(0, 0, 5);
     r.draw();
     c.draw();
     current_picture.output(Projections::PARALLEL_X_Y);
     

Fig. 53.

If the objects do not lie in the x-y plane, or a plane parallel to the x-y plane, then the projection will be distorted:

     current_picture.output(Projections::PARALLEL_X_Y);
     

Fig. 54.

Picture::output() can be called with an additional real argument factor for magnifying or shrinking the Picture.

     Rectangle r(origin, 4, 4, 90, 60);
     Circle c(origin, 4, 90, 60);
     c *= r.shift(0, 0, 5);
     r.filldraw(black, gray);
     c.unfilldraw(black);
     current_picture.output(Projections::PARALLEL_X_Y, .5);
     current_picture.shift(2.5);
     current_picture.output(Projections::PARALLEL_X_Y);
     current_picture.shift(1);
     current_picture.output(Projections::PARALLEL_X_Y, 2);
     

Fig. 55.

Parallel projection onto the x-z and z-y planes are completely analogous to parallel projection onto the x-y plane.

The Perspective Projection

The perspective projection obeys the laws of linear perspective. In 3DLDF, it is performed by means of a transformation, whose effect is, to the best of my knowledge, exactly equivalent to the result of a perspective projection done by hand using vanishing points and rulers.

It is very helpful to the artist to understand the laws of linear perspective, and to know how to make a perspective drawing by hand.²⁰ However, it is a very tedious and error-prone procedure (I know, I've done it). One of my main motivations for writing 3DLDF was so I wouldn't have to do it anymore.

[next figure] shows a perspective construction, the way it could be done by hand. The point of view, or focus is located 6cm from the picture plane, and 4cm above the ground (or x-z) plane at the point (0, 4, -6). The rectangle R lies in the ground plane, with the point r_0 at (2, 0, 1.5). The right side of R, with length = 2cm lies at an angle of 40 to the ground line, which corresponds to the intersection line of the ground plane with the picture plane, and the left side, with length = 5cm, at an angle of 90 degrees - 40 degrees = 50 degrees to the ground line.

Fig. 56.

While it's possible to use 3DLDF to make a perspective construction in the traditional way, as [the previous figure] shows, the code for [next figure]

achieves the same result more efficiently:

     default_focus.set(0, 4, -6, 0, 4, 6, 6);
     Rectangle r(origin, 2, 5, 0, 40);
     Point p(2, 0, 1.5);
     r.shift(p - r.get_point(0));
     r.draw();
     

Fig. 57.

In [the second-to-last figure] , it was convenient to start with the corner point r_0; if we needed the center of R, it would have to be found from the corner points. However, in 3DLDF, Rectangles are most often constructed about the center. Therefore, in [next figure] , R is first constructed about the origin, with the rotation about the y-axis passed as an argument to the constructor. It is then shifted such that *(R.points[0]), the first (or zeroth, if you will) Point on R comes to lie at (2, 0, 1.5).

Unlike the other transformations currently used in 3DLDF, the perspective transformation is non-affine. Affine transformations maintain parallelity of lines, while the rules of perspective state that parallel lines, with one exception, appear to recede toward a vanishing point.²¹

In [the second-to-last figure] , the lines from r_0 to r_1 and from r_3 to r_2 appear to vanish toward the right-hand 40 degrees vanishing point, while the lines from r_0 to r_3 and from r_1 to r_2 appear to vanish toward the left-hand 50 degrees vanishing point. The lower the angle of a vanishing point, the further away it is from the center of vision, as [next figure] shows:

Fig. 58.

In [the previous figure] , the 0.5 degrees vanishing point is nearly 5 and 3/4 meters away from the CV, and a line receding to it will be very nearly horizontal. However, the distance from the focus to the CV is only 5cm. As this distance increases, the distance from the CV to a given vanishing point increases proportionately. If the distance is 30cm, a more reasonable value for a drawing, then the x-coordinate of VP 10 degrees is 170.138cm, that of VP 5 degrees is 342.902cm, and that of VP 0.5 degrees is 3437.66cm! This is the reason why perspective drawings done by hand rarely contain lines receding to the horizon at low angles.

This problem doesn't arise when the perspective transformation is used. In this case, any angle can be calculated as easily as any other:

     default_focus.set(0, 4, -6, 0, 4, 6, 6);
     Rectangle r;
     Point center(0, 2);
     r.set(center, 2, 5, 0, 0, 0.5);
     r.draw();
     
     r.set(center, 2, 5, 0, 0, 2.5);
     r.draw();
     
     r.set(center, 2, 5, 0, 0, 5);
     r.draw();
     current_picture.output();
     

Fig. 59.

Focuses

The perspective transformation requires a focus; as a consequence, outputting a Picture requires an object of class Focus. Picture::output() takes an optional pointer-to-Focus argument, which is 0 by default. If the default is used, (or 0 is passed explicitly), the global variable default_focus is used. See Focus Reference; Global Variables.

A Focus can be thought of as the observer of a scene, or a camera. It contains a Point position for its location with respect to 3DLDF's coordinate system, and a Point direction, specifying the direction where the observer is looking, or where the camera is pointed. The Focus can be rotated freely about the line PD, where P stands for position and D for direction, so a Focus contains a third Point up, to indicate which direction will be "up" on the projection, when a Picture is projected.

The projection plane q will always be perpendicular to the line PD, or to put it another way, the line PD, is normal to q.

Unlike the traditional perspective construction, where the distance from the focus to the center of vision fixes both the location of the focus in space, and its distance to the picture plane,²² these two parameters can be set independently when the perspective transformation is used. The distance from a Focus to the picture plane is stored in the data member distance, of type real.

A Focus can be declared using two Point arguments for position and direction, and a real argument for distance, in that order.

     Point pos(0, 5, -10);
     Point dir(0, 5, 10);
     Focus f(pos, dir, 10);
     
     Point center(2, 0, 3);
     Rectangle r(center, 3, 3);
     r.draw();
     current_picture.output(f);
     

Fig. 60.

The "up" direction is calculated by the Focus constructor automatically. An optional argument can be used to specify the angle by which to rotate the Focus about the line PD.

     Point pos(0, 5, -10);
     Point dir(0, 5, 10);
     Focus f(pos, dir, 10, 30);
     Point center(2, 0, 3);
     Rectangle r(center, 3, 3);
     r.draw();
     current_picture.output(f);
     

Fig. 61.

Alternatively, a Focus can be declared using three real arguments each for the x, y, and z-coordinates of position and direction, respectively, followed by the real arguments for distance and the angle of rotation:

     Focus f(3, 5, -5, 0, 3, 0, 10, 10);
     Point center(2, 0, 3);
     Rectangle r(center, 3, 3);
     r.draw();
     current_picture.output(f);
     

Fig. 62.

Focuses contain two Transforms, transform and persp. A Focus can be located anywhere in 3DLDF's coordinate system. However, performing the perspective projection is more convenient, if position and direction both lie on one of the major axes, and the plane of projection corresponds to one of the major planes. transform is the transformation which would have this affect on the Focus, and is calculated by the Focus constructor. When a Picture is output using that Focus, transform is applied to all of the Shapes on the Picture, maintaining the relationship between the Focus and the Shapes, while making it easier to calculate the projection. The Focus need never be transformed by transform. The actual perspective transformation is stored in persp.

Focuses can be moved by using one of the setting functions, which take the same arguments as the constructors. Currently, there are no affine transformation functions for moving Focuses, but I plan to add them soon. If 3DLDF is used for making animation, resetting the Focus can be used to simulate camera movements:

     beginfig(1);
     Point pos(2, 10, 3);
     Point dir(2, -10, 3);
     Focus f;
     Point center(2, 0, 3);
     for (int i = 0; i < 5; ++i)
       {
         f.set(pos, dir, 10, (15 * i));
         Rectangle r(center, 3, 3);
         r.draw();
         current_picture.output(f);
         current_picture.clear();
         pos.shift(1, 1, 0);
         dir.rotate(0, 0, 10);
       }
     endfig(1);
     

Fig. 63.

In [the previous figure] , current_picture is output 5 times within a single MetaPost figure. Since the file passed to MetaPost is called persp.mp, the file of Encapsulated PostScript (EPS) code containing [the previous figure] is called persp.1. To use this technique for making an animation, it's necessary to output the Picture into multiple MetaPost figures.

     Point pos(2, 10, 3);
     Point dir(2, -10, 3);
     Focus f;
     Point center(2, 0, 3);
     for (int i = 0; i < 5; ++i)
       {
         f.set(pos, dir, 10, (15 * i));
         Rectangle r(center, 3, 3);
         r.draw();
         beginfig(i+1);
         current_picture.output(f);
         endfig();
         current_picture.clear();
         pos.shift(1, 1, 0);
         dir.rotate(0, 0, 10);
       }
     

Now, running MetaPost on persp.mp generates the EPS files persp.1, persp.2, persp.3, persp.4, and persp.5, containing the five separate drawings of r.

Surface Hiding

In [next figure] , Circle c lies in front of Rectangle r. Since c is drawn and not filled, r is visible behind c.

     default_focus.set(1, 3, -5, 0, 3, 5, 10);
     Point p(0, -2, 5);
     Rectangle r(p, 3, 4, 90);
     r.draw();
     Point q(2, -2, 3);
     Circle c(q, 3, 90);
     c.draw();
     current_picture.output();
     

Fig. 64.

If instead, c is filled or filldrawn, only the parts of r that are not covered by c should be visible:

     r.draw();
     c.filldraw();
     

Fig. 65.

What parts of r are covered depend on the point of view, i.e., the position and direction of the Focus used for outputting the Picture:

     default_focus.set(8, 0, -5, 5, 3, 5, 10);
     

Fig. 66.

Determining what objects cover other objects in a program for 3D graphics is called surface hiding, and is performed by a hidden surface algorithm. 3DLDF currently has a very primitive hidden surface algorithm that only works for the most simple cases.

The hidden surface algorithm used in 3DLDF is a painter's algorithm, which means that the objects that are furthest away from the Focus are drawn first, followed by the objects that are closer, which may thereby cover them. In order to make this possible, the Shapes on a Picture must be sorted before they are output. They are sorted according to the z-values in the projective_coordinates of the Points belonging to the Shape. This may seem strange, since the projection is two-dimensional and only the x and y-values from projective_coordinates are written to out_stream. However, the perspective transformation also produces a z-coordinate, which indicates the distance of the Points from the Focus in the z-dimension.

The problem is, that all Shapes, except Points themselves, consist of multiple Points, that may have different z-coordinates. 3DLDF currently does not yet have a satisfactory way of dealing with this situtation. In order to try to cope with it, the user can specify four different ways of sorting the Shapes: They can be sorted according to the maximum z-coordinate, the minimum z-coordinate, the mean of the maximum and minimum z-coordinate (max + min) / 2, and not sorted. In the last case, the Shapes are output in the order of the drawing and filling commands in the user code. The z-coordinates referred to are those in projective_coordinates, and will have been calculated for a particular Focus.

The function Picture::output() takes a const unsigned short sort_value argument that specifies which style of sorting should be used. The namespace Sorting contains the following constants which should be used for sort_value: MAX_Z, MIN_Z, MEAN_Z, and NO_SORT. The default is MAX_Z.

3DLDF's primitive hidden surface algorithm cannot work for objects that intersect. The following examples demonstrate why not:

     using namespace Sorting;
     using namespace Colors;
     using namespace Projections;
     default_focus.set(5, 3, -10, 3, 1, 1, 10, 180);
     Rectangle r0(origin, 3, 4, 45);
     Rectangle r1(origin, 2, 6, -45);
     r0.draw();
     r1.draw();
     current_picture.output(default_focus, PERSP, 1, MAX_Z);
     r0.show("r0:");
     -| r0:
     fill_draw_value == 0
     (-1.5, -1.41421, -1.41421) -- (1.5, -1.41421, -1.41421) --
     (1.5, 1.41421, 1.41421) -- (-1.5, 1.41421, 1.41421)
     -- cycle;
     
     r0.show("r0:", 'p');
     -| r0:
     fill_draw_value == 0
     Perspective coordinates.
     (-5.05646, -4.59333, -0.040577) -- (-2.10249, -4.86501, -0.102123) --
     (-1.18226, -1.33752, 0.156559) -- (-3.51276, -1.2796, 0.193084)
     -- cycle;
     
     r1.show("r1:");
     -| r1:
     fill_draw_value == 0
     (-1, 2.12132, -2.12132) -- (1, 2.12132, -2.12132) --
     (1, -2.12132, 2.12132) -- (-1, -2.12132, 2.12132)
     -- cycle;
     
     r1.show("r1:", 'p');
     -| r1:
     fill_draw_value == 0
     Perspective coordinates.
     (-5.09222, -0.995681, -0.133156) -- (-2.98342, -1.03775, -0.181037) --
     (-1.39791, -4.05125, 0.208945) -- (-2.87319, -3.93975, 0.230717)
     -- cycle;
     

Fig. 67.

In [the previous figure] , the Rectangles r_0 and r_1 intersect along the x-axis. The z-values of the world_coordinates of r_0 are -1.41421 and 1.41421 (two Points each), while those of r_1 are 2.12132 and -2.12132. So r_1 has two Points with z-coordinates greater than the z-coordinate of any Point on r_0, and two Points with z-coordinates less than the z-coordinate of any Point on r_0. The Points on r_0 and r_1 all have different z-values in their projective_coordinates, but r_1 still has a Point with a z-coordinate greater than that of any of the Points on r_0, and one with a z-coordinate less than that of any of the Points on r_0.

In [next figure] , the Shapes on current_picture are sorted according to the maximum z-values of the projective_coordinates of the Points belonging to the Shapes. r_1 is filled and drawn first, because it has the Point with the positive z-coordinate of greatest magnitude. When subsequently r_0 is drawn, it covers part of the top of r_1, which lies in front of r_0, and should be visible:

     current_picture.output(default_focus, PERSP, 1, MAX_Z);
     

Fig. 68.

In [next figure] , the Shapes on current_picture are sorted according to the minimum z-values of the projective_coordinates of the Points belonging to the Shapes. r1 is drawn and filled last, because it has the Point with the negative z-coordinate of greatest magnitude. It thereby covers the bottom part of r0, which lies in front of r1, and should be visible.

     current_picture.output(default_focus, PERSP, 1, MIN_Z);
     

Fig. 69.

Neither sorting by the mean z-value in the projective_coordinates, nor suppressing sorting does any good. In each case, one Rectangle is always drawn and filled last, covering parts of the other that lie in front of the first.

3DLDF's hidden surface algorithm will fail wherever objects intersect, not just where one extends past the other in both the positive and negative z-directions.

     Rectangle r(origin, 3, 4, 45);
     Circle c(origin, 2, -45);
     r.filldraw();
     c.filldraw(black, gray);
     current_picture.output(default_focus, PERSP, 1, NO_SORT);
     

Fig. 70.

Even where objects don't intersect, their projections may. In order to handle these cases properly, it is necessary to break up the Shapes on a Picture into smaller Shapes, until there are none that intersect or whose projections intersect. Then, any of the three methods of sorting described above can be used to sort the Shapes, and they can be output.

Before this can be done, 3DLDF must be able to find the intersections of all of the different kinds of Shapes. If 3DLDF converted solids to polyhedra and curves to sequences of line segments, this would reduce to the problem of finding the intersections of lines and planes, however it does not yet do this.

Even if it did, a fully functional hidden surface algorithm must compare each Shape on a Picture with every other Shape. Therefore, for n Shapes, there will be n! / ((n - r)! r!) (possibly time-consuming) comparisons.

Fig. 71.

Clearly, such a hidden surface algorithm would considerably increase run-time.

Currently, all of the Shapes on a Picture are output, as long as they lie completely within the boundaries passed as arguments to Picture::output(). See Pictures; Outputting. It would be more efficient to suppress output for them, if they are completely covered by other objects. This also requires comparisions, and could be implemented together with a fully-functional hidden surface algorithm.

Shadows, reflections, highlights and shading are all effects requiring comparing each Shape with every other Shape, and could greatly increase run-time.

Intersections

There are no functions for finding the intersection points of two (or more) arbitrary Paths. This is impossible, so long as 3DLDF outputs MetaPost code. 3DLDF only "knows" about the Points on a Path; it doesn't actually generate the curve or other figure that passes through the Points, and consequently doesn't "know" how it does this.

In addition, an arbitrary Path can contain connectors. In 3DLDF, the connectors are merely strings and are written verbatim to the output file, however, in MetaPost they influence the form of a Path.

3DLDF can, however, find the intersection points of some non-arbitrary Paths. So far, it can find the intersection point of the following combinations of Paths:

1. Two linear Paths, i.e., Paths for which Path::is_linear() returns true (see Path Reference; Querying). In addition, the static Point member function Point::intersection_points() can be called with four Point arguments. The first and second arguments are treated as the end points of one line, and the third and fourth arguments as the end points of the other.
2. A line and a Polygon. Currently, Reg_Polygon and Rectangle are the only classes derived from Polygon.
3. Two Polygons.
4. A line and a Regular Closed Plane Curve (Reg_Cl_Plane_Curve, see Regular Closed Plane Curve Reference; Intersections). Currently, Ellipse and Circle are the only classes derived from Reg_Cl_Plane_Curve.
5. Two Ellipses. Since a Circle is also an Ellipse, one or both of the Ellipses may be a Circle. See Ellipse Reference; Intersections.

Adding more functions for finding the intersections of various geometric figures is one of my main priorities with respect to extending 3DLDF.

There are currently no special functions for finding the intersection points of a line and a Circle or two Circles. Since the class Circle is derived from class Ellipse, Circle::intersection_points() resolves to Ellipse::intersection_points(), which, in turn, calls Reg_Cl_Plane_Curve::intersection_points(). This does the trick, but it's much easier to find the intersections for Circles that it is for Ellipses. In particular, the intersections of two coplanar Circles can be found algebraically, whereas I've had to implement a numerical solution for the case of two coplanar Ellipses with different centers and/or axis orientation. It may also be worthwhile to write a specialization for finding the intersection points of a Circle and an Ellipse.

The theory of intersections is a fascinating and non-trivial branch of mathematics.²³ As I learn more about it, I plan to define more classes to represent various curves (two-dimensional ones to start with) and functions for finding their intersection points.

Installing and Running 3DLDF

• Installing 3DLDF:
• Running 3DLDF:

Installing 3DLDF

3DLDF is available for downloading from http://ftp.gnu.org/gnu/3dldf. The official 3DLDF website is http://www.gnu.org/software/3dldf. The "tarball", i.e., the compressed archive file 3DLDF-1.1.5.1.tar.gz unpacks into a directory called /3DLDF-1.1.5.1/.

On a typical Unix-like system, entering the following commands at the command line in a shell will unpack the 3DLDF distribution. Please note that the form of the commands may differ on your system.

     gunzip 3DLDF-1.1.5.1.tar.gz
     tar xpvf 3DLDF-1.1.5.1.tar
     

The p option to tar ensures that the files will have the same permissions as when they were packed.

The directory 3DLDF-1.1.5.1/ contains a configure script, which should be called from the command line in the shell, using the absolute path of 3DLDF-1.1.5.1/ as the prefix argument. For example, if the path is /usr/local/mydir/3DLDF-1.1.5.1/, configure should be invoked as follows:

     cd 3DLDF-1.1.5.1
     configure --prefix=/usr/local/mydir/3DLDF-1.1.5.1/
     

configure generates a Makefile from the Makefile.in in 3DLDF-1.1.5.1/, and in each of the subdirectories 3DLDF-1.1.5.1/CWEB, 3DLDF-1.1.5.1/DOC, and 3DLDF-1.1.5.1/DOC/TEXINFO. Now, make install causes the 3DLDF to be built. The executable is called 3dldf.

See the files README and INSTALL in the 3DLDF distribution for more information.

• Template Functions:

Template Functions

3DLDF 1.1.5 is the first release that contains template functions, namely template <class C> C* create_new(), which is defined in creatnew.web, and template <class Real> Real get_second_largest(), which is defined in gsltmplt.web. See Dynamic Allocation of Shapes, and Get Second Largest Real.

In order for template functions to be instantiated correctly, their definitions must be available in each compilation unit where specializations are declared or used. For non-template functions, it suffices for their declarations to be available, and their definitions are found at link-time. For this reason, the definitions of create_new() and get_second_largest() are in their own CWEB files, and are written to their own header files. The latter are included in the other CWEB files that need them.

In addition, AM_CXXFLAGS = -frepo has been added to the file Makefile.am in 3DLDF-1.1.5/CWEB/, so that the C++

compiler is called using the -frepo option. The manual Using and Porting the GNU Compiler Collection explains this as follows:

"Compile your template-using code with -frepo. The compiler will generate files with the extension .rpo listing all of the template instantiations used in the corresponding object files which could be instantiated there; the link wrapper, collect2, will then update the .rpo files to tell the compiler where to place those instantiations and rebuild any affected object files. The link-time overhead is negligible after the first pass, as the compiler will continue to place the instantiations in the same files."²⁴

The first time the executable 3dldf is built, the files that use the template functions are recompiled one or more times, and the linker is also called several times. This doesn't happen anymore, once the .rpo files exist.

Template instantiation differs from compiler to compiler, so using template functions will tend to make 3DLDF less portable. I am no longer able to compile it on the DECalpha Personal Workstation I had been using with the DEC C++ compiler. See Ports, for more information.

Running 3DLDF

To use 3DLDF, call make run from the command line in the shell. The working directory should be 3DLDF-1.1.5.1/ or 3DLDF-1.1.5.1/CWEB. Either will work, but the latter may be more convenient, because this is the location of the CWEB, TeX and MetaPost files that you'll be editing. Alternatively, call ldfr, which is merely a shell script that calls make run. This takes care of running 3dldf, MetaPost, TeX, and dvips, producing a PostScript file containing your drawings. You can display the latter on your terminal using Ghostview or some other PostScript viewer, print it out, and whatever else you like to do with PostScript files.

However, you can also perform the actions performed by make run by hand, by writing your own shell scripts, by defining Emacs-Lisp commands, or in other ways. Even if you choose to use make run, it's important to understand what it does. The following explains how to do this by hand.

The CWEB source files for 3DLDF are in the subdirectory 3DLDF-1.1.5.1/CWEB/. They must be ctangled, and the resulting C++ files must be compiled and linked, in order to create the executable file 3dldf. The C++ files and header files generated by ctangle, the object files generated by the compiler, and the executable 3dldf all reside in 3DLDF-1.1.5.1/CWEB/. Therefore, the latter must be your working directory.

Since 3DLDF has no input routine as yet, as explained in No Input Routine, it is necessary to add C++ code to the function main() in main.web, and/or in a separate function in another file. In the latter case, the function containing the user code must be invoked in main(). Look for the line "Your code here!" in main.web.

This is an example of what you could write in main(). Feel free to make it more complicated, if you wish.

     beginfig(1);
     default_focus.set(2, 3, -10, 2, 3, 10, 20);
     Rectangle R(origin, 5, 3);
     Circle C(origin, 3, 90);
     C.half(180).filldraw(black, light_gray);
     R.filldraw();
     C.half().filldraw(black, light_gray);
     Point p = C.get_point(4);
     p.shift(0, -.5 * p.get_y());
     p.label("$C$", "");
     Point q = R.get_mid_point(0);
     q.shift(0, 0, -.5 * q.get_z());
     q.label("$R$", "");
     current_picture.output(default_focus, PERSP, 1, NO_SORT);
     endfig(1);
     

Fig. 72.

1. Save main.web, and any other CWEB files you've changed. Since these files have changed, they must be ctangled, and the resulting C++ files must be recompiled. If you've changed any files other than main.web, ctangle will also generate a header file for each of these files. If a header file differs from the version that existed before ctangle was run, all of the C++ files that depend on it must be recompiled. Then 3dldf must be relinked. To do this, call make 3dldf from the command line.

   If you've made any errors in typing your code, the compiler should have issued error messages, so go back into the appropriate CWEB file and correct your errors. Then call make 3dldf again.

2. Call CWEB/3dldf at the command line. It writes a file of MetaPost code called 3DLDFput.mp.
3. Run MetaPost on the file 3DLDFmp.mp, which inputs 3DLDFput.mp.

             mpost 3DLDFput
             

   The result is an Encapsulated PostScript file 3DLDFput.<integer> for each figure in your drawing.

4. The file 3DLDFtex.tex should contain code for including the 3DLDFput.<integer> files. This is an example taken from the 3DLDFtex.tex included in the distribution. You may change it to suit your purposes.

             \vbox to \vsize{\vskip 2cm
             \line{\hskip 2cm Figure 1.\hss}%
             \vfil
             \line{\hskip 2cm\epsffile{3DLDFmp.1}\hss}%
             \vss}
             

5. Run TeX on 3DLDFtex.tex to produce the DVI file, 3DLDFtex.dvi.

             tex 3DLDFtex
             

6. Run dvips on the DVI file to produce the PostScript file, 3DLDFtex.ps.

             dvips -o 3DLDFtex.ps 3DLDFtex
             

7. 3DLDFtex.ps can be viewed using Ghostview, it can be printed using lpr (on a Unix-like system), you can convert it to PDF with ps2pdf, or to some other format using the appropriate program.

I sincerely hope that it worked. If it didn't, ask your local computer wizard for help.

On the computer I'm using, I found that special arguments for setting landscape and papersize in TeX files for DIN A3 landscape didn't work. Ghostview cut off the right sides of the drawings. Nor did it work to call dvips -t landscape -t a3. This caused an error message which said that landscape would be ignored. When I called dvips with the -t landscape option alone, it worked, and Ghostview showed the entire drawing.

Another problem was Adobe Acrobat. It would display the entire DIN A3 page, but not always in landscape format. I was unable to find a way of rotating the pages in Acrobat. I finally found out, that if I included even a single letter of text in a label, Acrobat would display the document correctly.

• Converting EPS Files:
• Command Line Arguments:

Converting EPS Files

It is possible to have MetaPost generate structured PostScript directly by including the command prologues:=1; at the beginning of the MetaPost input. However, this "generally doesn't work when you use TeX fonts."²⁵ This is a significant problem if your labels contain math mode material, and you haven't already taken steps to ensure that appropriate fonts will be used in the PS output.

In the following, I describe the only way I've found to convert an EPS image to PNG format while still using TeX fonts. There may be other and better ways of doing this, but I haven't found them.

1. Assume the EPS image is in the file 3DLDFmp.1 Include the EPS image in a TeX file which looks like this:

             \advance\voffset by -1in
             \advance\hoffset by -1in
             \nopagenumbers
             \input epsf
             \epsfverbosetrue
             \def\epsfsize#1#2{#1}
             \setbox0=\vbox{\epsffile{3DLDFmp.1}}
             \vsize=\ht0
             \hsize=\wd0
             \special{papersize=\the\wd0,\the\ht0}
             \box0
             \bye
             

   Do not name this file 3DLDFmp.1.tex! While this worked fine for me on a DECalpha Personal Workstation running under Tru64 Unix 5.1, with TeX, Version 3.1415 (C version 6.1), and dvipsk 5.58f, it failed on a PC Pentium II XEON under Linux 2.4, with TeX, Version 3.14159 (Web2C 7.4.5), and dvips(k) 5.92b, kpathsea version 3.4.5, with the following error message:

             ``No BoundingBox comment found in file examples.1; using defaults''
             

   The resulting PS image had the wrong size and the the graphic was positioned improperly.

   Apparently, it confuses the EPSF macros when the name of an included image is the same as \jobname. So, for this example, let's call it 3DLDFmp.1_.tex.

   You don't really need to call the macro \epsfverbosetrue. If you do, it will print the measurements of the bounding box and other information to standard output.²⁶

2. Run tex 3DLDFmp.1_.tex.
3. Run dvips -o 3DLDF.1.ps 3DLDFmp.1_.dvi.
4. Run convert 3DLDF.1.ps 3DLDFmp.1.png.

     display 3DLDFmp.1.png
     

It can be included in an HTML document as follows:

     <img src="3DLDFmp.1.png"
              alt="[Fig. 1]."
     

Please note! The PNG files for this manual are now called filename 3DLDF1.png, 3DLDF2.png, ..., 3DLDF199.png, because I wasn't able to write files with names like 3DLDFmp.<number>.png to a CD-R (Compact Disk, Recordable), when `number' had more than one digit.

• Converting EPS Files ELISP:

Emacs-Lisp Functions

The file 3DLDF-1.1.5.1/CWEB/cnepspng.el contains definitions of two Emacs-Lisp functions that can be used to convert Encapsulated PostScript (EPS) files to structured PostScript (PS) and Portable Network Graphics (PNG) files.

convert-eps filename do-not-delete-files

Emacs-Lisp function

Converts an EPS image file to the PS and PNG formats. If called interactively, convert-eps prompts for the filename, including the extension, of an EPS image file. It follows the procedure described above in Converting EPS Files, to create filename.ps and filename.png. If do-not-delete-files is nil, the .tex, .dvi, and .log files will be deleted. This is the case when convert-eps is called interactively with no prefix argument. If convert-eps is called interactively with a prefix argument, or non-interactively with a non-nil do-not-delete-files argument, these files will not be deleted.

convert-eps-loop arg start end

Emacs-Lisp function

Converts a set of EPS image files to the PS and PNG formats. The files must all have the same filename, and the extensions must form a range of positive integers. For example, convert-eps-loop can be used to convert the files 3DLDFmp.1, 3DLDFmp.2, and 3DLDFmp.3 to 3DLDFmp.1.ps, 3DLDFmp.2.ps, and 3DLDFmp.3.ps on the one hand, and 3DLDFmp.1.png, 3DLDFmp.2.png, 3DLDFmp.3.png on the other. If convert-eps-loop is called interactively, it prompts for filename with no extension and the starting and ending numbers of the range. For all i \in \INT and start \le i \le end, convert-eps-loop checks whether a file named filename.i exists. If it does, it calls convert-eps, passing filename.i as the latter's filename argument. do-not-delete-files is also passed to convert-eps. If it's nil, the .tex, .dvi, and .log files will be deleted. This is the case when convert-eps-loop is called interactively with no prefix argument. If convert-eps-loop is called interactively with a prefix argument, or non-interactively with a non-nil do-not-delete-files argument, these files will not be deleted.

Command Line Arguments

3dldf can be called with the following command line arguments.

--help

Prints information about the valid command line options to standard output and exits with return value 0.

--silent

Suppresses some output to standard output and standard error when 3dldf is run

--verbose

Causes status information to be printed to standard output when 3dldf is run.

--version

Prints the version number of 3DLDF to standard output and exits with return value 0.

Currently, 3dldf can only handle long options. - cannot be substituted for --. However, the names of the options themselves can be abbreviated, as long as the abbreviation is unambigous. For example, 3dldf --h and 3dldf --verb are valid, but 3dldf --ver is not.

Typedefs and Utility Structures

3DLDF defines a number of data types for various reasons, e.g., for the sake of convenience, for use in conditional compilation, or as return values of functions. Some of these data types can be defined using typedef, while others are defined as structs.

The typedefs and utility structures described in this chapter are found in pspglb.web. Others, that contain objects of types defined in 3DLDF, are described in subsequent chapters.

real

typedef

Synonymous either with float or double, depending on the values of the preprocessor variables LDF_REAL_FLOAT and LDF_REAL_DOUBLE. The meaning of real is determined by means of conditional compilation. If real is float, 3DLDF will require less memory than if real is double, but its calculations will be less precise. real is "typedeffed" to float by default.

real_pair first second

typedef

Synonymous with pair<real, real>.

real_triple first second third

struct

All three data elements of real_triple are reals. It also has two constructors, described below. There are no other member functions.

void real_triple (void)	Constructor
void real_triple (real a, real b, real c)	Constructor

The constructor taking no arguments sets first, second, and third to 0. The constructor taking three real arguments sets first to a, second to b, and third to c.

Matrix

typedef

A Matrix is a 4 X 4 array of real, e.g., Matrix M; == real M[4][4]. It is used in class Transform for storing transformation matrices. See Transforms, and See Transform Reference, for more information.

real_short first second

typedef

Synonymous with pair<real, signed short>. It is the return type of Plane::get_distance().

bool_pair first second

typedef

Synonymous with pair<bool, bool>.

bool_real first second

typedef

Synonymous with pair<bool, real>.

Global Constants and Variables

The global constants and variables described in this chapter are found in pspglb.web. Others, of types defined in 3DLDF, are described in subsequent chapters.

bool ldf_real_float	Constants
bool ldf_real_double

Set to 0 or 1 to match the values of the preprocessor macros LDF_REAL_FLOAT and LDF_REAL_DOUBLE. The latter are used for conditional compilation and determine whether real is "typedeffed" to float or double, i.e., whether real is made to be a synonym of float or double using typedef. ldf_real_float and ldf_real_double can be used to control conditional expressions in non-conditionally compiled code.

real PI

Constant

The value of PI is calculated as 4.0 * arctan(1.0). I believe that a preprocessor macro "PI" was available when I compiled 3DLDF using the DEC C++ compiler, and that it wasn't, when I used GNU CC under Linux, but I'm no longer sure.

valarray <real> null_coordinates

Variable

Contains four elements, all 0. Used for resetting the sets of coordinates belonging to Points, but only when the DEC C++ compiler is used. This doesn't work when GCC is used.

real INVALID_REAL

Constant

Actually, INVALID_REAL is the largest possible real value (i.e., float or double) on a given machine. So, from the point of view of the compiler, it's not invalid at all. However, 3DLDF uses it to indicate failure of some kind. For example, the return value of a function returning real can be compared with INVALID_REAL to check whether the function succeeded or failed. An alternative approach would be to use the exception handling facilities of C++ . I do use these, but only in a couple of places, so far.

real_pair INVALID_REAL_PAIR

Constant

first and second are both INVALID_REAL.

real INVALID_REAL_SHORT

Constant

first is INVALID_REAL and second is 0.

real MAX_REAL

Variable

The largest real value permitted in the the elements of Transforms and the coordinates of Points. It is the second largest real value (i.e., float or double) on a given machine (INVALID_REAL is the largest). MAX_REAL is a variable, but it should be used like a constant. In other words, users should never reset its value. It can't be declared const because its value must be calculated using function calls, which can't be done before the entry point of the program, i.e., main(). Therefore, the value of MAX_REAL is calculated at the beginning of main().

real MAX_REAL_SQRT

Variable

The square root of MAX_REAL. MAX_REAL_SQRT is a variable, but it should be used like a constant. In other words, users should never reset its value. It can't be declared const because its value is calculated using the sqrt() function, which can't be done before the entry point of the program, i.e., main(). Therefore, the value of MAX_REAL_SQRT is set after MAX_REAL is calculated, at the beginning of main(). MAX_REAL_SQRT is used in Point::magnitude() (see Vector Operations). The magnitude of a Point is found by using the formula \sqrtx^2 + y^2 + z^2. x, y, and z are all tested against MAX_REAL_SQRT to ensure that x^2, y^2, and z^2 will all be less than or equal to MAX_REAL before trying to calculate them. Metafont implements an operation called Pythagorean addition, notated as "++"which can be used to calculate distances without first squaring and then taking square roots:27 a++b == \sqrt(a^2 + b^2) and a++b++c == \sqrt(a^2 + b^2 + c^2). This makes it possible to calculate distances for greater values of a, b, and c, that would otherwise cause floating point errors. Metafont also implements the inverse operation Pythagorean subtraction, notated as "+-+": a+-+b == \sqrt(a^2 - b^2). Unfortunately, 3DLDF implements neither Pythagorean addition nor subtraction as yet, but it's on my list of "things to do".

Dynamic Allocation of Shapes

template <class C> C* create_new (const C* arg)	Template function
template <class C> C* create_new (const C& arg)	Template function

These functions dynamically allocate an object derived from Shape on the free store, returning a pointer to the type of the Shape and setting on_free_store to true. If a non-zero pointer or a reference is passed to create_new(), the new object will be a copy of arg. It is not possible to instantiate more than one specialization of create_new() that takes no argument, because calls to these functions would be ambiguous. If the new object is not meant to be a copy of an existing one, 0 must be passed to create_new() as its argument. create_new is called like this: Point* p = create_new<Point>(0); p->show("*p:"); -| *p: (0, 0, 0) Color c(.3, .5, .25); Color* d = create_new<Color>(c); d->show("*d:"); -| *d: name == use_name == 0 red_part == 0.3 green_part == 0.5 blue_part == 0.25 Point a0(3, 2.5, 6); Point a1(10, 11, 14); Path q(a0, a1); Path* r = create_new<Path>(&q); r->show("*r:"); -| *r: points.size() == 2 connectors.size() == 1 (3, 2.5, 6) -- (10, 11, 14); Specializations of this template function are currently declared for Color, Point, Path, Reg_Polygon, Rectangle, Ellipse, Circle, Solid, and Cuboid.

System Information

The functions described in this chapter are all declared in the namespace System. They are for finding out information about the system on which 3DLDF is being run. They are declared and defined in pspglb.web, except for the template function get_second_largest(), which is declared and defined in gsltmplt.web.

There are two reasons for this. The first is that template definitions must be available in the compilation units where specializations are instantiated. I therefore write the template definition of get_second_largest() to gsltmplt.h, so it can be included by the CWEB files that need it, currently main.web only. If I wrote it to pspglb.h, it would be included by all of the CWEB files except for loader.web, causing unnecessarily bloated object code.

The other reason is because of the way way 3DLDF is built using Automake and make. I originally tried to define get_second_largest() in pspglb.web and wrote the definition to gsltmplt.cc, which is no problem with CWEB. However, I was unable to express the dependencies among the CWEB, C++ , and object files in such a way that 3DLDF was built properly.

Therefore all template functions will be put into files either by themselves, or in small groups.

• Endianness:
• Register Width:
• Get Second Largest Real:

Endianness

signed short get_endianness ([const bool verbose = false])

Function

Returns the following values: 0 if the processor is little-endian. 1 if the processor is big-endian. -1 if the endianness cannot be determined. It is called by is_little_endian() and is_big_endian(). If verbose is true, messages are printed to standard output. This function has been adapted from Harbison, Samuel P., and Guy L. Steele Jr. C, A Reference Manual, pp. 163-164. This book has the clearest explanation of endianness that I've found so far. This is the C++ code: signed short System::get_endianness(const bool verbose) { union { long Long; char Char[sizeof(long)]; } u; u.Long = 1; if (u.Char[0] == 1) { if (verbose) cout << "Processor is little-endian." << endl << endl << flush; return 0; } else if (u.Char[sizeof(long) - 1] == 1) { if (verbose) cout << "Processor is big-endian." << endl << endl << flush; return 1; } else { cerr << "ERROR! In System::get_endianness():\n" << "Can't determine endianness. Returning -1" << endl << endl << flush; return -1; } }

bool is_big_endian ([const bool verbose = false])

Function

Returns true if the processor is big-endian, otherwise false. If verbose is true, messages are printed to standard output.

bool is_little_endian ([const bool verbose = false])

Function

Returns true if the processor is little-endian, otherwise false. If verbose is true, messages are printed to standard output.

Register Width

unsigned short get_register_width (void)

Function

Returns the register width of the CPU of the system on which 3DLDF is being run. This will normally be either 32 or 64 bits. This is the C++ code: return (sizeof(void*) * CHAR_BIT); This assumes that an address will be the same size as the processor's registers, and that CHAR_BIT will be the number of bits in a byte. These are reasonable assumptions that apply to all architectures I know about. This function is called by is_32_bit() and is_64_bit().

bool is_32_bit (void)

Function

Returns true if the CPU of the system on which 3DLDF is being run has a register width of 32 bits, otherwise false.

bool is_64_bit (void)

Function

Returns true if the CPU of the system on which 3DLDF is being run has a register width of 64 bits, otherwise false.

Get Second Largest Real

template <class Real> Real get_second_largest (Real MAX_VAL, [bool verbose = false])

Template function

float get_second_largest (float, bool)	Template specialization

double get_second_largest (double, bool)	Template specialization

get_second_largest returns the second largest floating point number of the type specified the template paramater Real. If verbose is true, messages are printed to standard output. This function is used for setting the value of MAX_REAL. See Global Constants and Variables. get_second_largest depends on there being an unsigned integer type with the same length as Real. This should always be the case for float and double, but may not be long double. MAX_VAL should be the largest number of type Real on a given architecture. The GNU C++ compiler GCC 3.3 does not currently supply the numeric_limits template, so it is necessary to pass one of the macros FLT_MAX or DBL_MAX explicitly, depending on which specialization you use28. When and if GCC supplies the numeric_limits template, I will eliminate the MAX_REAL argument.

Color Reference

Class Color is defined in colors.web.

• Color Data Members:
• Color Constructors and Setting Functions:
• Color Operators:
• Modifying Colors:
• Showing Colors:
• Querying Colors:
• Defining and Initializing Colors:
• Namespace Colors:

Data Members

string name

Variable

The name of the Color.

bool use_name

Variable

If true, name is written to out_stream when the Color is used for drawing or filling. Otherwise, the RGB (red-green-blue) values are written to out_stream.

bool on_free_store

Variable

true, if the Color has been created by create_new<Color>(), which allocates memory for the Color on the free store. Otherwise false. Colors should only ever be dynamically allocated by using create_new<Color>(). See Color Reference;;Constructors and Setting Functions.

real red_part	Variable
real green_part	Variable
real blue_part	Variable

The RGB (red-green-blue) values of the Color. A real value r is valid for these variables if and only if 0 <= r <= 1.

Constructors and Setting Functions

void Color (void)

Default constructor

Creates a Color and initializes its red_part, green_part, and blue_part to 0. use_name and on_free_store are set to false.

void Color (const Color& c, [const string n = "", [const bool u = true]])

Copy constructor

Creates a Color and makes it a copy of c. If n is not the empty string and u is true, use_name is set to true. Otherwise, its set to false.

void Color (const string n, const unsigned short r, const unsigned short g, const unsigned short b, [const bool u = true])

Constructor

Creates a Color with name n. Its red_part, green_part, and blue_part are set to r/255.0, g/255.0, and b/255.0, respectively. use_name is set to u.

void set (const string n, const unsigned short r, const unsigned short g, const unsigned short b, [const bool u = false])

Setting function

Corresponds to the constructor above, except that u is false by default.

void Color (const real r, const real g, const real b)

Constructor

Creates an unnamed Color using the real values r, g, and b for its red_part, green_part, and blue_part, respectively.

void set (const real r, const real g, const real b)

Setting function

Corresponds to the constructor above.

Color* create_new<Color> (const Color* c)

Template specializations

Color* create_new<Color> (const Color& c)

Pseudo-constructors for dynamic allocation of Colors. They create a Color on the free store and allocate memory for it using new(Color). They return a pointer to the new Color. If c is a non-zero pointer or a reference, the new Color will be a copy of c. If the new object is not meant to be a copy of an existing one, 0 must be passed to create_new<Color>() as its argument. See Dynamic Allocation of Shapes, for more information. This function is used in the drawing and filling functions for Path and Solid. Point::drawdot() should be changed to use it too, but I haven't gotten around to doing this yet.

Operators

void operator= (const Color& c)

Assignment operator

Sets name to the empty string, use_name to false, and red_part, green_part, and blue_part to c.red_part, c.green_part, and c.blue_part, respectively.

bool operator== (const Color& c)

const operator

Equality operator. Returns true, if the red_parts, green_parts, and blue_parts of *this and c are equal, otherwise false. The names and use_names are not compared.

bool operator!= (const Color& c)

const operator

Inequality operator. Returns false, if the red_parts, green_parts, and blue_parts of *this and c are equal, otherwise true. The names and use_names are not compared.

ostream& operator<< (ostream& o, const Color& c)

Non-member function

Output operator. Writes the MetaPost code for the Color to out_stream when a Picture is output. This occurs when the Color has been used as an argument to drawing or filling functions. If use_name is true, name is written to out_stream. Otherwise, "(red_part, green_part, blue_part)" is written to out_stream.

Modifying

void set_name (const string s)

Function

Sets name to s. use_name is not reset.

void set_use_name (const bool b)

Function

Sets use_name to b.

void modify (const real r, [const real g = 0, [const real b = 0]])

Function

Adds r, g, and b to red_part, green_part, and blue_part, respectively. Following the addition, if red_part, green_part, and/or blue_part is greater than 1, it is reduced to 1. If it is less than 0, it is increased to 0.

void set_red_part (const real q)	Function
void set_green_part (const real q)	Function
void set_blue_part (const real q)	Function

Let p stand for red_part, green_part, or blue_part, depending upon which function is used. If 0 <= q <= 1, p is set to q. If q < 0, p is set to 0. If q > 1, p is set to 1.

Showing

void show ([string text = ""])

const function

Prints information about the Color to standard output. If text is not the empty string, prints text on a line of its own. Otherwise, it prints "Color:". Then it prints name, use_name, red_part, green_part, and blue_part.

Querying

bool is_on_free_store (void)

const function

Returns on_free_store. This will only be true, if the Color was created by create_new<Color>(). See Color Reference; Constructors and Setting Functions.

real get_red_part ([bool decimal = false])	Inline const function
real get_green_part ([bool decimal = false])	Inline const function
real get_blue_part ([bool decimal = false])	Inline const function

These functions return the red_part, green_part, or blue_part of the Color, respectively. If decimal is false (the default), the actual real value of the "part" is returned. Otherwise, the corresponding whole number n such that 0 <= n <= 255 is returned.

bool get_use_name (void)

const function

Returns use_name.

string get_name (void)

Inline const function

Returns name.

Defining and Initializing Colors

void define_color_mp ()

const function

Writes MetaPost code to out_stream, in order to define objects of type color within MetaPost, and set their redparts, greenparts, and blueparts.

void initialize_colors (void)

Static function

Calls define_color_mp() (described above) for the Colors that are defined in namespace Colors (see Namespace Colors).

Namespace Colors.

const Color red	Constant
const Color green	Constant
const Color blue	Constant
const Color cyan	Constant
const Color yellow	Constant
const Color magenta	Constant
const Color orange_red	Constant
const Color violet_red	Constant
const Color pink	Constant
const Color green_yellow	Constant
const Color orange	Constant
const Color violet	Constant
const Color purple	Constant
const Color blue_violet	Constant
const Color yellow_green	Constant
const Color black	Constant
const Color white	Constant
const Color gray	Constant
const Color light_gray	Constant

These constant Colors can be used in drawing and filling commands.

const Color default_background

Constant

The default background color. Equal to white per default.

const Color* background_color

Pointer

Points to default_background by default.

const Color* default_color

Pointer

Points to black by default.

const Color* help_color

Pointer

Points to green by default.

The following vectors of pointers to Color can be used in the drawing and filling functions for Solid (see Solid Reference; Drawing and Filling).

const vector <const Color*> default_color_vector

Vector

Contains one pointer, namely default_color.

const vector <const Color*> help_color_vector

Vector

Contains one pointer, namely help_color.

const vector <const Color*> background_color_vector

Vector

Contains one pointer, namely background_color.

Input and Output

• I/O Global Variables:
• I/O Functions:

Global Variables

ifstream in_stream

Variable

Intended for inputting files of input code. However, 3DLDF does not currently have a routine for reading input code. in_stream is currently attached to the file ldfinput.ldf by initialize_io() (see I/O Functions). in_stream is read in character-by-character in main(), however this serves no useful purpose as yet.

ofstream out_stream

Variable

Used for writing the file of MetaPost code, which is 3DLDF's output. Currently attached to the file subpersp.mp by initialize_io() (see I/O Functions).

ofstream tex_stream

Variable

TeX code can be written to a file through tex_stream, if desired. 3DLDF makes no use of it itself. Currently attached to subpersp.tex by initialize_io() (see I/O Functions).

I/O Functions

void initialize_io (string in_stream_name, string out_stream_name, string tex_stream_name, char* program_name)

Function

Opens files with names specified by the first three arguments, and attaches them to the file streams in_stream, out_stream, and tex_stream, respectively. Comments are written at the beginning of the files, containing their names, a datestamp, and the name of the program used to generate them.

void write_footers (void)

Function

Writes code at the end of the files attached to in_stream, out_stream, and tex_stream, before the streams are closed. Currently, they write comments containing local variable lists for use in Emacs.

void beginfig (unsigned short i)

Inline function

Writes "beginfig(i)" to out_stream.

void endfig ([unsigned short i = 0])

Inline function

Writes "endfig()" to out_stream. The argument i is "syntactic sugar"; it's ignored by endfig(), but may help the user keep track of what figure is being ended.

Shape Reference

Class Shape is defined in shapes.web.

Shape is an abstract class, which means that all of its member functions are pure virtual functions, and that it's only used as a base class, i.e., no objects of type Shape may be declared.

All of the "drawable" types in 3DLDF, Point, Path, Ellipse, etc., are derived from Shape.

Deriving all of the drawable types from Shape makes it possible to handle objects of different types in the same way. This is especially important in the Picture functions, where objects of various types (but all derived from Shape) are accessed through pointers to Shape. See Picture Reference.

• Shape Data Members:
• Shape Operators:
• Copying Shapes:
• Modifying Shapes:
• Affine Transformations for Shapes:
• Applying Transformations to Shapes:
• Clearing Shapes:
• Querying Shapes:
• Showing Shapes:
• Outputting Shapes:

Data Members

signed short DRAWDOT	Protected static constants
signed short DRAW
signed short FILL
signed short FILLDRAW
signed short UNDRAWDOT
signed short UNDRAW
signed short UNFILL
signed short UNFILLDRAW

Values used in the output() functions of the classes derived from Shape. For example, in Path, if the data member fill_draw_value = DRAW, then the MetaPost command draw is written to out_stream when that Path is output.

Operators

Transform operator*= (const Transform& t)

Pure virtual function

Copying

Shape* get_copy (void)

const pure virtual function

Copies an object, allocating memory on the free store for the copy, and returns a pointer to Shape for accessing the copy. Used in the drawing and filling functions for copying the Shape, and putting a pointer to the copy onto the vector<Shape*> shapes of the Picture.

Modifying

bool set_on_free_store (bool b = true)

Pure virtual function

Sets the data member on_free_store to b. All classes derived from Shape must therefore also have a data member on_free_store. This function is used in the template function create_new<type>. See Dynamic Allocation of Shapes, for more information.

Affine Transformations

Transform rotate (const real x, const real y, const real z)	Pure virtual functions
Transform scale (real x, real y, real z)
Transform shear (real xy, real xz, real yx, real yz, real zx, real zy)
Transform shift (real x, real y, real z)
Transform rotate (const Point& p0, const Point& p1, const real r)

See Point Reference; Affine Transformations.

Applying Transformations

void apply_transform (void)

Pure virtual function

Applies the Transform stored in the transform data member of the Points belonging to the Shape to their world_coordinates. The transforms are subsequently reset to the identity Transform.

Clearing

void clear (void)

Pure virtual function

The precise definition of this function will depend on the nature of the derived class. In general, it will call the destructor on dynamically allocated objects belonging to the Shape, and deallocate the memory they occupied.

Querying

bool is_on_free_store (void)

const pure virtual function

Returns true if the object was allocated on the free store, otherwise false.

Showing

void show ([string text = "", [char coords = 'w', [const bool do_persp = true, [const bool do_apply = true, [Focus* f = 0, [const unsigned short proj = 0, [const real factor = 1]]]]]]])

const pure virtual function

Prints information about an object to standard output. See the descriptions of show() for the classes derived from Shape for more information.

Outputting

void output (void)

Pure virtual function

Called by Picture::output() for writing MetaPost code to out_stream for a Shape pointed to by a pointer on the vector<Shape*> shapes belonging to the Picture. Such a Shape will have been created by a drawing or filling function.

vector<Shape*> extract (const Focus& f, const unsigned short proj, real factor)

Pure virtual function

Called in Picture::output(). It determines whether a Shape can be output. If it can, and an output() function for the type of the Shape exists, a vector<Shape*> containing a pointer to the Shape is returned. On the other hand, it is possible to define a type derived from Shape, without an output() function of its own, and not derived from a type that has one. It may then consist of one or more objects of types that do have output() functions. In this case, the vector<Shape*> returned by extract() will contain pointers to all of these subsidiary Shapes, and Picture::output() will treat them as independent objects. In particular, if any one of them cannot be projected using the arguments passed to Picture::output(), this will have no effect on whether the others are outputted or not. Currently, there are no Shapes without an output() function, either belonging to the class, or inherited. However, it's useful to be able to define Shapes in this way, so that they can be tested without having to define an output() function first.

bool set_extremes (void)

Pure virtual function

Sets the values of projective_extremes for the Shape. This is needed in Picture::output() for determining the order in which objects are output.

real get_minimum_z (void)	const pure virtual functions
real get_maximum_z (void)
real get_mean_z (void)

These functions return the minimum, maximum, and mean z-value respectively of the projected Points belonging to the Shape, i.e., from projective_extremes. The values for the Shapes on the Picture are used for determining the order in which they are output

const valarray<real> get_extremes (void)

const pure virtual function

Returns projective_extremes.

void suppress_output (void)

Pure virtual function

Sets do_output to false. This function is called in Picture::output(), if a Shape on a Picture cannot be output using the arguments passed to Picture::output().

void unsuppress_output (void)

Pure virtual function

Sets do_output to true. Called in Picture::output() after output() is called on the Shapes. This way, output of Shapes that couldn't be output when Picture::output() was called with a particular set of arguments won't necessarily be suppressed when Picture::output() is called again with different arguments.

f

Transform Reference

Class Transform is defined in transfor.web. Point is a friend of Transform.

• Transform Data Members:
• Transform Global Variables and Constants:
• Transform Constructors :
• Transform Operators:
• Matrix Inversion:
• Setting Values Transforms:
• Querying Transforms:
• Returning Information for Transforms:
• Showing Transforms:
• Affine Transformations for Transforms:
• Alignment with an Axis for Transforms:
• Resetting Transforms:
• Cleaning Transforms:

Data Members

Matrix matrix

Private variable

A 4 X 4 matrix of real representing the actual transformation matrix.

Global Variables and Constants

Transform user_transform

Variable

Currently has no function. It is intended to be used for transforming the coordinates of Points between the world coordinate system (WCS) and a user coordinate system (UCS), when routines for managing user coordinate systems are implemented.

const Transform INVALID_TRANSFORM

Constant

Every member of matrix in INVALID_TRANSFORM is equal to INVALID_REAL.

const Transform IDENTITY_TRANSFORM

Constant

Homogeneous coordinates and Transforms are unchanged by multiplication with IDENTITY_TRANSFORM. matrix is an identity matrix: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 See Transforms.

Constructors

void Transform (void)

Default constructor

Creates a Transform containing the identity matrix.

void Transform (real r)

Constructor

Creates a Transform and sets all of the elements of matrix to r. Currently, this constructor is never used, but who knows? Maybe someday it will be useful for something.

void Transform (real r0_0, real r0_1, real r2, real r0_2, real r0_3, real r1_0, real r1_1, real r1_2, real r1_3, real r2_0, real r2_1, real r2_2, real r2_3, real r3_0, real r3_1, real r3_2, real r3_3)

Constructor

Each of the sixteen real arguments is assigned to the corresponding element of matrix: matrix[0][0] = r0_0, matrix[0][1] = r0_1, etc. Useful for specifying a transformation matrix completely.

Operators

Transform operator= (const Transform& t)

Assignment operator

Sets *this to t and returns t. Returning *this would, of course, have exactly the same effect.

real operator*= (real r)

Operator

Multiplication with assignment by a scalar. This operator multiplies each element E of matrix by the scalar r. The return value is r. This makes it possible to chain invocations of this function: For a_x, b_x, c_x, ..., p_x in R , x in N Transform T0(a_0, b_0, c_0, d_0, e_0, f_0, g_0, h_0, i_0, j_0, k_0 l_0, m_0, n_0, o_0, p_0); Transform T1(a_1, b_1, c_1, d_1, e_1, f_1, g_1, h_1, i_1, j_1, k_1 l_1, m_1, n_1, o_1, p_1); Transform T2(a_2, b_2, c_2, d_2, e_2, f_2, g_2, h_2, i_2, j_2, k_2 l_2, m_2, n_2, o_2, p_2); real r = 5; Let M_0, M_1, and M_2 stand for T0.matrix, T1.matrix, and T2.matrix respectively: M_0 = a_0 b_0 c_0 d_0 e_0 f_0 g_0 h_0 i_0 j_0 k_0 l_0 m_0 m_0 o_0 p_0 M_1 = a_1 b_1 c_1 d_1 e_1 f_1 g_1 h_1 i_1 j_1 k_1 l_1 m_1 m_1 o_1 p_1 M_2 = a_2 b_2 c_2 d_2 e_2 f_2 g_2 h_2 i_2 j_2 k_2 l_2 m_2 m_2 o_2 p_2 T0 *= T1 *= T2 *= r; Now, M_0 = 5a_0 5b_0 5c_0 5d_0 5e_0 5f_0 5g_0 5h_0 5i_0 5j_0 5k_0 5l_0 5m_0 5m_0 5o_0 5p_0 M_1 = 5a_1 5b_1 5c_1 5d_1 5e_1 5f_1 5g_1 5h_1 5i_1 5j_1 5k_1 5l_1 5m_1 5m_1 5o_1 5p_1 M_2 = 5a_2 5b_2 5c_2 5d_2 5e_2 5f_2 5g_2 5h_2 5i_2 5j_2 5k_2 5l_2 5m_2 5m_2 5o_2 5p_2 This function is not currently used anywhere, but it may turn out to be useful for something.

Transform operator* (const real r)

const operator

Multiplication of a Transform by a scalar without assignment. The return value is a Transform A. If this.matrix has elements E_T, then A.matrix has elements E_A such that E_A = r * E_T for all E.

Transform operator*= (const Transform& t)

Operator

Performs matrix multiplication on matrix and t.matrix. The result is assigned to matrix. t is returned, not *this! This makes it possible to chain invocations of this function: Transform a; a.shift(1, 1, 1); Transform b; b.rotate(0, 90); Transform c; c.shear(5, 4); Transform d; d.scale(3, 4, 5); Let a_m, b_m, and c_m stand for a.matrix, b.matrix, c.matrix, and d.matrix respectively: a_m = 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 b_m = 0.5 0.5 0.707 0 0.146 0.854 -0.5 0 -0.854 0.146 0.5 0 0 0 0 1 c_m = 1 12 14 0 10 1 15 0 11 13 1 0 0 0 0 1 d_m = 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 1 a *= b *= c *= d; a, b, and c are transformed by d, which remains unchanged. Now, a_m = 3 0 0 0 0 4 0 0 0 0 5 0 3 4 5 1 b_m = 1.5 2 3.54 0 -0.439 3.41 -2.5 0 -2.56 0.586 2.5 0 0 0 0 1 c_m = 3 48 70 0 30 4 75 0 33 52 5 0 0 0 0 1 d_m is unchanged.

Transform operator* (const Transform t)

const operator

Multiplication of a Transform by another Transform without assignment. The return value is a Transform whose matrix contains values that are the result of the matrix multiplication of matrix and t.matrix.

Matrix Inversion

Transform inverse (void)	const function
Transform inverse ([bool assign = false])	Function

Returns a Transform T with a T.matrix that is the inverse of matrix. If assign==true, then matrix is set to its inverse. In the const version, matrix remains unchanged. The second should only ever be called with true as its assign argument. If you're tempted call inverse(false), you might as well just leave out the argument, which issues a warning message, and calls the const version.

Setting Values

void set_element (const unsigned short row, const unsigned short col, real r)

Function

Sets the element of matrix indicated by the arguments to r. Transform t; t.set_element(0, 2, -3.45569); t.show("t:"); -| t: 1 0 -3.46 0 0 1 0 0 0 0 1 0 0 0 0 1

Querying

bool is_identity (void)

Function

Returns true if *this is the identity Transform, otherwise false. This function has both a const and a non-const version. In the non-const version, clean() is called on *this before comparing the elements of matrix with 1 (for the main diagonal) and 0 (for the other elements). In the const version, *this is copied, clean() is called on the copy, and the elements of the copy's matrix are compared with 0 and 1.

real get_element (const unsigned short row, const unsigned short col)

const function

Returns the value stored in the element of matrix indicated by the arguments. Transform t; t.shift(1, 2, 3); t.scale(2.5, -1.2, 4); t.rotate(30, 15, 60); t.show("t:"); -| t: 1.21 2.09 0.647 0 0.822 -0.654 0.58 0 -2.18 0.224 3.35 0 -3.69 1.45 11.8 1 cout << t.get_element(2, 1); -| 0.224

Returning Information

real epsilon (void)

Static function

Returns the positive real value of smallest magnitude \epsilon which an element of a Transform should contain. An element of a Transform may also contain -\epsilon. The value \epsilon is used for in the function clean() (see Transform Reference; Cleaning). It will also be used for comparing Transforms, when I've added the equality operator Transform::operator==(). epsilon() returns different values, depending on whether real is float or double: If real is float (the default), epsilon() returns 0.00001. If real is double, it returns 0.000000001. Please note: I haven't tested whether 0.000000001 is a good value yet, so users should be aware of this if they set real to double!29 The way to test this is to transform two different Transforms t_1 and t_2 using different rotations in such a way that the end result should be the same for both Transforms. Let \epsilon stand for the value returned by epsilon(). If for all sets of corresponding elements E_1 and E_2 of t_1 and t_2, ||E_1| - |E_2|| \le \epsilon, then \epsilon is a good value. It will be easier to test this when I've added Transform::operator==(). Rotation causes a significant loss of precision to due to the use of the sin() and cos() functions. Therefore, neither Transform::epsilon() nor Point::epsilon() (see Point Reference; Returning Information) can be as small as I'd like them to be. If they are two small, operations that test for equality of Transforms and Points will return false for objects that should be equal.

Showing

void show ([string text = ""])

const function

If the optional argument text is used, and is not the empty string (""), text is printed on a line of its own to the standard output first. Otherwise, "Transform:" is printed on a line of its own to the standard output. Then, the elements of matrix are printed to standard output. Transform t; t.show("t:"); -| t: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 t.scale(1, 2, 3); t.shift(1, 1, 1); t.rotate(90, 90, 90); t.show("t:"); -| t: 0 0 1 0 0 2 0 0 -3 0 0 0 -1 1 1 1

Affine Transformations

The affine transformation functions use their arguments to create a new Transform t (local to the function) representing the appropriate transformation. Then, *this is multiplied by t and t is returned. Returning t instead of *this makes it possible to put the affine transformation function at the end of a chain of invocations of Transform::operator*=():

     Transform t0, t1, t2, t3;
     ...
     t0 *= t1 *= t2 *= t3.scale(2, 3.5, 9);
     

t0, t1, and t2 are all multiplied by the Transform with

     matrix =
     2   0    0  0
     0   3.5  0  0
     0   0    9  0
     0   0    0  1
     

Transform scale (real x, [real y = 1, [real z = 1]])

Function

Creates a Transform t representing the scaling operation locally, multiplies *this by t, and returns t. A Transform representing scaling only, when applied to a Point p, will cause its x-coordinate to be multiplied by x, its y-coordinate to be multiplied by y, and its z-coordinate to be multiplied by z. Transform t; t.scale(x, y, z); => t.matrix = x 0 0 0 0 y 0 0 0 0 z 0 0 0 0 1 Transform t; t.scale(12.5, 20, 1.3); t.show("t:"); -| t: 12.5 0 0 0 0 20 0 0 0 0 1.3 0 0 0 0 1

Transform shear (real xy, [real xz = 0, [real yx = 0, [real yz = 0, [real zx = 0, [real zy = 0]]]]])

Function

Creates a Transform t representing the shearing operation locally, multiplies *this by t, and returns t. When applied to a Point, shearing causes each coordinate to be modified according to the values of the other coordinates and the arguments to shear: Point p(x,y,z); Transform t; t.shear(a, b, c, d, e, f); p *= t; => p = ((x + ay + bz), (y + cx + dz), (z + ex + fy)) Transform t; t.shear(2, 3, 4, 5, 6, 7); t.show("t:"); -| t: 1 4 6 0 2 1 7 0 3 5 1 0 0 0 0 1

Transform shift (real x, [real y = 0, [real z = 0]])	Function
Transform shift (const Point& p)	Function

These functions create a Transform t representing the shifting operation locally, multiplies *this by t, and returns t. The version with the argument const Point& p passes the updated x, y, and z-coordinates of p (from world_coordinates) to the version with three real arguments. When a Transform representing a single shifting operation only is applied to a Point, the x, y, and z arguments are added to the corresponding coordinates of the Point: Point p(x,y,z); Transform t; t.shift(a, b, c); p *= t; => p = (x + a, y + b, z + c)

Transform shift_times (real x, [real y = 1, [real z = 1]])

Function

Multiplies the corresponding elements of matrix by the real arguments, i.e., matrix[3][0] is multiplied by x, matrix[3][1] is multiplied by y, and matrix[3][2] is multiplied by z. Returns *this. Ordinary shifting is additive, so a special function is needed to multiply the elements of matrix responsible for shifting. The effect of shift_times() is to modify a Transform representing a shifting operation such that the direction of the shift is maintained, while changing the distance. If the Transform represents other operations in addition to shifting, e.g., scaling and/or shearing, the effect of shift_times() may be unpredictable.30 Transform t; t.shift(1, 2, 3); => t.matrix = 1 0 0 0 0 1 0 0 0 0 1 0 1 2 3 1 t.shift_times(2, 2, 2); => t.matrix = 1 0 0 0 0 1 0 0 0 0 1 0 2 4 6 1 Rectangle r[4]; r[0].set(origin, 1, 1, 90); r[3] = r[2] = r[1] = r[0]; Transform t; t.shift(1.5, 1.5); r[0] *= t; r[0].draw(); t.shift_times(1.5, 1.5); r[1] *= t; r[1].draw(); t.shift_times(1.5, 1.5); r[2] *= t; r[2].draw(); t.shift_times(1.5, 1.5); r[3] *= t; r[3].draw(); Fig. 73. Cuboid c(origin, 1, 1, 1); c.draw(); Transform t; t.rotate(30, 30, 30); t.shift(1, 0, 1); c *= t; c.draw(); t.shift_times(1.5, 0, 1.5); c *= t; c.draw(); t.shift_times(1.5, 0, 1.5); c *= t; c.draw(); t.shift_times(1.5, 0, 1.5); c *= t; c.draw(); t.shift_times(1.5, 0, 1.5); c *= t; c.draw(); Fig. 74.

Transform rotate (real x, [real y = 0, [real z = 0]])

Function

Rotation around the main axes. Creates a Transform t representing the rotation, multiplies *this by t, and returns t.

Transform rotate (Point p0, Point p1, [const real angle = 180])

Function

Rotation around an arbitrary axis. The Point arguments represent the end points of the axis, and angle is the angle of rotation. Since 180 degrees rotation is needed so often, 180 is the default for angle.

Transform rotate (const Path& p, [const real angle = 180])

Function

Rotation around an arbitrary axis. Path argument. The Path p must be linear, i.e., p.is_linear() must return true. See Path Reference; Querying.

Alignment with an Axis

Transform align_with_axis (Point p0, Point p1, [char axis = 'z'])

Function

Returns the Transform that would align the line through p0 and p1 with the major axis denoted by the axis argument. The default is the z-axis. This function is used in the functions that find intersections. Point P0(1, 1, 1); Point P1(2, 3, 4); P0.draw(P1); P0.dotlabel("$P_0$"); P1.dotlabel("$P_1$"); Transform t; t.align_with_axis(P0, P1, 'z'); P0 *= P1 *= t; t.show("t:"); -| t: 0.949 -0.169 0.267 0 0 0.845 0.535 0 -0.316 -0.507 0.802 0 -0.632 -0.169 -1.6 1 P0.show("P0:"); -| P0: (0, 0, 0) P1.show("P1:"); -| P1: (0, 0, 3.74) The following example shows how align_with_axis() can be used for putting plane figures into a major plane. default_focus.set(2, 3, -10, 2, 3, 10, 10); Circle c(origin, 3, 75, 25, 6); c.shift(2, 3); c.draw(); Point n = c.get_normal(); n.shift(c.get_center()); Transform t; t.align_with_axis(c.get_center(), n, 'y'); t.show("t:"); -| t: 0.686 0.379 -0.621 0 0.543 0.3 0.784 0 0.483 -0.875 0 0 -3 -1.66 -1.11 1 n *= c *= t; c.draw(); c.show("c:"); -| c: fill_draw_value == 0 (1.31, 0, -0.728) .. (1.49, 0, -0.171) .. (1.44, 0, 0.413) .. (1.17, 0, 0.933) .. (0.728, 0, 1.31) .. (0.171, 0, 1.49) .. (-0.413, 0, 1.44) .. (-0.933, 0, 1.17) .. (-1.31, 0, 0.728) .. (-1.49, 0, 0.171) .. (-1.44, 0, -0.413) .. (-1.17, 0, -0.933) .. (-0.728, 0, -1.31) .. (-0.171, 0, -1.49) .. (0.413, 0, -1.44) .. (0.933, 0, -1.17) .. cycle; n.show("n:"); -| n: (0, 1, 0) Fig. 75.

Resetting

void reset (void)

Function

Resets matrix to the identity matrix.

Cleaning

void clean (void)

Function

Sets elements in matrix whose absolute values are < epsilon() to 0.

Label Reference

Class Label is defined in pictures.web. Point and Picture are friends of Label.

Labels can be included in drawings by using the label() and dotlabel() functions, which are currently defined for the classes Point and Path, and the classes derived from them. See Point Reference; Labelling, and See Path Reference; Labelling. They are currently not defined for Solid, and its derived classes. I plan to add them for Solid soon.

Users will normally never need to declare objects of type Label, access its data members or call its member functions directly.

When label() or dotlabel() is invoked, one or more Labels is allocated dynamically and pointers to the new Labels are placed onto the vector<Label*> labels of a Picture: current_picture, by default. There are no explicitly defined constructors for Label, nor is it intended that Labels ever be created in any way other than through label() or dotlabel(). When a Picture is copied, the Labels are copied, too, and when a Picture is cleared (using Picture::clear()) or destroyed, the Labels are deallocated and destroyed.

• Label Data Members:
• Copying Labels:
• Outputting Labels:

Data Members

Point* pt

Private variable

A pointer to the Point representing the location of the Label.

bool dot

Private variable

true if the label should be dotted, otherwise false. dot will be false, if the label was generated by a call to label() with the "dot" argument false (the default), true, if the label was generated by a call to dotlabel(), or to label() with the "dot" argument true.

string text

Private variable

The text of the label. text is always put between "btex" and "etex" in the MetaPost code, so that TeX will be used to format the labels. In particular, this means that TeX's math mode can be used. However, double backslashes must be used instead of single backslashes, in order that single backslashes be written to out_stream. Point P(1, 1, 2); origin.drawarrow(P); P.label("$\\vec{P}$"); Fig. 76.

string position

Private variable

The position of the text with respect to *pt. Valid values are as in MetaPost: "top", "bot" (bottom), "lft" (left), "rt" (right), "ulft" (upper left), "llft" (lower left), "urt" (upper right), "lrt" (lower right).

bool DO_LABELS

Public static variable

Enables or disables creation of Labels. If true, label and dotlabel() cause Labels to be created and put onto a Picture. If false, they are not. Note that it is also possible to suppress output of existing Labels when outputting a Picture.

Copying

Label* get_copy (void)

const Function

Creates a copy of the Label and returns a pointer to the copy. Called in Picture::operator=() and Picture::operator+=() where Pictures are copied. Users should never need to call this function directly. See Picture Reference; Operators. This function dynamically allocates a new Label and a new Point within the Label, and copies the strings from *this to the new Label. The standard library functions for strings take care of the allocation for the string members of Label.

Outputting

void output (const Focus& f, const unsigned short proj, real factor, const Transform& t)

Function

Writes MetaPost code for the labels to out_stream. It is called in Picture::output() (see Picture Reference; Outputting). Users should never need to call this function directly. When Picture::output() is invoked, the MetaPost code for Labels is written to out_stream after the code for the drawing and filling commands. This prevents the Labels from being covered up. However, they can still be covered by other Labels, or by Shapes or Labels from subsequent invocations of Picture::output() within the same figure (see I/O Functions, for descriptions of beginfig() and endfig()).

Picture Reference

Class Picture is defined in pictures.web.

• Picture Data Members:
• Picture Global Variables:
• Picture Constructors:
• Picture Operators:
• Affine Transformations for Pictures:
• Modifying Pictures:
• Showing Pictures:
• Outputting Pictures:

Data Members

Transform transform

Private variable

Applied to the Shapes on the Picture when the latter is output. It is initialized as the identity Transform, and can be modified by the transformation functions, by Picture::operator*=(const Transform&) (see Picture Reference; Operators), and by Picture::set_transform() (see Picture Reference; Modifying).

vector<Shape*> shapes

Private variable

Contains pointers to the Shapes on the Picture. When a drawing or filling function is invoked for a Shape, a copy is dynamically allocated and a pointer to the copy is placed onto shapes.

vector<Label*> labels

Private variable

Contains pointers to the Labels on the Picture. When a Point is labelled, either directly or through a call to label() or dotlabel() for another type of Shape31, a Label is dynamically allocated, the Point is copied to *Label::pt, and a pointer to the Label is placed onto labels.

bool do_labels

Private variable

Used for enabling or disabling output of Labels when outputting a Picture. The default value is true. It is set to false by using suppress_labels() and can be reset to true by using unsuppress_labels(). See Picture Reference; Output Functions. Often, when a Picture is copied, transformed, and output again in a single figure, it's undesirable to have the Labels output again in their new positions. To avoid this, use suppress_labels() after outputting the Picture the first time.

Global Variables

Variable Picture current_picture

Variable

The Picture used as the default by the drawing and filling functions.

Constructors

void Picture (void)

Default constructor

Creates an empty Picture.

void Picture (const Picture& p)

Copy constructor

Creates a copy of Picture p. Circle c(origin, 3); c.draw(); current_picture.output(Projections::PARALLEL_X_Z); Picture new_picture(current_picture); new_picture.shift(2); new_picture.output(Projections::PARALLEL_X_Z); Fig. 77.

Operators

void operator= (const Picture& p)

Assignment operator

Makes *this a copy of p, destroying the old contents of *this.

void operator+= (const Picture& p)

Operator

Adds the contents of p to *this. p remains unchanged.