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Graphics Algorithms FAQ

This article is Copyright 2002 by Joseph O'Rourke.  It may be freely 
redistributed in its entirety provided that this copyright notice is 
not removed.

  Changed items this posting (|): 3.14
  New     items this posting (+): none

History of Changes (approx. last six months):
Changes in  1 Jul 02 posting:
  3.14: Correct GIF author info, add URL. [Thanks to Greg Roelofs.]
Changes in  1 May 02 posting:
  0.04: Errata for Watt & Watt book added. [Thanks to Jacob Marner.]
  5.14: 3D viewing revised by Ken Shoemake.
  5.23: Remove (erroneous) 3D medial axis info.
  5.25: New article on quaternions by Ken Shoemake.
  5.26: New article on camera aiming and quaternions by Ken Shoemake.
  6.01: Add (correct) 3D medial axis info.  (Thanks to Tamal Dey.)
  6.09: Plucker coordinates article revised by Ken Shoemake.
Changes in 15 Apr 02 posting:
  3.05: Scaling bitmaps revised by Ken Shoemake.
  3.09: Morphing article written by Ken Shoemake.
  6.08: Added references on random points on a sphere (Ken Shoemake).
Changes in  1 Apr 02 posting:
  1.01: 2D point rotation revised by Ken Shoemake.
  1.01: 2D segment intersection revised by Ken Shoemake.
  5.01: 3D point rotation revised by Ken Shoemake.
  0.07: Greg Ferrar's 3D rendering library no longer available.
Changes in 15 Mar 02 posting:
  2.03: Reference Dan Sunday's winding number algorithm.
  4.04: More detail on Beziers approximating a circle.
        (Thanks to William Gibbons.)
  5.22: Added NASA's "Intersect" code for intersecting triangulated
  5.23: Updated Cocone software description.
Changes in 15 Feb 02 posting:
  5.03: Noted that Sutherland-Hodgman can clip against any convex polygon.
        (Thanks to Ben Landon.)
  5.15: More links on simplifying meshes. (Thanks to Stefan Krause.)
Changes in  1 Jan 02 posting:
  2.03: Fixed link to Franklin's code. (Thanks to Keith M. Briggs.)
  5.13: Update to SWIFT++; add Terdiman's collision lib.
        (Thanks to Pierre Terdiman.)
Changes in  1 Nov 01 posting:
  6.01,02,03: Update to Qhull 3.1 release (Thanks to Brad Barber.)
Changes in 15 Sep 01 posting:
  0.04: "Radiosity: A Programmer's Perspective" out of print.
  0.05: CQUANT97 link no longer available; RADBIB info updated.
        (Thanks to Ian Ashdown for both.)
  2.01: Explained indices in more efficient formula, and restored
        Sunday's version. (Thanks to Dan Sunday.)
  4.04: Link for approximating a circle via a Bezier curve
        (Thanks to John McDonald, Jr.)
  5.10: Add in link to Jules Bloomenthal's list of papers for algorithms
        that could substitute for the marching cubes algorithm.
  5.11: Refer to 5.10. (Thanks to Eric Haines for both.)
Changes in  1 Sep 01 posting:
  2.01: Fixed indices in efficient area formula 
	(Thanks to peter@Glaze.phys.dal.ca.)
  2.03: Link to classic "Point in Polygon Strategies" article.
        (Thanks to Eric Haines.)
  5.09: Additional references for caustics (Thanks to Lars Brinkhoff.)
  5.11: New links for marching cubes patent (Thanks to John Stone.)
  5.17: Stale link notice.
  5.23: New Cocone link for surface reconstruction.
Table of Contents

0. General Information
   0.01: Charter of comp.graphics.algorithms
   0.02: Are the postings to comp.graphics.algorithms archived?
   0.03: How can I get this FAQ?
   0.04: What are some must-have books on graphics algorithms?
   0.05: Are there any online references?
   0.06: Are there other graphics related FAQs?
   0.07: Where is all the source?

1. 2D Computations: Points, Segments, Circles, Etc.
   1.01: How do I rotate a 2D point?
   1.02: How do I find the distance from a point to a line?
   1.03: How do I find intersections of 2 2D line segments?
   1.04: How do I generate a circle through three points?
   1.05: How can the smallest circle enclosing a set of points be found?
   1.06: Where can I find graph layout algorithms?

2. 2D Polygon Computations
   2.01: How do I find the area of a polygon?
   2.02: How can the centroid of a polygon be computed?
   2.03: How do I find if a point lies within a polygon?
   2.04: How do I find the intersection of two convex polygons?
   2.05: How do I do a hidden surface test (backface culling) with 2D points?
   2.06: How do I find a single point inside a simple polygon?
   2.07: How do I find the orientation of a simple polygon?
   2.08: How can I triangulate a simple polygon?
   2.09: How can I find the minimum area rectangle enclosing a set of points?

3. 2D Image/Pixel Computations
   3.01: How do I rotate a bitmap?
   3.02: How do I display a 24 bit image in 8 bits?
   3.03: How do I fill the area of an arbitrary shape?
   3.04: How do I find the 'edges' in a bitmap?
   3.05: How do I enlarge/sharpen/fuzz a bitmap?
   3.06: How do I map a texture on to a shape?
   3.07: How do I detect a 'corner' in a collection of points?
   3.08: Where do I get source to display (raster font format)?
   3.09: What is morphing/how is it done?
   3.10: How do I quickly draw a filled triangle?
   3.11: D Noise functions and turbulence in Solid texturing.
   3.12: How do I generate realistic sythetic textures?
   3.13: How do I convert between color models (RGB, HLS, CMYK, CIE etc)?
   3.14: How is "GIF" pronounced?

4. Curve Computations
   4.01: How do I generate a Bezier curve that is parallel to another Bezier?
   4.02: How do I split a Bezier at a specific value for t?
   4.03: How do I find a t value at a specific point on a Bezier?
   4.04: How do I fit a Bezier curve to a circle?

5. 3D computations
   5.01: How do I rotate a 3D point?
   5.02: What is ARCBALL and where is the source?
   5.03: How do I clip a polygon against a rectangle?
   5.04: How do I clip a polygon against another polygon?
   5.05: How do I find the intersection of a line and a plane?
   5.06: How do I determine the intersection between a ray and a triangle?
   5.07: How do I determine the intersection between a ray and a sphere?
   5.08: How do I find the intersection of a ray and a Bezier surface?
   5.09: How do I ray trace caustics?
   5.10: What is the marching cubes algorithm?
   5.11: What is the status of the patent on the "marching cubes" algorithm?
   5.12: How do I do a hidden surface test (backface culling) with 3D points?
   5.13: Where can I find algorithms for 3D collision detection?
   5.14: How do I perform basic viewing in 3D?
   5.15: How do I optimize/simplify a 3D polygon mesh?
   5.16: How can I perform volume rendering?
   5.17: Where can I get the spline description of the famous teapot etc.?
   5.18: How can the distance between two lines in space be computed?
   5.19: How can I compute the volume of a polyhedron?
   5.20: How can I decompose a polyhedron into convex pieces?
   5.21: How can the circumsphere of a tetrahedron be computed?
   5.22: How do I determine if two triangles in 3D intersect?
   5.23: How can a 3D surface be reconstructed from a collection of points?
   5.24: How can I find the smallest sphere enclosing a set of points in 3D? 
   5.25: What's the big deal with quaternions?
   5.26: How can I aim a camera in a specific direction?

6. Geometric Structures and Mathematics
   6.01: Where can I get source for Voronoi/Delaunay triangulation?
   6.02: Where do I get source for convex hull?
   6.03: Where do I get source for halfspace intersection?
   6.04: What are barycentric coordinates?
   6.05: How do I generate a random point inside a triangle?
   6.06: How do I evenly distribute N points on (tesselate) a sphere?
   6.07: What are coordinates for the vertices of an icosohedron?
   6.08: How do I generate random points on the surface of a sphere?
   6.09: What are Plucker coordinates?

7. Contributors
   7.01: How can you contribute to this FAQ?
   7.02: Contributors.  Who made this all possible.

Search e.g. for "Section 6" to find that section.
Search e.g. for "Subject 6.04" to find that item.
Section 0. General Information
Subject 0.01: Charter of comp.graphics.algorithms

    comp.graphics.algorithms is an unmoderated newsgroup intended as a forum
    for the discussion of the algorithms used in the process of generating
    computer graphics.  These algorithms may be recently proposed in
    published journals or papers, old or previously known algorithms, or
    hacks used incidental to the process of computer graphics.  The scope of
    these algorithms may range from an efficient way to multiply matrices,
    all the way to a global illumination method incorporating raytracing,
    radiosity, infinite spectrum modeling, and perhaps even mirrored balls
    and lime jello.

    It is hoped that this group will serve as a forum for programmers and
    researchers to exchange ideas and ask questions on recent papers or
    current research related to computer graphics.

    comp.graphics.algorithms is not:

     - for requests for gifs, or other pictures
     - for requests for image translator or processing software; see
            alt.binaries.pictures* FAQ  
            alt.binaries.pictures.utilities [now degenerated to pic postings]
            alt.graphics.pixutils (image format translation)
            comp.sources.misc (image viewing source code)
     - for requests for compression software; for these try:
     - specifically for game development; for this try:

Subject 0.02: Are the postings to comp.graphics.algorithms archived?

    Archives may be found at: http://www.faqs.org/

Subject 0.03: How can I get this FAQ?

    The FAQ is posted on the 1st and 15th of every month.  The easiest
    way to get it is to search back in your news reader for the most
    recent posting, with Subject: 
          comp.graphics.algorithms Frequently Asked Questions
    It is posted to comp.graphics.algorithms, and cross-posted to
    news.answers and comp.answers.  

    If you can't find it on your newsreader,
    you can look at a recent HTML version at the "official" FAQ archive site:
    The maintainer also keeps a copy of the raw ASCII, always the
    latest version, accessible via http://cs.smith.edu/~orourke/FAQ.html .

    Finally, you can ftp the FAQ from several sites, including:


    The (busy) rtfm.mit.edu site lists many alternative "mirror" sites.
    Also can reach the FAQ from http://www.geom.umn.edu/software/cglist/,
    which is worth visiting in its own right.

Subject 0.04: What are some must-have books on graphics algorithms?

    The keywords in brackets are used to refer to the books in later
    questions.  They generally refer to the first author except where
    it is necessary to resolve ambiguity or in the case of the Gems.

    Basic computer graphics, rendering algorithms,

    Computer Graphics: Principles and Practice (2nd Ed.),
    J.D. Foley, A. van Dam, S.K. Feiner, J.F. Hughes, Addison-Wesley
    1990, ISBN 0-201-12110-7;
    Computer Graphics: Principles and Practice, C version
    J.D. Foley,  A. van Dam, S.K. Feiner, J.F. Hughes, Addison-Wesley 
    ISBN: 0-201-84840-6, 1996, 1147 pp.

    Procedural Elements for Computer Graphics, Second Edition
    David F. Rogers, WCB/McGraw Hill 1998, ISBN 0-07-053548-5

    Mathematical Elements for Computer Graphics 2nd Ed.,
    David F. Rogers and J. Alan Adams, McGraw Hill 1990, ISBN

    _3D Computer Graphics, 2nd Edition_,
    Alan Watt, Addison-Wesley 1993, ISBN 0-201-63186-5

    An Introduction to Ray Tracing,
    Andrew Glassner (ed.), Academic Press 1989, ISBN 0-12-286160-4

    [Gems I]
    Graphics Gems,
    Andrew Glassner (ed.), Academic Press 1990, ISBN 0-12-286165-5
    http://www.graphicsgems.org/ for all the Gems.

    [Gems II]
    Graphics Gems II,
    James Arvo (ed.), Academic Press 1991, ISBN 0-12-64480-0

    [Gems III]
    Graphics Gems III,
    David Kirk (ed.), Academic Press 1992, ISBN 0-12-409670-0 (with
    IBM disk) or 0-12-409671-9 (with Mac disk)
    See also "AP Professional Graphics CD-ROM Library,"
    Academic Press,  ISBN 0-12-059756-X, which contains Gems I-III.

    [Gems IV]
    Graphics Gems IV,
    Paul S. Heckbert (ed.), Academic Press 1994, ISBN 0-12-336155-9
    (with IBM disk) or 0-12-336156-7 (with Mac disk)

    [Gems V]
    Graphic Gems V,
    Alan W. Paeth (ed.), Academic Press 1995, ISBN 0-12-543455-3
    (with IBM disk)

    Advanced Animation and Rendering Techniques,
    Alan Watt, Mark Watt, Addison-Wesley 1992, ISBN 0-201-54412-1
    (Unofficial) errata: http://www.rolemaker.dk/other/AART/

    An Introduction to Splines for Use in Computer Graphics and
        Geometric Modeling,
    Richard H. Bartels, John C. Beatty, Brian A. Barsky, 1987, ISBN

    Curves and Surfaces for Computer Aided Geometric Design:
    A Practical Guide, 4th Edition, Gerald E. Farin, Academic Press
    1996. ISBN 0122490541.

    The Algorithmic Beauty of Plants,
    Przemyslaw W. Prusinkiewicz, Aristid Lindenmayer, Springer-Verlag,
    1990, ISBN 0-387-97297-8, ISBN 3-540-97297-8

    Tricks of the Graphics Gurus,
    Dick Oliver, et al. (2) 3.5 PC disks included, $39.95 SAMS Publishing

    Introduction to computer graphics,
    Hearn & Baker

    Radiosity and Realistic Imange Sythesis,
    Michael F. Cohen, John R. Wallace, Academic Press Professional
    1993, ISBN 0-12-178270-0 [limited reprint 1999]

    Radiosity: A Programmer's Perspective
    Ian Ashdown, John Wiley & Sons 1994, ISBN 0-471-30444-1, 498 pp.
    Now (Sep 2001) out of print.

    Radiosity & Global Illumination
    Francois X. Sillion snd Claude Puech, Morgan Kaufmann 1994, ISBN
    1-55860-277-1, 252 pp.

    Texturing and Modeling - A Procedural Approach (2nd Ed.)
    David S. Ebert (ed.), F. Kenton Musgrave, Darwyn Peachey, Ken Perlin,
    Steven Worley, Academic Press 1998, ISBN 0-12-228730-4, Includes CD-ROM.

    Visualization Toolkit, 2nd Edition, The: An Object-Oriented Approach to
    3-D Graphics (Bk/CD) (Professional Description)
    William J. Schroeder,  Kenneth Martin, and Bill Lorensen,
    Prentice-Hall 1998, ISBN: 0-13-954694-4
    See Subject 0.07 for source.

    PC Graphics Unleashed
    Scott Anderson. SAMS Publishing, ISBN 0-672-30570-4

    Computer Graphics for Java Programmers,
    Leen Ammeraal, John Wiley 1998, ISBN 0-471-98142-7.
    Additional information at http://home.wxs.nl/~ammeraal/ .

    3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics.
    David Eberly, Morgan Kaufmann/Academic Press, 2001.

    For image processing,

    Fractal Image Compression,
    Michael F. Barnsley and Lyman P. Hurd, AK Peters, Ltd, 1993 ISBN

    Fundamentals of Image Processing,
    Anil K. Jain, Prentice-Hall 1989, ISBN 0-13-336165-9

    Digital Image Processing,
    Kenneth R. Castleman, Prentice-Hall 1996, ISBN(Cloth): 0-13-211467-4
    (Description and errata at: "http://www.phoenix.net/~castlman")

    Digital Image Processing, Second Edition,
    William K. Pratt, Wiley-Interscience 1991, ISBN 0-471-85766-1

    Digital Image Processing (3rd Ed.),
    Rafael C. Gonzalez, Paul Wintz, Addison-Wesley 1992, ISBN

    The Image Processing Handbook (3rd Ed.),
    John C. Russ, CRC Press and IEEE Press 1998, ISBN 0-8493-2532-3
    [Russ & Russ]
    The Image Processing Tool Kit v. 3.0
    Chris Russ and John Russ, Reindeer Games Inc. 1999, ISBN 1-928808-00-X

    Digital Image Warping,
    George Wolberg, IEEE Computer Society Press Monograph 1990, ISBN

    Computational geometry

    A Programmer's Geometry,
    Adrian Bowyer, John Woodwark, Butterworths 1983, 
    ISBN 0-408-01242-0 Pbk
    Out of print, but see:
    Introduction to Computing with Geometry,
    Adrian Bowyer and John Woodwark, 1993
    ISBN 1-874728-03-8.  Available in PDF:

    [Farin & Hansford]
    The Geometry Toolbox for Graphics and Modeling
    by Gerald E. Farin, Dianne Hansford
    A K Peters Ltd; ISBN: 1568810741 

    [O'Rourke (C)]
    Computational Geometry in C (2nd Ed.)
    Joseph O'Rourke, Cambridge University Press 1998, 
    ISBN 0-521-64010-5 Pbk, ISBN 0-521-64976-5 Hbk
    Additional information and code at http://cs.smith.edu/~orourke/ .

    [O'Rourke (A)]
    Art Gallery Theorems and Algorithms
    Joseph O'Rourke, Oxford University Press 1987,
    ISBN 0-19-503965-3.

    [Goodman & O'Rourke]
    Handbook of Discrete and Computational Geometry
    J. E. Goodman and J. O'Rourke, editors.
    CRC Press LLC, July 1997.
    Additional information at http://cs.smith.edu/~orourke/ .

    Applications of Spatial Data Structures:  Computer Graphics, 
    Image Processing, and GIS, 
    Hanan Samet, Addison-Wesley, Reading, MA, 1990.
    ISBN 0-201-50300-0.

    [Samet:Design & Analysis]
    The Design and Analysis of Spatial Data Structures,
    Hanan Samet, Addison-Wesley, Reading, MA, 1990.
    ISBN 0-201-50255-0.

    Geometric Modeling,
    Michael E. Mortenson, Wiley 1985, ISBN 0-471-88279-8

    Computational Geometry: An Introduction,
    Franco P. Preparata, Michael Ian Shamos, Springer-Verlag 1985,
    ISBN 0-387-96131-3

    Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
    A. Okabe and B. Boots and K. Sugihara,
    John Wiley, Chichester, England, 1992.

    Computational Geometry: Algorithms and Applications
    M. de Berg and M. van Kreveld and M. Overmars and O. Schwarzkopf
    Springer-Verlag, Berlin, 1997.

    Oriented Projective Geometry: A Framework for Geometric Computations
    Academic Press, 1991.

    Methods of Algebraic Geometry, Volume 1
    W.V.D. Hodge and D. Pedoe, Cambridge, 1994.
    ISBN 0-521-469007-4 Paperback

    [Tamassia et al 199?]
    Graph Drawing: Algorithms for the Visualization of Graphs
    Prentice Hall; ISBN: 0133016153

    Algorithms books with chapters on computational geometry

    [Cormen et al.]
    Introduction to Algorithms,
    T. H. Cormen, C. E. Leiserson, R. L. Rivest,
    The MIT Press, McGraw-Hill, 1990.

    Data Structures and Algorithms,
    K. Mehlhorn,
    Springer-Verlag, 1984.

    R. Sedgewick,
    Addison-Wesley, 1988.

    Solid Modelling

    Introduction to Solid Modeling
    Martti Mantyla, Computer Science Press 1988,
    ISBN 07167-8015-1

Subject 0.05: Are there any online references?

    The computational geometry community maintains its own
    bibliography of publications in or closely related to that
    subject.  Every four months, additions and corrections are
    solicited from users, after which the database is updated and
    released anew.  As of 7 Nov 200, it contained 13485 bib-tex
    entries.  See Jeff Erickson's page on "Computational Geometry
    The bibliography can be retrieved from:

    ftp://ftp.cs.usask.ca/pub/geometry/geombib.tar.gz - bibliography proper
    ftp://ftp.cs.usask.ca/pub/geometry/o-cgc19.ps.gz  - overview published
        in '93 in SIGACT News and the Internat. J. Comput.  Geom. Appl.
    ftp://ftp.cs.usask.ca/pub/geometry/ftp-hints      - detailed retrieval info

    Universitat Politecnica de Catalunya maintains a search engine at:

    The ACM SIGGRAPH Online Bibliography Project, by
    Stephen Spencer (biblio@siggraph.org).
    The database is available for anonymous FTP from the
    ftp://siggraph.org/publications/bibliography directory.  Please
    download and examine the file READ_ME in that directory for more
    specific information concerning the database.

    'netlib' is a useful source for algorithms, member inquiries for
    SIAM, and bibliographic searches.  For information, send mail to
    netlib@ornl.gov, with "send index" in the body of the mail

    You can also find free sources for numerical computation in C via
    ftp://ftp.usc.edu/pub/C-numanal/ . In particular, grab
    numcomp-free-c.gz in that directory.

    Check out Nick Fotis's computer graphics resources FAQ -- it's
    packed with pointers to all sorts of great computer graphics
    stuff.  This FAQ is posted biweekly to comp.graphics.

    This WWW page contains links to a large number
    of computer graphic related pages:

    There's a Computer Science Bibliography Server at:
    with Computer Graphics, Vision and Radiosity sections

    A comprehensive bibliography of color quantization papers and articles 
    (CQUANT97) was available at http://www.ledalite.com/library-/cgis.htm.
    [Link no longer available -- replacement? --JOR]

    Modelling physically based systems for animation:

    The University of Manchester NURBS Library:

    For an implementation of Seidel's algorithm for fast trapezoidation
    and triangulation of polygons. You can get the code from:

    Ray tracing bibliography:

    Quaternions and other comp sci curiosities:

    Directory of Computational Geometry Software,
    collected by Nina Amenta (nina@cs.utexas.edu)
    Nina Amenta is maintaining a WWW directory to computational 
    geometry software. The directory lives at The Geometry Center. 
    It has pointers to lots of convex hull and voronoi diagram programs, 
    triangulations, collision detection, polygon intersection, smallest 
    enclosing ball of a point set and other stuff.

    A compact reference for real-time 3D computer graphics programming:

    RADBIB is a comprehensive bibliography of radiosity and
    related global illumination papers, articles, and
    books. It currently includes 1,972 references.
    This bibliography is available in BibTex format 
    (with a release date of 15 Jul 01) from:
      http://www.helios32.com/   under "Resources."

    The "Electronic Visualization Library" (EVlib) is a domain-
    secific digital library for Scientific Visualization and 
    Computer Graphics:  http://visinfo.zib.de/

    3D Object Intersection: http://www.realtimerendering.com/int/ 
    This page presents information about a wide variety of 3D object/object
    intersection tests. Presented in grid form, each axis lists ray, plane,
    sphere, triangle, box, frustum, and other objects. For each combination
    (e.g. sphere/box), references to articles, books, and online resources
    are given.

    Ray Tracing News, ed. Eric Haines:  http://www.raytracingnews.com .

Subject 0.06: Are there other graphics related FAQs?

    BSP Tree FAQ by Bretton Wade

    Gamma and Color FAQs by Charles A. Poynton has

    The documents are mirrored in Darmstadt, Germany at

Subject 0.07: Where is all the source?

    Graphics Gems source code.
    This site is now the offical distribution site for Graphics Gems code.

    Master list of Computational Geometry software:
    Described in [Goodman & O'Rourke], Chap. 52.

    Jeff Erikson's software list:

    Dave Eberly's extensive collection of free geometry, graphics,
    and image processing software:

    General 'stuff'

    There are a number of interesting items in
    http://graphics.lcs.mit.edu/~seth including:
    - Code for 2D Voronoi, Delaunay, and Convex hull
    - Mike Hoymeyer's implementation of Raimund Seidel's
      O( d! n ) time linear programming algorithm for
      n constraints in d dimensions
    - geometric models of UC Berkeley's new computer science

    Sources to "Computational Geometry in C", by J. O'Rourke
    can be found at http://cs.smith.edu/~orourke/books/compgeom.html
    or ftp://cs.smith.edu/pub/compgeom .

    Greg Ferrar's C++ 3D rendering library seems no longer available
    at ftp://ftp.math.ohio-state.edu/pub/users/gregt . 
    See http://www.flowerfire.com/ADSODA/ instead.

    TAGL is a portable and extensible library that provides a subset
    of Open-GL functionalities.

    Try ftp://x2ftp.oulu.fi  for /pub/msdos/programming/docs/graphpro.lzh by
    Michael Abrash. His XSharp package has an implementation of Xiaoulin
    Wu's anti-aliasing algorithm (in C).

    Example sources for BSP tree algorithms can be found at
    http://reality.sgi.com/bspfaq/, item 24.

    Mel Slater (mel@dcs.qmw.ac.uk) also made some implementations of
    BSP trees and shadows for static scenes using shadow volumes
    code available

    The Visualization Toolkit (A visualization textbook, C++ library 
    and Tcl-based interpreter) (see [Schroeder]): 

    WINGED.ZIP, a C++ implementation of Baumgart's winged-edge data structure:

    CGAL, the Computational Geometry Algorithms Library, is written in C++ 
    and is available at URL http://www.cs.ruu.nl/CGAL/ .  It consists of 
    three parts. The first part is the kernel, which consists of constant 
    size non-modifiable geometric primitive objects.  The second part is 
    a collection of basic geometric datastructures and algorithms, which 
    are parameterized by traits classes that define the interface between 
    the datastructure or algorithm, and the primitives they use. 
    The third part consists of non-geometric support facilities.

    A C++ NURBS library written by Lavoie Philippe. Version 2.1. 
    Results may be exported as POV-Ray, RIB (renderman) or VRML files.
    It also offers wrappers to OpenGL:

    Paul Bourke has code for several problems, including isosurface 
    generation and Delauney triangulation, at:

    A nearly comprehensive list of available 3D engines 
    (most with source code):

    See also 5.17: 
        Where can I get the spline description of the famous teapot etc.?

    Interactive Geometry Software called "Cinderella":

Section 1. 2D Computations: Points, Segments, Circles, Etc.
Subject 1.01: How do I rotate a 2D point?

    In 2D, you make (X,Y) from (x,y) with a rotation by angle t so:
        X = x cos t - y sin t
        Y = x sin t + y cos t
    As a 2x2 matrix this is very simple.  If you want to rotate a
    column vector v by t degrees using matrix M, use
        M = [cos t  -sin t]
            [sin t   cos t]
    in the product M v.

    If you have a row vector, use the transpose of M (turn rows into
    columns and vice versa).  If you want to combine rotations, in 2D
    you can just add their angles, but in higher dimensions you must
    multiply their matrices.

Subject 1.02: How do I find the distance from a point to a line?

    Let the point be C (Cx,Cy) and the line be AB (Ax,Ay) to (Bx,By).
    Let P be the point of perpendicular projection of C on AB.  The parameter
    r, which indicates P's position along AB, is computed by the dot product 
    of AC and AB divided by the square of the length of AB:
    (1)     AC dot AB
        r = ---------  
    r has the following meaning:
        r=0      P = A
        r=1      P = B
        r<0      P is on the backward extension of AB
        r>1      P is on the forward extension of AB
        00      C is right of AB
           s=0      C is on AB

    Compute s as follows:

        s = -----------------------------

    Then the distance from C to P = |s|*L.

Subject 1.03: How do I find intersections of 2 2D line segments?

    This problem can be extremely easy or extremely difficult; it
    depends on your application. If all you want is the intersection
    point, the following should work:

    Let A,B,C,D be 2-space position vectors.  Then the directed line
    segments AB & CD are given by:

        AB=A+r(B-A), r in [0,1]
        CD=C+s(D-C), s in [0,1]

    If AB & CD intersect, then

        A+r(B-A)=C+s(D-C), or

        Ay+r(By-Ay)=Cy+s(Dy-Cy)  for some r,s in [0,1]

    Solving the above for r and s yields

        r = -----------------------------  (eqn 1)

        s = -----------------------------  (eqn 2)

    Let P be the position vector of the intersection point, then

        P=A+r(B-A) or


    By examining the values of r & s, you can also determine some
    other limiting conditions:

        If 0<=r<=1 & 0<=s<=1, intersection exists
            r<0 or r>1 or s<0 or s>1 line segments do not intersect

        If the denominator in eqn 1 is zero, AB & CD are parallel
        If the numerator in eqn 1 is also zero, AB & CD are collinear.

    If they are collinear, then the segments may be projected to the x- 
    or y-axis, and overlap of the projected intervals checked.

    If the intersection point of the 2 lines are needed (lines in this
    context mean infinite lines) regardless whether the two line
    segments intersect, then

        If r>1, P is located on extension of AB
        If r<0, P is located on extension of BA
        If s>1, P is located on extension of CD
        If s<0, P is located on extension of DC

    Also note that the denominators of eqn 1 & 2 are identical.


    [O'Rourke (C)] pp. 249-51
    [Gems III] pp. 199-202 "Faster Line Segment Intersection,"

Subject 1.04: How do I generate a circle through three points?

    Let the three given points be a, b, c.  Use _0 and _1 to represent
    x and y coordinates. The coordinates of the center p=(p_0,p_1) of
    the circle determined by a, b, and c are:

        A = b_0 - a_0;
        B = b_1 - a_1;
        C = c_0 - a_0;
        D = c_1 - a_1;
        E = A*(a_0 + b_0) + B*(a_1 + b_1);
        F = C*(a_0 + c_0) + D*(a_1 + c_1);
        G = 2.0*(A*(c_1 - b_1)-B*(c_0 - b_0));
        p_0 = (D*E - B*F) / G;
        p_1 = (A*F - C*E) / G;
    If G is zero then the three points are collinear and no finite-radius
    circle through them exists.  Otherwise, the radius of the circle is:

            r^2 = (a_0 - p_0)^2 + (a_1 - p_1)^2


    [O' Rourke (C)] p. 201. Simplified by Jim Ward.

Subject 1.05: How can the smallest circle enclosing a set of points be found?

    This circle is often called the minimum spanning circle.  It can be
    computed in O(n log n) time for n points.  The center lies on
    the furthest-point Voronoi diagram.  Computing the diagram constrains
    the search for the center.  Constructing the diagram can be accomplished
    by a 3D convex hull algorithm; for that connection, see, e.g.,
    [O'Rourke (C), p.195ff].  For direct algorithms, see:
      S. Skyum, "A simple algorithm for computing the smallest enclosing circle"
      Inform. Process. Lett. 37 (1991) 121--125.
      J. Rokne, "An Easy Bounding Circle" [Gems II] pp.14--16.

Subject 1.06: Where can I find graph layout algorithms?

    See the paper by Eades and Tamassia, "Algorithms for Drawing
    Graphs: An Annotated Bibliography," Tech Rep CS-89-09, Dept.  of
    CS, Brown Univ, Feb. 1989.

    A revised version of the annotated bibliography on graph drawing
    algorithms by Giuseppe Di Battista, Peter Eades, and Roberto
    Tamassia is now available from

    ftp://wilma.cs.brown.edu/pub/papers/compgeo/gdbiblio.tex.gz and

    Commercial software includes the Graph Layout Toolkit from Tom Sawyer
    Software http://www.tomsawyer.com and Northwoods Software's GO++
    http://www.nwoods.com/go/ .

Section 2. 2D Polygon Computations
Subject 2.01: How do I find the area of a polygon?

    The signed area can be computed in linear time by a simple sum.
    The key formula is this:

        If the coordinates of vertex v_i are x_i and y_i,
        twice the signed area of a polygon is given by

        2 A( P ) = sum_{i=0}^{n-1} (x_i y_{i+1} - y_i x_{i+1}).

    Here n is the number of vertices of the polygon, and index
    arithmetic is mod n, so that x_n = x_0, etc. A rearrangement
    of terms in this equation can save multiplications and operate on
    coordinate differences, and so may be both faster and more

       2 A(P) = sum_{i=0}^{n-1} ( x_i  (y_{i+1} - y_{i-1}) )

    Here again modular index arithmetic is implied, with n=0 and -1=n-1.
    This can be avoided by extending the x[] and y[] arrays up to [n+1]
    with x[n]=x[0], y[n]=y[0] and y[n+1]=y[1], and using instead

       2 A(P) = sum_{i=1}^{n} ( x_i  (y_{i+1} - y_{i-1}) )

    References: [O' Rourke (C)] Thm. 1.3.3, p. 21; [Gems II] pp. 5-6:
    "The Area of a Simple Polygon", Jon Rokne.  Dan Sunday's explanation:
       http://GeometryAlgorithms.com/Archive/algorithm_0101/  where

    To find the area of a planar polygon not in the x-y plane, use:
       2 A(P) = abs(N . (sum_{i=0}^{n-1} (v_i x v_{i+1})))
       where N is a unit vector normal to the plane. The `.' represents the
       dot product operator, the `x' represents the cross product operator,
       and abs() is the absolute value function.
    [Gems II] pp. 170-171:
    "Area of Planar Polygons and Volume of Polyhedra", Ronald N. Goldman.

Subject 2.02: How can the centroid of a polygon be computed?

    The centroid (a.k.a. the center of mass, or center of gravity)
    of a polygon can be computed as the weighted sum of the centroids
    of a partition of the polygon into triangles.  The centroid of a
    triangle is simply the average of its three vertices, i.e., it
    has coordinates (x1 + x2 + x3)/3 and (y1 + y2 + y3)/3.  This 
    suggests first triangulating the polygon, then forming a sum
    of the centroids of each triangle, weighted by the area of
    each triangle, the whole sum normalized by the total polygon area.
    This indeed works, but there is a simpler method:  the triangulation
    need not be a partition, but rather can use positively and
    negatively oriented triangles (with positive and negative areas),
    as is used when computing the area of a polygon.  This leads to
    a very simple algorithm for computing the centroid, based on a
    sum of triangle centroids weighted with their signed area.
    The triangles can be taken to be those formed by any fixed point,
    e.g., the vertex v0 of the polygon, and the two endpoints of 
    consecutive edges of the polygon: (v1,v2), (v2,v3), etc.  The area 
    of a triangle with vertices a, b, c is half of this expression:
                (b[X] - a[X]) * (c[Y] - a[Y]) -
                (c[X] - a[X]) * (b[Y] - a[Y]);
    Code available at ftp://cs.smith.edu/pub/code/centroid.c (3K).
    Reference: [Gems IV] pp.3-6; also includes code.

Subject 2.03: How do I find if a point lies within a polygon?

    The definitive reference is "Point in Polygon Strategies" by
    Eric Haines [Gems IV]  pp. 24-46.  Now also at 
    The code in the Sedgewick book Algorithms (2nd Edition, p.354) fails
    under certain circumstances.  See 
    for a discussion.

    The essence of the ray-crossing method is as follows.
    Think of standing inside a field with a fence representing the polygon. 
    Then walk north. If you have to jump the fence you know you are now 
    outside the poly. If you have to cross again you know you are now 
    inside again; i.e., if you were inside the field to start with, the total 
    number of fence jumps you would make will be odd, whereas if you were 
    ouside the jumps will be even.

    The code below is from Wm. Randolph Franklin 
    (see URL below) with some minor modifications for speed.  It returns 
    1 for strictly interior points, 0 for strictly exterior, and 0 or 1 
    for points on the boundary.  The boundary behavior is complex but 
    determined; in particular, for a partition of a region into polygons, 
    each point is "in" exactly one polygon.  
    (See p.243 of [O'Rourke (C)] for a discussion of boundary behavior.)

    int pnpoly(int npol, float *xp, float *yp, float x, float y)
      int i, j, c = 0;
      for (i = 0, j = npol-1; i < npol; j = i++) {
        if ((((yp[i]<=y) && (y
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