Frequently Asked Questions

Skip to first unread message

Joseph O'Rourke

Feb 15, 2003, 11:16:44 AM2/15/03
Posted-By: auto-faq 3.3.1 beta (Perl 5.006)
Archive-name: graphics/algorithms-faq
Posting-Frequency: bi-weekly

Welcome to the FAQ for!

Thanks to all who have contributed. Corrections and contributions
(to always welcome.

This article is Copyright 2003 by Joseph O'Rourke. It may be freely
redistributed in its entirety provided that this copyright notice is
not removed.

Changed items this posting (|): 5.04
New items this posting (+): none

History of Changes (approx. last six months):
Changes in 15 Feb 03 posting:
5.04: Fixed broken link in clipping article. [Thanks to Keith Forbes.]
Changes in 1 Feb 03 posting:
0.04: Ashdown Radiosity back in print. [Thanks to Ian Ashdown.]
0.06: Update BSP FAQ links [Thanks to Ken Shoemake.]
0.07: Update CGAL links in source article.
3.11: Broken link re course based on Perlin's Noise book. [Thanks to Mikkel Gjoel.]
6.01: Add CGAL link to Voronoi source article. [Thanks to Andreas Fabri.]
7.02: All contributor email addresses removed to protect them from spam.
Changes in 15 Jan 03 posting:
0.06: Query re [Thanks to <>.]
0.07: Update moved link on WINGED.ZIP. [Thanks to Ben Landon.]
0.07: Update Ferrar's ++ 3D rendering library link. [Thanks to F.Iannarilli, Jr.]
1.06: Added ref to AT&T Graphviz. [Thanks to Michael Meire.]
2.08: Fix sloan ear-clipping link. [Thanks to]
5.09: Update moved link re caustics. [Thanks to Ben Landon.]
5.18: Formula for distance between two 3D lines. [Thanks to Daniel Zwick.]
5.27: New article on transforming normals by Ken Shoemake.
6.08: Random points on sphere in terms of longitude & latitude [Thanks to Uffe Kousgaard.]
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
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
0.02: Are the postings to 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?
5.27: How can I transform normals?

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 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. is not:

- for requests for gifs, or other pictures
- for requests for image translator or processing software; see* FAQ [now degenerated to pic postings] (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 archived?

Archives may be found at:

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: Frequently Asked Questions
It is posted to, 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 .

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

The (busy) site lists many alternative "mirror" sites.
Also can reach the FAQ from,
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 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:

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.
Back in print, Jan 2003. See

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 .

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: "")

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 .

[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 .

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: - bibliography proper - overview published
in '93 in SIGACT News and the Internat. J. Comput. Geom. Appl. - detailed retrieval info

Universitat Politecnica de Catalunya maintains a search engine at:

The ACM SIGGRAPH Online Bibliography Project, by
Stephen Spencer (
The database is available for anonymous FTP from the 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, with "send index" in the body of the mail

You can also find free sources for numerical computation in C via . 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

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
[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 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: under "Resources."

The "Electronic Visualization Library" (EVlib) is a domain-
secific digital library for Scientific Visualization and
Computer Graphics:

3D Object Intersection:
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: .

Subject 0.06: Are there other graphics related FAQs?

BSP Tree FAQ by Bretton Wade
and see

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 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
or .

Greg Ferrar's C++ 3D rendering library is available at

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

Try 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, item 24.

Mel Slater ( 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
CGAL contains algorithms and data structures for 2D computations
(convex hull, Delaunay, constrained Delaunay, Voronoi diagram,
regular traingulation, (weighted) Alpha shapes, polytope distance,
boolean operations on polygons, decomposition of polygons in
monotone or convex parts, arrangements, etc.), 3D, and arbitrary

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
0<r<1 P is interior to AB

The length of a line segment in d dimensions, AB is computed by:

L = sqrt( (Bx-Ax)^2 + (By-Ay)^2 + ... + (Bd-Ad)^2)

so in 2D:

L = sqrt( (Bx-Ax)^2 + (By-Ay)^2 )

and the dot product of two vectors in d dimensions, U dot V is computed:

D = (Ux * Vx) + (Uy * Vy) + ... + (Ud * Vd)

so in 2D:

D = (Ux * Vx) + (Uy * Vy)

So (1) expands to:

(Cx-Ax)(Bx-Ax) + (Cy-Ay)(By-Ay)
r = -------------------------------

The point P can then be found:

Px = Ax + r(Bx-Ax)
Py = Ay + r(By-Ay)

And the distance from A to P = r*L.

Use another parameter s to indicate the location along PC, with the
following meaning:
s<0 C is left of AB
s>0 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 and

Commercial software includes the Graph Layout Toolkit from Tom Sawyer
Software and Northwoods Software's GO++ .

Perhaps the best code is the AT&T Research Labs open C source:

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: 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 (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<yp[j])) ||
((yp[j]<=y) && (y<yp[i]))) &&
(x < (xp[j] - xp[i]) * (y - yp[i]) / (yp[j] - yp[i]) + xp[i]))

c = !c;
return c;

The code may be further accelerated, at some loss in clarity, by
avoiding the central computation when the inequality can be deduced,
and by replacing the division by a multiplication for those processors
with slow divides. For code that distinguishes strictly interior
points from those on the boundary, see [O'Rourke (C)] pp. 239-245.
For a method based on winding number, see Dan Sunday,
"Fast Winding Number Test for Point Inclusion in a Polygon,", March 2001.

Franklin's code:
[Gems IV] pp. 24-46
[O'Rourke (C)] Sec. 7.4.

Subject 2.04: How do I find the intersection of two convex polygons?

Unlike intersections of general polygons, which might produce a
quadratic number of pieces, intersection of convex polygons of n
and m vertices always produces a polygon of at most (n+m) vertices.
There are a variety of algorithms whose time complexity is proportional
to this size, i.e., linear. The first, due to Shamos and Hoey, is
perhaps the easiest to understand. Let the two polygons be P and
Q. Draw a horizontal line through every vertex of each. This
partitions each into trapezoids (with at most two triangles, one
at the top and one at the bottom). Now scan down the two polygons
in concert, one trapezoid at a time, and intersect the trapezoids
if they overlap vertically.
A more difficult-to-describe algorithm is in [O'Rourke (C)], pp. 252-262.
This walks around the boundaries of P and Q in concert, intersecting
edges. An implementation is included in [O'Rourke (C)].

Subject 2.05: How do I do a hidden surface test (backface culling) with 2D points?

c = (x1-x2)*(y3-y2)-(y1-y2)*(x3-x2)

x1,y1, x2,y2, x3,y3 = the first three points of the polygon.

If c is positive the polygon is visible. If c is negative the
polygon is invisible (or the other way).

Subject 2.06: How do I find a single point inside a simple polygon?

Given a simple polygon, find some point inside it. Here is a method
based on the proof that there exists an internal diagonal, in
[O'Rourke (C), 13-14]. The idea is that the midpoint of a diagonal
is interior to the polygon.

1. Identify a convex vertex v; let its adjacent vertices be a and b.
2. For each other vertex q do:
2a. If q is inside avb, compute distance to v (orthogonal to ab).
2b. Save point q if distance is a new min.
3. If no point is inside, return midpoint of ab, or centroid of avb.
4. Else if some point inside, qv is internal: return its midpoint.

Code for finding a diagonal is in [O'Rourke (C), 35-39], and is part
of many other software packages. See Subject 0.07: Where is all the

Subject 2.07: How do I find the orientation of a simple polygon?

Compute the signed area (Subject 2.01). The orientation is
counter-clockwise if this area is positive.

A slightly faster method is based on the observation that it isn't
necessary to compute the area. Find the lowest vertex (or, if
there is more than one vertex with the same lowest coordinate,
the rightmost of those vertices) and then take the cross product
of the edges fore and aft of it. Both methods are O(n) for n vertices,
but it does seem a waste to add up the total area when a single cross
product (of just the right edges) suffices. Code for this is
available at (2K).

The reason that the lowest, rightmost (or any other such extreme) point
works is that the internal angle at this vertex is necessarily convex,
strictly less than pi (even if there are several equally-lowest points).

Subject 2.08: How can I triangulate a simple polygon?

Triangulation of a polygon partitions its interior into triangles
with disjoint interiors. Usually one restricts corners of the
triangles to coincide with vertices of the polygon, in which case
every polygon of n vertices can be triangulated, and all triangulations
contain n-2 triangles, employing n-3 "diagonals" (chords between
vertices that otherwise do not touch the boundary of the polygon).

Triangulations can be constructed by a variety of algorithms,
ranging from a naive search for noncrossing diagonals, which is
O(n^4), to "ear" clipping, which is O(n^2) and relatively straightforward
to implement [O'Rourke (C), Chap. 1], to partitioning into
monotone polygons, which leads to O(n log n) time complexity
[O'Rourke (C), Chap. 2; Overmars, Chap. 3], to several randomized
algorithms---by Clarkson et al., by Seidel, and by Devillers, which
lead to O(n log* n) complexity---to Chazelle's linear-time algorithm,
which has yet to be implemented.

There is a tradeoff between code complexity and time complexity.
Fortunately, several of the algorithms have been implemented and are
Seidel's Alg:
See also the collection of triangulation links at


[O'Rourke (C)]
[Gems V]
Clarkson, K., Tarjan, R., and VanWyk, C. A fast Las Vegas algorithm for
triangulating a simple polygon. Discrete and Computational Geometry,
4(1):423--432, 1989.
Clarkson, K., Cole, R., Tarjan, R. Randomized parallel algorithms for
trapezoidal diagrams. Int. J. Comp. Geom. Appl., 117--133, 1992.
Seidel, R. (1991), A simple and fast incremental randomized algorithm for
computing trapezoidal decompositions and for triangulating polygons,
Comput. Geom. Theory Appl. 1, 51--64.
Devillers, O. (1992), Randomization yields simple O(n log* n)
algorithms for difficult Omega(n) problems,
Internat. J. Comput. Geom. Appl. 2(1), 97--111.
Chazelle, B. (1991), Triangulating a simple polygon in linear time,
Discrete Comput. Geom. 6, 485--524.
Held, M. (1998) "FIST: Fast Industrial-Strength Triangulation".

Subject 2.09: How can I find the minimum area rectangle enclosing a set of points?
First take the convex hull of the points. Let the resulting convex
polygon be P. It has been known for some time that the minimum
area rectangle enclosing P must have one rectangle side flush with
(i.e., collinear with and overlapping) one edge of P. This geometric
fact was used by Godfried Toussaint to develop the "rotating calipers"
algorithm in a hard-to-find 1983 paper, "Solving Geometric Problems
with the Rotating Calipers" (Proc. IEEE MELECON). The algorithm
rotates a surrounding rectangle from one flush edge to the next,
keeping track of the minimum area for each edge. It achieves O(n)
time (after hull computation). See the "Rotating Calipers Homepage" for a description
and applet.

Section 3. 2D Image/Pixel Computations
Subject 3.01: How do I rotate a bitmap?

The easiest way, according to the faq, is to take
the rotation transformation and invert it. Then you just iterate
over the destination image, apply this inverse transformation and
find which source pixel to copy there.

A much nicer way comes from the observation that the rotation

R(T) = { { cos(T), -sin(T) }, { sin(T), cos(T) } }

is formed my multiplying three matrices, namely:

R(T) = M1(T) * M2(T) * M3(T)


M1(T) = { { 1, -tan(T/2) },
{ 0, 1 } }
M2(T) = { { 1, 0 },
{ sin(T), 1 } }
M3(T) = { { 1, -tan(T/2) },
{ 0, 1 } }

Each transformation can be performed in a separate pass, and
because these transformations are either row-preserving or
column-preserving, anti-aliasing is quite easy.

Another fast approach is to perform first a column-preserving roation,
and then a row-preserving rotation. For an image W pixels wide and
H pixels high, this requires W+H BitBlt operations in comparison to
the brute-force rotation, which uses W*H SetPixel operations (and a
lot of multiplying).


Paeth, A. W., "A Fast Algorithm for General Raster Rotation",
Proceedings Graphics Interface '89, Canadian Information
Processing Society, 1986, 77-81
[Note - e-mail copies of this paper are no longer available]

[Gems I]

Subject 3.02: How do I display a 24 bit image in 8 bits?

[Gems I] pp. 287-293, "A Simple Method for Color Quantization:
Octree Quantization"

B. Kurz. Optimal Color Quantization for Color Displays.
Proceedings of the IEEE Conference on Computer Vision and Pattern
Recognition, 1983, pp. 217-224.

[Gems II] pp. 116-125, "Efficient Inverse Color Map Computation"

This describes an efficient technique to
map actual colors to a reduced color map,
selected by some other technique described
in the other papers.

[Gems II] pp. 126-133, "Efficient Statistical Computations for
Optimal Color Quantization"

Xiaolin Wu. Color Quantization by Dynamic Programming and
Principal Analysis. ACM Transactions on Graphics, Vol. 11, No. 4,
October 1992, pp 348-372.

Subject 3.03: How do I fill the area of an arbitrary shape?

"A Fast Algorithm for the Restoration of Images Based on Chain
Codes Description and Its Applications", L.W. Chang & K.L. Leu,
Computer Vision, Graphics, and Image Processing, vol.50,
pp296-307 (1990)

Heckbert, Paul S., Generic Convex Polygon Scan Conversion and Clipping,
Graphics Gems, p. 84-86, code: p. 667-680, PolyScan/.
Heckbert, Paul S., Concave Polygon Scan Conversion, Graphics Gems, p.
87-91, code: p. 681-684, ConcaveScan.c.

For filling a region of a bitmap, see
Heckbert, Paul S., A Seed Fill Algorithm, Graphics Gems, p. 275-277,
code: p. 721-722, SeedFill.c. The code is at

[Gems I]

Subject 3.04: How do I find the 'edges' in a bitmap?

A simple method is to put the bitmap through the filter:

-1 -1 -1
-1 8 -1
-1 -1 -1

This will highlight changes in contrast. Then any part of the
picture where the absolute filtered value is higher than some
threshold is an "edge".

A more appropriate edge detector for noisy images is
described by Van Vliet et al. "A nonlinear Laplace operator
as edge detector in noisy images", in Computer Vision,
Graphics, and image processing 45, pp. 167-195, 1989.

Subject 3.05: How do I enlarge/sharpen/fuzz a bitmap?

Sharpening of bitmaps can be done by the following algorithm:

I_enh(x,y) = I_fuz(x,y)-k*Laplace(I_fuz(x,y))

or in words: An image can be sharpened by subtracting a positive
fraction k of the Laplace from the fuzzy image.

One "Laplace" kernel, approximating a Laplacian operator, is:
1 1 1
1 -8 1
1 1 1

The following library implements Fast Gaussian Blurs:

MAGIC: An Object-Oriented Library for Image Analysis by David Eberly

The library source code and the documentation (in Latex) are at
The code compiles on Unix systems using g++ and on PCs using
Microsoft Windows 3.1 and Borland C++. The fast Gaussian blurring
is based on a finite difference method for solving s u_s = s^2 \nabla^2 u
where s is the standard deviation of the Gaussian (t = s^2/2). It
takes advantage of geometrically increasing steps in s (rather than
linearly increasing steps in t), thus getting to a larger "time" rapidly,
but still retaining stability. Section 4.5 of the documentation contains
the algorithm description and implementation.

A bitmap is a sampled image, a special case of a digital signal,
and suffers from two limitations common to all digital signals.
First, it cannot provide details at fine enough spacing to exactly
reproduce every continuous image, nor even more detailed sampled
images. And second, each sample approximates the infinitely fine
variability of ideal values with a discrete set of ranges encoded
in a small number of bits---sometimes just one bit per pixel. Most
bitmaps have another limitation imposed: The values cannot be
negative. The resolution limitation is especially important, but
see "How do I display a 24 bit image in 8 bits?" for range issues.

The ideal way to enlarge a bitmap is to work from the original
continuous image, magnifying and resampling it. The standard way
to do it in practice is to (conceptually) reconstruct a continuous
image from the bitmap, and magnify and resample that instead. This
will not give the same results, since details of the original have
already been lost, but it is the best approach possible given an
already sampled image. More details are provided below.

Both sharpening and fuzzing are examples of filtering. Even more
specifically, they can be both be accomplished with filters which
are linear and shift invariant. A crude way to sharpen along a row
(or column) is to set output pixel B[n] to the difference of input
pixels, A[n]-A[n-1]. A similarly crude way to fuzz is to set B[n]
to the average of input pixels, 1/2*A[n]+1/2*A[n-1]. In each case
the output is a weighted sum of input pixels, a "convolution". One
important characteristic of such filters is that a sinusoid going
in produces a sinusoid coming out, one of the same frequency. Thus
the Fourier transform, which decomposes a signal into sinusoids of
various frequencies, is the key to analysis of these filters. The
simplest (and most efficient) way to handle the two dimensions of
images is to operate on first the rows then the columns (or vice
versa). Fourier transforms and many filters allow this separation.

A filter is linear if it satisfies two simple relations between the
input and output: scaling the input by a factor scales the output
by the same factor, and the sum of two inputs gives the sum of the
two outputs. A filter is shift invariant if shifting the input up,
down, left, or right merely shifts the output the same way. When a
filter is both linear and shift invariant, it can be implemented as
a convolution, a weighted sum. If you find the output of the filter
when the input is a single pixel with value one in a sea of zeros,
you will know all the weights. This output is the impulse response
of the filter. The Fourier transform of the impulse response gives
the frequency response of the filter. The pattern of weights read
off from the impulse response gives the filter kernel, which will
usually be displayed (for image filters) as a 2D stencil array, and
it is almost always symmetric around the center. For example, the
following filter, approximating a Laplacian (and used for detecting
edges), is centered on the negative value.
1/6 4/6 1/6
4/6 -20/6 4/6
1/6 4/6 1/6
The symmetry allows a streamlined implementation. Suppose the input
image is in A, and the output is to go into B. Then compute
B[i][j] = (A[i-1][j-1]+A[i-1][j+1]+A[i+1][j-1]+A[i+1][j+1]

Ideal blurring is uniform in all directions, in other words it has
circular symmetry. Gaussian blurs are popular, but the obvious code
is slow for wide blurs. A cheap alternative is the following filter
(written for rows, but then applied to the columns as well).
B[i][j] = ((A[i][j]*2+A[i][j-1]+A[i][j+1])*4
For sharpening, subtract the results from the original image, which
is equivalent to using the following.
B[i][j] = ((A[i][j]*2-A[i][j-1]-A[i][j+1])*4
Credit for this filter goes to Ken Turkowski and Steve Gabriel.

Reconstruction is impossible without some assumptions, and because
of the importance of sinusoids in filtering it is traditional to
assume the continuous image is made of sinusoids mixed together.
That makes more sense for sounds, where signal processing began,
than it does for images, especially computer images of character
shapes, sharp surface features, and halftoned shading. As pointed
out above, often image values cannot be negative, unlike sinusoids.
Also, real world images contain noise. The best noise suppressors
(and edge detectors) are, ironically, nonlinear filters.

The simplest way to double the size of an image is to use each of
the original pixels twice in its row and in its column. For much
better results, try this instead. Put zeros between the original
pixels, then use the blurring filter given a moment ago. But you
might want to divide by 8 instead of 16 (since the zeros will dim
the image otherwise). To instead shrink the image by half (in both
vertical and horizontal), first apply the filter (dividing by 16),
then throw away every other pixel. Notice that there are obvious
optimizations involving arithmetic with powers of two, zeros which
are in known locations, and pixels which will be discarded.

Subject 3.06: How do I map a texture on to a shape?

Paul S. Heckbert, "Survey of Texture Mapping", IEEE Computer
Graphics and Applications V6, #11, Nov. 1986, pp 56-67 revised
from Graphics Interface '86 version

Eric A. Bier and Kenneth R. Sloan, Jr., "Two-Part Texture
Mappings", IEEE Computer Graphics and Applications V6 #9, Sept.
1986, pp 40-53 (projection parameterizations)

Subject 3.07: How do I detect a 'corner' in a collection of points?

[Currently empty entry.]

Subject 3.08: Where do I get source to display (raster font format)?
See also James Murray's graphics file formats FAQ:

Subject 3.09: What is morphing/how is it done?

Morphing is the name that has come to be applied to the technique
ILM used in the movie "Willow", where one object changes into
another by changing both its shape and picture detail. It was a
2D image manipulation, and has been done in different ways. The
first method published was by Thad Beier at PDI. Michael Jackson
famously used morphing in his music videos, notably "Black or
White". The word is now used more generally.

For more, see [Anderson], Chapter 3, page 59-90, and Beier's

Subject 3.10: How do I quickly draw a filled triangle?

The easiest way is to render each line separately into an edge
buffer. This buffer is a structure which looks like this (in C):

struct { int xmin, xmax; } edgebuffer[YDIM];

There is one entry for each scan line on the screen, and each
entry is to be interpreted as a horizontal line to be drawn from
xmin to xmax.

Since most people who ask this question are trying to write fast
games on the PC, I'll tell you where to find code. Look at:

ftp::/ (Sweden)

See also Subject 3.03, which describes methods for filling polygons.

Subject 3.11: 3D Noise functions and turbulence in Solid texturing.


In it there are implementations of Perlin's noise and turbulence
functions, (By the man himself) as well as Lewis' sparse
convolution noise function (by D. Peachey) There is also some of
other stuff in there (Musgrave's Earth texture functions, and some
stuff on animating gases by Ebert).

SPD (Standard Procedural Databases) package:
Now moved to


Noise, Hypertexture, Antialiasing and Gesture, (Ken Perlin) in
Chapter 6, (p.193-), The disk accompanying the book is available

For more info on this text/code see:

For examples from a current course based on this book, see:
Linke broken 21Jan03; will remove eventually if not fixed.

Three-dimensional Nocie, Chapter 7.2.1
Simulating turbulance, Chapter 7.2.2

Subject 3.12: How do I generate realistic sythetic textures?

For fractal mountains, trees and sea-shells:

SPD (Standard Procedural Databases) package:
Now moved to

Reaction-Diffusion Algorithms:
For an illustartion of the parameter space of a reaction
diffusion system, check out the Xmorphia page at


Entire book devoted to this subject, with RenderMan(TM) and C code.

Procedural texture mapping and modelling, Chapter 7

"Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion"
Greg Turk, Computer Graphics, Vol. 25, No. 4, pp. 289-298
July 1991 (SIGGRAPH '91)

A list of procedural texture synthesis related web pages

Subject 3.13: How do I convert between color models (RGB, HLS, CMYK, CIE etc)?

[Watt:3D] pp. 313-354
[Foley] pp. 563-603

Subject 3.14: How is "GIF" pronounced?

"GIF" is an acronymn for "Graphics Interchange Format." Despite the
hard "G" in "Graphics," GIF is pronounced "JIF." Although we don't
have a direct quote from the official CompuServe specification
released June 1987, here is a quote from related CompuServe documentation,
for CompuShow, a DOS-based image viewer used shortly thereafter:
"The GIF (Graphics Interchange Format), pronounced "JIF", was
designed by CompuServe ..."
We also have a report that the principal author of the GIF spec,
Steve Wilhite, says "it's pronounced JIF (like the peanut butter."

See also

Section 4. Curve Computations
Subject 4.01: How do I generate a Bezier curve that is parallel to another Bezier?

You can't. The only case where this is possible is when the
Bezier can be represented by a straight line. And then the
parallel 'Bezier' can also be represented by a straight line.

The situation is different for the broader class of rational
Bezier curves. For example, these can represent circular arcs,
and a parallel offset is just a concentric circular arc, also
representable as a rational Bezier.

Subject 4.02: How do I split a Bezier at a specific value for t?

A Bezier curve is a parametric polynomial function from the
interval [0..1] to a space, usually 2D or 3D. Common Bezier
curves use cubic polynomials, so have the form

f(t) = a3 t^3 + a2 t^2 + a1 t + a0,

where the coefficients are points in 3D. Blossoming converts this
polynomial to a more helpful form. Let s = 1-t, and think of
tri-linear interpolation:

F([s0,t0],[s1,t1],[s2,t2]) =
s0(s1(s2 c30 + t2 c21) + t1(s2 c21 + t2 c12)) +
t0(s1(s2 c21 + t2 c12) + t1(s2 c12 + t2 c03))
c30(s0 s1 s2) +
c21(s0 s1 t2 + s0 t1 s2 + t0 s1 s2) +
c12(s0 t1 t2 + t0 s1 t2 + t0 t1 s2) +
c03(t0 t1 t2).

The data points c30, c21, c12, and c03 have been used in such a
way as to make the resulting function give the same value if any
two arguments, say [s0,t0] and [s2,t2], are swapped. When [1-t,t]
is used for all three arguments, the result is the cubic Bezier
curve with those data points controlling it:

f(t) = F([1-t,t],[1-t,t],[1-t,t])
= (1-t)^3 c30 +
3(1-t)^2 t c21 +
3(1-t) t^2 c12 +
t^3 c03.

Notice that
F([1,0],[1,0],[1,0]) = c30,
F([1,0],[1,0],[0,1]) = c21,
F([1,0],[0,1],[0,1]) = c12, _
F([0,1],[0,1],[0,1]) = c03.

In other words, cij is obtained by giving F argument t's i of
which are 0 and j of which are 1. Since F is a linear polynomial
in each argument, we can find f(t) using a series of simple steps.
Begin with

f000 = c30, f001 = c21, f011 = c12, f111 = c03.

Then compute in succession:

f00t = s f000 + t f001, f01t = s f001 + t f011,
f11t = s f011 + t f111,
f0tt = s f00t + t f01t, f1tt = s f01t + t f11t,
fttt = s f0tt + t f1tt.

This is the de Casteljau algorithm for computing f(t) = fttt.

It also has split the curve for the intervals [0..t] and [t..1].
The control points for the first interval are f000, f00t, f0tt,
fttt, while those for the second interval are fttt, f1tt, f11t,

If you evaluate 3 F([1-t,t],[1-t,t],[-1,1]) you will get the
derivate of f at t. Similarly, 2*3 F([1-t,t],[-1,1],[-1,1]) gives
the second derivative of f at t, and finally using 1*2*3
F([-1,1],[-1,1],[-1,1]) gives the third derivative.

This procedure is easily generalized to different degrees,
triangular patches, and B-spline curves.

Subject 4.03: How do I find a t value at a specific point on a Bezier?

In general, you'll need to find t closest to your search point.
There are a number of ways you can do this. Look at [Gems I, 607],
there's a chapter on finding the nearest point on the Bezier
curve. In my experience, digitizing the Bezier curve is an
acceptable method. You can also try recursively subdividing the
curve, see if you point is in the convex hull of the control
points, and checking is the control points are close enough to a
linear line segment and find the nearest point on the line
segment, using linear interpolation and keeping track of the
subdivision level, you'll be able to find t.

Subject 4.04: How do I fit a Bezier curve to a circle?

Interestingly enough, Bezier curves can approximate a circle but
not perfectly fit a circle.
A common approximation is to use four beziers to model a circle, each
with control points a distance d=r*4*(sqrt(2)-1)/3 from the end points
(where r is the circle radius), and in a direction tangent to the
circle at the end points. This will ensure the mid-points of the
Beziers are on the circle, and that the first derivative is continuous.
The radial error in this approximation will be about 0.0273% of the
circle's radius.

Michael Goldapp, "Approximation of circular arcs by cubic
polynomials" Computer Aided Geometric Design (#8 1991 pp.227-238)

Tor Dokken and Morten Daehlen, "Good Approximations of circles by
curvature-continuous Bezier curves" Computer Aided Geometric
Design (#7 1990 pp. 33-41).

See also .

Section 5. 3D computations
Subject 5.01: How do I rotate a 3D point?

Let's assume you want to rotate vectors around the origin of your
coordinate system. (If you want to rotate around some other point,
subtract its coordinates from the point you are rotating, do the
rotation, and then add back what you subtracted.) In 3D, you need
not only an angle, but also an axis. (In higher dimensions it gets
much worse, very quickly.) Actually, you need 3 independent
numbers, and these come in a variety of flavors. The flavor I
recommend is unit quaternions: 4 numbers that square and add up to
+1. You can write these as [(x,y,z),w], with 4 real numbers, or
[v,w], with v, a 3D vector pointing along the axis. The concept
of an axis is unique to 3D. It is a line through the origin
containing all the points which do not move during the rotation.
So we know if we are turning forwards or back, we use a vector
pointing out along the line. Suppose you want to use unit vector u
as your axis, and rotate by 2t degrees. (Yes, that's twice t.)
Make a quaternion [u sin t, cos t]. You can use the quaternion --
call it q -- directly on a vector v with quaternion
multiplication, q v q^-1, or just convert the quaternion to a 3x3
matrix M. If the components of q are {(x,y,z),w], then you want
the matrix

M = {{1-2(yy+zz), 2(xy-wz), 2(xz+wy)},
{ 2(xy+wz),1-2(xx+zz), 2(yz-wx)},
{ 2(xz-wy), 2(yz+wx),1-2(xx+yy)}}.

Rotations, translations, and much more are explained in all basic
computer graphics texts. Quaternions are covered briefly in
[Foley], and more extensively in several Graphics Gems, and the
SIGGRAPH 85 proceedings.

/* The following routine converts an angle and a unit axis vector
* to a matrix, returning the corresponding unit quaternion at no
* extra cost. It is written in such a way as to allow both fixed
* point and floating point versions to be created by appropriate
* The following is an example of floating point definitions.
#define FPOINT double
typedef struct {FPOINT x,y,z,w;} QUAT;
enum Indices {X,Y,Z,W};
typedef FPOINT MATRIX[4][4];
#define MUL(a,b) ((a)*(b))
#define HALF(a) ((a)*0.5)
#define TWICE(a) ((a)*2.0)
#define COS cos
#define SIN sin
#define ONE 1.0
#define ZERO 0.0
QUAT MatrixFromAxisAngle(VECTOR axis, ANGLE theta, MATRIX m)
ANGLE halfTheta = HALF(theta);
FPOINT cosHalfTheta = COS(halfTheta);
FPOINT sinHalfTheta = SIN(halfTheta);
FPOINT xs, ys, zs, wx, wy, wz, xx, xy, xz, yy, yz, zz;
q.x = MUL(axis.x,sinHalfTheta);
q.y = MUL(axis.y,sinHalfTheta);
q.z = MUL(axis.z,sinHalfTheta);
q.w = cosHalfTheta;
xs = TWICE(q.x); ys = TWICE(q.y); zs = TWICE(q.z);
wx = MUL(q.w,xs); wy = MUL(q.w,ys); wz = MUL(q.w,zs);
xx = MUL(q.x,xs); xy = MUL(q.x,ys); xz = MUL(q.x,zs);
yy = MUL(q.y,ys); yz = MUL(q.y,zs); zz = MUL(q.z,zs);
m[X][X] = ONE - (yy + zz); m[X][Y] = xy - wz; m[X][Z] = xz + wy;
m[Y][X] = xy + wz; m[Y][Y] = ONE - (xx + zz); m[Y][Z] = yz - wx;
m[Z][X] = xz - wy; m[Z][Y] = yz + wx; m[Z][Z] = ONE - (xx + yy);
/* Fill in remainder of 4x4 homogeneous transform matrix. */
m[W][X] = m[W][Y] = m[W][Z] = m[X][W] = m[Y][W] = m[Z][W] = ZERO;
m[W][W] = ONE;
return (q);
/* The routine just given, MatrixFromAxisAngle, performs rotation about
* an axis passing through the origin, so only a unit vector was needed
* in addition to the angle. To rotate about an axis not containing the
* origin, a point on the axis is also needed, as in the following. For
* mathematical purity, the type POINT is used, but may be defined as:
QUAT MatrixFromAnyAxisAngle(POINT o, VECTOR axis, ANGLE theta, MATRIX m)
q = MatrixFromAxisAngle(axis,theta,m);
m[X][W] = o.x-(MUL(m[X][X],o.x)+MUL(m[X][Y],o.y)+MUL(m[X][Z],o.z));
m[Y][W] = o.y-(MUL(m[Y][X],o.x)+MUL(m[Y][Y],o.y)+MUL(m[Y][Z],o.z));
m[Z][W] = o.x-(MUL(m[Z][X],o.x)+MUL(m[Z][Y],o.y)+MUL(m[Z][Z],o.z));
return (q);
/* An axis can also be specified by a pair of points, with the direction
* along the line assumed from the ordering of the points. Although both
* the previous routines assumed the axis vector was unit length without
* checking, this routine must cope with a more delicate possibility. If
* the two points are identical, or even nearly so, the axis is unknown.
* For now the auxiliary routine which makes a unit vector ignores that.
* It needs definitions like the following for floating point.
#define SQRT sqrt
#define RECIPROCAL(a) (1.0/(a))
VECTOR Normalize(VECTOR v)
FPOINT norm = MUL(v.x,v.x)+MUL(v.y,v.y)+MUL(v.z,v.z);
/* Better to test for (near-)zero norm before taking reciprocal. */
u.x = MUL(v.x,scl); u.y = MUL(v.y,scl); u.z = MUL(v.z,scl);
return (u);
QUAT MatrixFromPointsAngle(POINT o, POINT p, ANGLE theta, MATRIX m)
VECTOR diff, axis;
diff.x = p.x-o.x; diff.y = p.y-o.y; diff.z = p.z-o.z;
axis = Normalize(diff);
return (MatrixFromAnyAxisAngle(o,axis,theta,m));

Subject 5.02: What is ARCBALL and where is the source?

Arcball is a general purpose 3D rotation controller described by
Ken Shoemake in the Graphics Interface '92 Proceedings. It
features good behavior, easy implementation, cheap execution, and
optional axis constraints. A Macintosh demo and electronic version
of the original paper (Microsoft Word format) may be obtained from

Complete source code appears in Graphics Gems IV pp. 175-192. A
fairly complete sketch of the code appeared in the original
article, in Graphics Interface 92 Proceedings, available from
Morgan Kaufmann.

The original arcball code was written for IRIS GL. A translation
into OpenGL/GLUT, and for IRIS Performer, may be found at:

Subject 5.03: How do I clip a polygon against a rectangle?

This is the Sutherland-Hodgman algorithm, an old favorite that
should be covered in any textbook. See the selected list below.
According to Vatti (q.v.) "This
[algorithm] produces degenerate edges in certain concave / self
intersecting polygons that need to be removed as a special
extension to the main algorithm" (though this is not a problem if
all you are doing with the results is scan converting them.)

It should be noted that the Sutherland-Hodgman algorithm
may be used to clip a polygon against any convex polygon.
Cf. also Subject 5.04.

[Foley, van Dam]: Section 3.14.1 (pp 124 - 126)
[Hearn]: Section 6-8, pp 237 - 242 (with actual C code!)
See also

|Subject 5.04: How do I clip a polygon against another polygon?

Klamer Schutte, has developed and implemented
some code in C++ to perform clipping of two possibly concave 2D
polygons. A description can be found at:
To compile the source code you will need a C++ compiler with templates,
such as g++. The source code is available at:
| See also, which extends
the above to permit holes.

Alan Murta released a polygon clipper library (in C) which uses a
modified version of the Vatti algorithm:


Weiler, K. "Polygon Comparison Using a Graph Representation", SIGGRAPH 80
pg. 10-18

Vatti, Bala R. "A Generic Solution to Polygon Clipping",
Communications of the ACM, July 1992, Vol 35, No. 7, pg. 57-63

Subject 5.05: How do I find the intersection of a line and a plane?

If the plane is defined as:

a*x + b*y + c*z + d = 0

and the line is defined as:

x = x1 + (x2 - x1)*t = x1 + i*t
y = y1 + (y2 - y1)*t = y1 + j*t
z = z1 + (z2 - z1)*t = z1 + k*t

Then just substitute these into the plane equation. You end up

t = - (a*x1 + b*y1 + c*z1 + d)/(a*i + b*j + c*k)

When the denominator is zero, the line is contained in the plane
if the numerator is also zero (the point at t=0 satisfies the
plane equation), otherwise the line is parallel to the plane.

Subject 5.06: How do I determine the intersection between a ray and a triangle?

First find the intersection between the ray and the plane in which
the triangle is situated. Then see if the point of intersection is
inside the triangle.
Details may be found in [O'Rourke (C)] pp.226-238, whose code is at .
Efficient code complete with statistical tests is described in the Mo:ller-
Trumbore paper in J. Graphics Tools (C code downloadable from there):
See also the full paper:
See also the "3D Object Intersection" page, described in Subject 0.05.

Subject 5.07: How do I determine the intersection between a ray and a sphere

Given a ray defined as:

x = x1 + (x2 - x1)*t = x1 + i*t
y = y1 + (y2 - y1)*t = y1 + j*t
z = z1 + (z2 - z1)*t = z1 + k*t

and a sphere defined as:

(x - l)**2 + (y - m)**2 + (z - n)**2 = r**2

Substituting in gives the quadratic equation:

a*t**2 + b*t + c = 0


a = i**2 + j**2 + k**2
b = 2*i*(x1 - l) + 2*j*(y1 - m) + 2*k*(z1 - n)
c = (x1-l)**2 + (y1-m)**2 + (z1-n)**2 - r**2;

If the discriminant of this equation is less than 0, the line does
not intersect the sphere. If it is zero, the line is tangential to
the sphere and if it is greater than zero it intersects at two
points. Solving the equation and substituting the values of t into
the ray equation will give you the points.


See also the "3D Object Intersection" page, described in Subject 0.05.

Subject 5.08: How do I find the intersection of a ray and a Bezier surface?

Joy I.K. and Bhetanabhotla M.N., "Ray tracing parametric surfaces
utilizing numeric techniques and ray coherence", Computer
Graphics, 16, 1986, 279-286

Joy and Bhetanabhotla is only one approach of three major method
classes: tessellation, direct computation, and a hybrid of the
two. Tessellation is relatively easy (you intersect the polygons
making up the tessellation) and has no numerical problems, but can
be inaccurate; direct is cheap on memory, but very expensive
computationally, and can (and usually does) suffer from precision
problems within the root solver; hybrids try to blend the two.


See also the "3D Object Intersection" page, described in Subject 0.05.

Subject 5.09: How do I ray trace caustics?

See the work of Henrik Wann Jensen at

, author = "Henrik Wann Jensen"
, title = "Rendering Caustics on Non-Lambertian Surfaces"
, booktitle = "Proc. Graphics Interface '96"
, pages = "116--121"
, location = "Toronto"
, year = 1996

Metropolis Light Transport handles this phenomenon well:

Bidirectional path tracing also handles caustics.
http://graphics.stanford.EDU/papers/veach_thesis/ (Chapter 10)

Some older references:

An expensive answer:
author = "Don P. Mitchell and Pat Hanrahan",
title = "Illumination From Curved Reflectors",
year = "1992",
month = "July",
volume = "26",
booktitle = "Computer Graphics (SIGGRAPH '92 Proceedings)",
pages = "283--291",
keywords = "caustics, interval arithmetic, ray tracing",
editor = "Edwin E. Catmull",

A cheat:
author = "Masa Inakage",
title = "Caustics and Specular Reflection Models for
Spherical Objects and Lenses ",
pages = "379--383",
journal = "The Visual Computer",
volume = "2",
number = "6",
year = "1986",
keywords = "ray tracing effects",

Very specialized:
author = "Ying Yuan and Tosiyasu L. Kunii and Naota
Inamato and Lining Sun ",
title = "Gemstone Fire: Adaptive Dispersive Ray Tracing
of Polyhedrons",
year = "1988",
month = "November",
journal = "The Visual Computer",
volume = "4",
number = "5",
pages = "259--70",
keywords = "caustics",

Subject 5.10: What is the marching cubes algorithm?

The marching cubes algorithm is used in volume rendering to
construct an isosurface from a 3D field of values.

The 2D analog would be to take an image, and for each pixel, set
it to black if the value is below some threshold, and set it to
white if it's above the threshold. Then smooth the jagged black
outlines by skinning them with lines.

The marching cubes algorithm tests the corner of each cube (or
voxel) in the scalar field as being either above or below a given
threshold. This yields a collection of boxes with classified
corners. Since there are eight corners with one of two states,
there are 256 different possible combinations for each cube.
Then, for each cube, you replace the cube with a surface that
meets the classification of the cube. For example, the following
are some 2D examples showing the cubes and their associated

- ----- + - ----- - - ----- + - ----- +
|:::' | |:::::::| |:::: | | ':::|
|:' | |:::::::| |:::: | |. ':|
| | | | |:::: | |::. |
+ ----- + + ----- + - ----- + + ----- -

The result of the marching cubes algorithm is a smooth surface
that approximates the isosurface that is constant along a given
threshold. This is useful for displaying a volume of oil in a
geological volume, for example.

"Marching Cubes: A High Resolution 3D Surface Construction Algorithm",
William E. Lorensen and Harvey E. Cline,
Computer Graphics (Proceedings of SIGGRAPH '87), Vol. 21, No. 4, pp. 163-169.

[Watt:Animation] pp. 302-305 and 313-321

For alternatives to the (patented; cf. Subj. 5.11) marching cubes algorithm,
under "Implicit Surface Polygonization."

Subject 5.11: What is the status of the patent on the "marching cubes" algorithm?

United States Patent Number: 4,710,876
Date of Patent: Dec. 1, 1987
Inventors: Harvey E. Cline, William E. Lorensen
Assignee: General Electric Company
Title: "System and Method for the Display of Surface Structures Contained
Within the Interior Region of a Solid Body"
Filed: Jun. 5, 1985
Type in "4710876" (w/o commas, w/o quotes) into their search engine.

United States Patent Number: 4,885,688
Date of Patent: Dec. 5, 1989
Inventor: Carl R. Crawford
Assignee: General Electric Company
Title: "Minimization of Directed Points Generated in Three-Dimensional
Dividing Cubes Method"
Filed: Nov. 25, 1987
Access as above.

For alternative, unpatented algorithms, cf. Subj. 5.10.

Subject 5.12: How do I do a hidden surface test (backface culling) with 3D points?

Just define all points of all polygons in clockwise order.

c = (x3*((z1*y2)-(y1*z2))+

x1,y1,z1, x2,y2,z2, x3,y3,z3 = the first three points of the

If c is positive the polygon is visible. If c is negative the
polygon is invisible (or the other way).

Subject 5.13: Where can I find algorithms for 3D collision detection?

Check out "proxima", from Purdue, which is a C++ library for 3D
collision detection for arbitrary polyhedra. It's a nice system;
the algorithms are sophisticated, but the code is of modest size.

For information about the I_COLLIDE 3D collision detection system

"Fast Collision Detection of Moving Convex Polyhedra", Rich Rabbitz,
Graphics Gems IV, pages 83-109, includes source in C.

SOLID: "a library for collision detection of three-dimensional
objects undergoing rigid motion and deformation. SOLID is designed to
be used in interactive 3D graphics applications, and is especially
suited for collision detection of objects and worlds described in VRML.
Written in standard C++, compiles under GNU g++ version 2.8.1 and
Visual C++ 5.0." See:

SWIFT++: a C++ library for collision detection, exact and approximate
distance computation, and contact determination of three-dimensional
polyhedral objects undergoing rigid motion.
Some preliminary results indicate that it is faster than I-COLLIDE and
V-CLIP, and more robust than I-COLLIDE.

ColDet: C++ library for 3D collison detection. Works on generic
polyhedra, and even polygon soups. Uses bounding box hierarchies
and triangle intersection tests. Released as open source under LGPL.
Tested on Windows, MacOS, and Linux. .

Terdiman's lib, which might need less RAM than the above:

Subject 5.14: How do I perform basic viewing in 3D?

We describe the shape and position of objects using numbers,
usually (x,y,z) coordinates. For example, the corners of a cube
might be {(0,0,0), (1,0,0), (1,1,0), (0,1,0), (0,0,1), (1,0,1),
(1,1,1), (0,1,1)}. A deep understanding of what we are saying with
these numbers requires mathematical study, but I will try to keep
this simple. At the least, we must understand that we have
designated some point in space as the origin--coordinates
(0,0,0)--and marked off lines in 3 mutually perpendicular
directions using equally spaced units to give us (x,y,z) values.
It might be helpful to know if we are using 1 to mean 1 foot, 1
meter, or 1 parsec; the numbers alone do not tell us.

A picture on a screen is two steps removed from the 3D world it
depicts. First, it is a 2D projection; and second, it is a finite
resolution approximation to the infinitely precise projection. I
will ignore the approximation (sampling) for now. To know what the
projection looks like, we need to know where our viewpoint is, and
where the plane of the projection is, both in the 3D world. Think
of it as looking out a window into a scene. As artists discovered
some 500 years ago, each point in the world appears to be at a
point on the window. If you move your head or look out a different
window, everything changes. When the mathematicians understood
what the artists were doing, they invented perspective geometry.

If your viewpoint is at the origin--(0,0,0)--and the window sits
parallel to the x-y plane but at z=1, projection is no more than
(x,y,z) in the world appears at (x/z,y/z,1) on the plane. Distant
objects will have large z values, causing them to shrink in the
picture. That's perspective.

The trick is to take an arbitrary viewpoint and plane, and
transform the world so we have the simple viewing situation.
There are two steps: move the viewpoint to the origin, then move
the viewplane to the z=1 plane. If the viewpoint is at (vx,vy,vz),
transform every point by the translation (x,y,z) -->
(x-vx,y-vy,z-vz). This includes the viewpoint and the viewplane.
Now we need to rotate so that the z axis points straight at the
viewplane, then scale so it is 1 unit away.

After all this, we may find ourselves looking out upside- down. It
is traditional to specify some direction in the world or viewplane
as "up", and rotate so the positive y axis points that way (as
nearly as possible if it's a world vector). Finally, we have acted
so far as if the window was the entire plane instead of a limited
portal. A final shift and scale transforms coordinates in the
plane to coordinates on the screen, so that a rectangular region
of interest (our "window") in the plane fills a rectangular region
of the screen (our "canvas" if you like).

Details of how to define and perform the rotation of the viewplane
have been left out, but see "How can I aim a camera in a specific
direction?" elsewhere in this FAQ. One simple way to designate a
plane is with the point closest to the origin, call it D. Then
a point P is on the plane if D.P = D.D; or using d = ||D|| and
N = D/d, if N.P = d. Aim the camera with N, and scale with d.

A further practical difficulty is the need to clip away parts of
the world behind us, so -(x,y,z) doesn't pop up at (x/z,y/z,1).
(Notice the mathematics of projection alone would allow that!) In
fact ordinarily a clipping box, the "viewing frustum", is used
to eliminate parts of the scene outside the window left or right,
top or bottom, and too close or too far.

All the viewing transformations can be done using translation,
rotation, scale, and a final perspective divide. If a 4x4
homogeneous matrix is used, it can represent everything needed,
which saves a lot of work.

Subject 5.15: How do I optimize/simplify a 3D polygon mesh?

"Mesh Optimization" Hoppe, DeRose Duchamp, McDonald, Stuetzle,
ACM COMPUTER GRAPHICS Proceedings, Annual Conference Series, 1993.

"Re-Tiling Polygonal Surfaces",
Greg Turk, ACM Computer Graphics, 26, 2, July 1992

"Decimation of Triangle Meshes", Schroeder, Zarge, Lorensen,
ACM Computer Graphics, 26, 2 July 1992

"Simplification of Objects Rendered by Polygonal Approximations",
Michael J. DeHaemer, Jr. and Michael J. Zyda, Computer & Graphics,
Vol. 15, No. 2, 1991, Great Britain: Pergamon Press, pp. 175-184.

"Topological Refinement Procedures for Triangular Finite Element
Procedures", S. A. Cannan, S. N. Muthukrishnan and R. P. Phillips,
Engineering With Computers, No. 12, p. 243-255, 1996.

"Progressive Meshes", Hoppe, SIGGRAPH 96,

Several papers by Michael Garland (quadric-based error metric):

By Stan Melax:
By Stefan Krause: [Gnu Open Source]
By "klaudius":

Subject 5.16: How can I perform volume rendering?

Two principal methods can be used:
- Ray casting or front-to-back, where the volume is behind the
projection plane. A ray is projected from each point in the projection
plane through the volume. The ray accumulates the properties of each
voxel it passes through.
- Object order or back-to-front, where the projection plane is behind
the volume. Each slice of the volume is projected on the projection
plane, from the farest plane to the nearest plane.

You can also use the marching-cubes algorithm, if the extraction of
surfaces from the data set is easy to do (see Subject 5.10).

Here is one algorithm to do front-to-back volume rendering:

Set up a projection plane as a plane tangent to a sphere that encloses
the volume. From each pixel of the projection plane, cast a ray
through the volume by using a Bresenham 3D algorithm.
The ray accumulates properties from each voxel intersected, stopping
when the ray exits the volume. The pixel value on
the projection plane determines the final color of the ray.

For unshaded images (i.e., without gradient and light computations),
if C is the ray color t the ray transparency
C' the new ray color t' the new ray transparency
Cv the voxel color tv the voxel transparency
C' = C + t*Cv and t' = t * tv
with initial values: C = 0.0 and t = 1.0

An alternate version: instead of C' = C + t*Cv , use :
C' = C + t*Cv*(1-tv)^p with p a float variable.
Sometimes this gives the best results.
When the ray has accumulated transparency, if it becomes negligible
(e.g., t<0.1), the process can stop and proceed to the next ray.


Bresenham 3D:
- [Gems IV] p. 366
Volume rendering:
- [Watt:Animation] pp. 297-321
- IEEE Computer Graphics and application
Vol. 10, Nb. 2, March 1990 - pp. 24-53
- "Volume Visualization"
Arie Kaufman - Ed. IEEE Computer Society Press Tutorial
- "Efficient Ray Tracing of Volume Data"
Marc Levoy - ACM Transactions on Graphics, Vol. 9, Nb 3, July 1990

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

See the Standard Procedural Databases software, whose official
distribution site is
This database contains much useful 3D code, including spline surface
tessellation, for the teapot.

Subject 5.18: How can the distance between two lines in space be computed?

Let x_i be points on the respective lines and n_i unit direction
vectors along the lines. Then the distance is
| (x_1 - x_0)^T (n_1 X n_0) | / || n_1 X n_0 ||.
Often one wants the points of closest approach as well as the distance.
The following is robust C code from Seth Teller that computes the
these points on two 3D lines. It also classifies
the lines as parallel, intersecting, or (the generic case) skew.
What's listed below shows the main ideas; the full code is at

// computes pB ON line B closest to line A
// computes pA ON line A closest to line B
// return: 0 if parallel; 1 if coincident; 2 if generic (i.e., skew)
line_line_closest_points3d (
POINT *pA, POINT *pB, // computed points
const POINT *a, const VECTOR *adir, // line A, point-normal form
const POINT *b, const VECTOR *bdir ) // line B, point-normal form
static VECTOR Cdir, *cdir = &Cdir;
static PLANE Ac, *ac = &Ac, Bc, *bc = &Bc;

// connecting line is perpendicular to both
vcross ( cdir, adir, bdir );

// check for near-parallel lines
if ( !vnorm( cdir ) ) {
*pA = *a; // all points are closest
*pB = *b;
return 0; // degenerate: lines parallel

// form plane containing line A, parallel to cdir
plane_from_two_vectors_and_point ( ac, cdir, adir, a );

// form plane containing line B, parallel to cdir
plane_from_two_vectors_and_point ( bc, cdir, bdir, b );

// closest point on A is line A ^ bc
intersect_line_plane ( pA, a, adir, bc );

// closest point on B is line B ^ ac
intersect_line_plane ( pB, b, bdir, ac );

// distinguish intersecting, skew lines
if ( edist( pA, pB ) < 1.0E-5F )
return 1; // coincident: lines intersect
return 2; // distinct: lines skew

Also Dave Eberly has code for computing distance between various
geometric primitives, including MinLineLine(), at

Subject 5.19: How can I compute the volume of a polyhedron?

Assume that the surface is closed, every face is a triangle, and
the vertices of each triangle are oriented ccw from the outside.
Let Volume(t,p) be the signed volume of the tetrahedron formed
by a point p and a triangle t. This may be computed by a 4x4
determinant, as in [O'Rourke (C), p.26].
Choose an arbitrary point p (e.g., the origin), and compute
the sum Volume(t_i,p) for every triangle t_i of the surface. That
is the volume of the object. The justification for this claim
is nontrivial, but is essentially the same as the justification for
the computation of the area of a polygon (Subject 2.01).

C Code available at
and .

For computing the volumes of n-d convex polytopes,
there is a C implementation by Bueeler and Enge of various
algorithms available at

http://www.Mathpool.Uni-Augsburg.DE/~enge/Volumen.html .

Subject 5.20: How can I decompose a polyhedron into convex pieces?

Usually this problem is interpreted as seeking a collection
of pairwise disjoint convex polyhedra whose set union is the
original polyhedron P. The following facts are known about
this difficult problem:

o Not every polyhedron may be partitioned by diagonals into
tetrahedra. A counterexample is due to Scho:nhardt
[O'Rourke (A), p.254].

o Determining whether a polyhedron may be so partitioned is
NP-hard, a result due to Seidel & Ruppert [Disc. Comput. Geom.
7(3) 227-254 (1992).]

o Removing the restriction to diagonals, i.e., permitting
so-called Steiner points, there are polyhedra of n vertices
that require n^2 convex pieces in any decomposition.
This was established by Chazelle [SIAM J. Comput.
13: 488-507 (1984)]. See also [O'Rourke (A), p.256]

o An algorithm of Palios & Chazelle guarantees at most
O(n+r^2) pieces, where r is the number of reflex edges
(i.e., grooves). [Disc. Comput. Geom. 5:505-526 (1990).]

o Barry Joe's geompack package, at,
includes a 3D convex decomposition Fortran program.

o There seems to be no other publicly available code.

Subject 5.21: How can the circumsphere of a tetrahedron be computed?

Let a, b, c, and d be the corners of the tetrahedron, with
ax, ay, and az the components of a, and likewise for b, c, and d.
Let |a| denote the Euclidean norm of a, and let a x b denote the
cross product of a and b. Then the center m and radius r of the
circumsphere are given by

| |
| |d-a|^2 [(b-a)x(c-a)] + |c-a|^2 [(d-a)x(b-a)] + |b-a|^2 [(c-a)x(d-a)] |
| |
r= -------------------------------------------------------------------------
| bx-ax by-ay bz-az |
2 | cx-ax cy-ay cz-az |
| dx-ax dy-ay dz-az |


|d-a|^2 [(b-a)x(c-a)] + |c-a|^2 [(d-a)x(b-a)] + |b-a|^2 [(c-a)x(d-a)]
m= a + ---------------------------------------------------------------------
| bx-ax by-ay bz-az |
2 | cx-ax cy-ay cz-az |
| dx-ax dy-ay dz-az |

Some notes on stability:

- Note that the expression for r is purely a function of differences between
coordinates. The advantage is that the relative error incurred in the
computation of r is also a function of the _differences_ between the
vertices, and is not influenced by the _absolute_ coordinates of the
vertices. In most applications, vertices are usually nearer to each other
than to the origin, so this property is advantageous.

Similarly, the formula for m incurs roundoff error proportional to the
differences between vertices, but not proportional to the absolute
coordinates of the vertices until the final addition.

- These expressions are unstable in only one case: if the denominator is
close to zero. This instability, which arises if the tetrahedron is
nearly degenerate, is unavoidable. Depending on your application, you
may want to use exact arithmetic to compute the value of the determinant.

Subject 5.22: How do I determine if two triangles in 3D intersect?

Let the two triangles be T1 and T2. If T1 lies strictly to one side
of the plane containing T2, or T2 lies strictly to one side of the
plane containing T1, the triangles do not intersect. Otherwise,
compute the line of intersection L between the planes. Let Ik
be the interval (Tk inter L), k=1,2. Either interval may be empty.
T1 and T2 intersect iff I1 and I2 overlap.

This method is decribed in Tomas Mo:ller, "A fast triangle-triangle
intersection test," J. Graphics Tools 2(2) 25-30 1997. C code at . See also

See also the "3D Object Intersection" page, described in Subject 0.05.

NASA's "Intersect" code will intersect any number of triangulated
surfaces provided that each of the surfaces is both closed and manifold.
Based on "Robust and Efficient Cartesian Mesh Generation for
Component-Based Geometry" AIAA Paper 97-0196. Michael Aftosmis.

Subject 5.23: How can a 3D surface be reconstructed from a collection of points?

This is a difficult problem. There are two main variants:
(1) when the points are organized into parallel slices through
the object; (2) when the points are unorganized.
For (1), see this survey:
D. Meyers, S. Skinner, K. Sloan. "Surfaces from Contours"
ACM Trans. Graph. 11(3) Jul 1992, 228--258.
Code (NUAGES) is available at
For (2), see this paper:
H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle
"Surface reconstruction from unorganized points"
Proc. SIGGRAPH '92, 71--78.
and P. Kumar's collection of links at
New code, Cocone, written with CGAL, based on recent work by
N. Amenta, S. Choi, T. K. Dey and N. Leekha:

Subject 5.24: How can I find the smallest sphere enclosing a set of points in 3D?

Although not obvious, the smallest sphere enclosing a set of points
in any dimension can be found by Linear Programming. This was proved
by Emo Welzl in the paper, "Smallest enclosing disks (balls and
ellipsoids)" [Lecture Notes Comput. Sci., 555, Springer-Verlag, 1991,
359--370]. + Code developed by Bernd Gaertner available (GNU
General Public License) at:
This code is remarkably efficient: e.g., 2 seconds for 10^6 points in
3D on an Ultra-Sparc II. See also Dave Eberly's direct implementation
of Welzl's algorithm:

Subject 5.25: What's the big deal with quaternions?

This could mean "Why do they evoke such heated debate?" or "What are
their virtues?"

The heat of debate is hard to explain, but it's been happening for many
decades. When Gibbs first deprecated the quaternion product and split it
into a cross product and a dot product, one outraged Victorian called
the result a "hermaphrodite monster" -- and that before the Internet's
flame wars. Generally, the quaternion advocates seem to feel the
opponents are lazy or thick-headed, and that deeper understanding of
quaternions would lead to deeper appreciation. The opponents don't
appreciate that attitude, and seem to feel the advocates are snooty or
sheep, and that matrices and such are less abstract and do just fine.
(Advocates of Clifford algebra would claim that both sides are mired in
the past.) Passion aside, quaternions have appropriate uses, as do their

Someone new to the debate first needs to know what quaternions are, and
what they're supposed to be good for. Quaternions are a quadruple of
numbers, used to represent 3D rotations.
q = [x,y,z,w] = [(x,y,z),w] = [V,w]

The "norm" of a quaternion N(q) is conventionally the sum of the squares
of the four components. Some writers confuse this with the magnitude,
which is the square root of the norm. Another common misconception is
that only quaternions of unit norm can be used, those with the sum of
the four squares equal to 1, but that is wrong (though they are
[U sin a,cos a] rotates by angle 2a around unit vector U

Popular non-quaternion options are 3x3 special orthogonal matrices (9
numbers with constraints), Euler angles (3 numbers), axis-angle (4
numbers), and angular velocity vectors (3 numbers). None of these
options actually _are_ rotations, which are physical; they _represent_
rotations. The distinction is important because, for example, it is
common to use an axis-angle with an angle greater than 360 degrees to
tell an animation system to spin an object more than a full turn,
something a matrix cannot say. In mathematics, the usual meaning of a
rotation would not allow the multiple spin version, which can lead to
confusing debates.
q and -q represent the same rotation

Two rotations, the physical things, can be applied one after the other.
Assuming the two rotation axes have a least one point in common, the
result will be another rotation. Some rotation representations handle
this gracefully, some don't. For quaternions and matrices, forms of
multiplication are defined such that the product gives the desired
result. For Euler angles especially there is no simple computation.
q = q2 q1 = [V2,w2][V1,w1] = [V2xV1+w2V1+w1V2,w2w1-V2.V1]

Every rotation has a reverse rotation, such that the combination of the
two leaves an object as it was. (Rotations are an algebraic "group".)
Euler angles make reversals difficult to compute. Other representations,
including quaternions, make them simple.
reverse([V,w]) = [V,-w]
q^(-1) = [-V,w]/(V.V+ww)

Two physical rotations are also more or less similar. Unit quaternions
do a particularly good job of representing similar rotations with
similar numbers.
similar(q1,q2) = |q1.q2| = | x1x2+y1y2+z1z2+w1w2 |

Points in space, the physical things, are normally represented as 3 or 4
numbers. The effect of a rotation on a collection of points can be
computed from the representation of the rotation, and here matrices seem
fastest, using three dot products. Using their own product twice,
quaternions are a bit less efficient. (They are usually converted to
matrices at the last minute.)
p2 = q p1 q^(-1)

Sequences of rotations can be interpolated, so that the object being
turned is rotated to specific poses at specific times. This motivated
Ken Shoemake's early use of quaternions in computer graphics, as
published in 1985. He used an analog of linear interpolation (sometimes
called "lerp") that he called "Slerp", and also introduced an analog of
a piecewise Bezier curve. A few years later in some course notes he
described another curve variation he called "Squad", which still seems
to be popular. Later authors have proposed many alternatives.
sin (1-t)A sin tA
Slerp(q1,q2;t) = q1 ---------- + q2 ------, cos A = q1.q2
sin A sin A

Squad(q1,a1,b2,q2;t) = Slerp(Slerp(q1,q2;t),

Physics simulation, aerospace control, and robotics are examples of
computations which also depend on rotation representation. Constrained
rotations like a wheel on an axle or the elbow bend of a robot typically
use specialized representations, such as an angle alone. In many general
situations, however, quaternions have proved valuable.
2 dq = W q dt, W is the angular velocity vector

User interfaces for 3D rotation also require a representation. Direct
manipulation interfaces typically use angles for jointed figures, but
for freer manipulation may use quaternions, as in Arcball or
through-the-lens camera control. As Shoemake's _full_ Graphics Gems code
for Arcball demonstrates (with the [CAPS LOCK] key), any rotation can be
graphed as an arc on a sphere. (Not to be confused with the quaternion
unit sphere in 4D.) Whether quaternions, or any other representation,
are helpful for numeric presentation and input seems a matter of taste
and circumstance.
q = U2 U1^(-1) = [U1xU2,U1.U2]

Because of their metric properties for representing rotations, unit
quaternions are most common. Advocates frequently point out that it is
far cheaper to normalize the length of a non-zero quaternion than to
bring a matrix back to rotation form. Also Shoemake's later conversion
code cheaply creates a correct rotation matrix from _any_ quaternion
(found with his Euler angle code from Graphics Gems, which does the same
for all 24 variations of that representation).
Normalize(q) = q/Sqrt(q.q)

Comparisons to Euler angles may mention "gimbal lock" (frequently
misspelled) as a disadvantage quaternions avoid. In the physical world
where gyroscopes are mounted on nested pivots, which are called gimbals,
locking is a real problem quaternions cannot help. What's usually meant
is that because the similarity of rotations organizes them somewhat like
a sphere, while similarity of vectors is quite different, an inevitable
misfit plagues Euler angles. This can show up in behavior much like
physical gimbal lock, but also in other ways. The difficulties are
topological, and aiming runs into them as well, even if quaternions are
used. Quaternion authors who propose using curves in the vector space of
quaternion logarithms often risk the misfit unawares. Frankly, you must
pick through the literature carefully, whether informal and online or
refereed and printed, because mistakes are tragically common.

To explore Graphics Gems code, see "Where is all the source?" in this
FAQ. To read more about quaternions, you have many options. Since they
were discovered in 1843 by Hamilton (the same Irish mathematician and
physicist whose name shows up in quantum mechanics), quaternions have
found many friends, as a web search will reveal. Quaternions can be
approached and applied in numerous different ways, so if you keep
looking it's likely you will find something that suits your taste and
your needs.

(Subject 0.04) [Eberly], Ch. 2.
Hamilton's original paper. Not easy for today's readers.
K. Shoemake. Animating Rotation with Quaternion Curves.
Proceedings of Siggraph 85.
Original animation method. Covers all the basics.
Later Shoemake tutorial. More complete than most authors.
Graphics Gems I-V, various authors, various articles.
As usual, a helpful source of code and discussion.
Henry Baker collects good quaternion stuff. Access spotty.
Henry Baker collection with more reliable access.
Visualizing quaternion rotation. May help build intuition.
The GL code implementing above Hart et al. paper.
Mathematical, but not too fast and not too fancy.
Laura Downs covers the fundamentals.
Ken Joy covers the fundamentals.
High-tech interpolation method. Demanding but rewarding.
Duff, Tom: Quaternion Splines for Animating Orientation,
Proceedings of the USENIX Association Second Computer Graphics
Workshop (held in Monterey, CA 12-13 Dec. 1985), pp. 54-62.
Subdivision paper in odd place. Author last seen at Pixar.

Subject 5.26: How can I aim a camera in a specific direction?

What's needed is a method for creating a rotation that turns one unit
vector to line up with another. To aim at an object, you can subtract
the position of the camera from the position of the object to get a
vector which you then normalize. The vector you want to turn is the
camera forward vector, commonly a unit vector along the camera -z axis.
Be warned that more than one rotation can achieve aim alone. (The issue
is rather complicated, as laid out in Ken Shoemake's article on twist
reduction in Graphics Gems IV.) For example, even if the camera is
already properly aimed you could rotate it around its z axis. The most
direct rotation is given by the non-unit quaternion
q = [(b,-a,0),1-c], to aim -z axis along unit vector (a,b,c)
Normalization is advised, but it will fail for aim vector (0,0,1). In
that case just rotate 180 degrees around the y axis, using
q = [(0,1,0),0]

If the camera is level after rotation by quaternion [(x,y,z),w], the y
component of its rotated x axis will be zero, so
xy+wz = 0
If it is upright, the y component of its rotated y axis will not be
negative, so
ww-xx+yy-zz >= 0

To ensure these two desirable properties, aim with a more sophisticated
non-unit quaternion
[(bs,-at,ab),st], where s = r-c, t = r+1, and r = sqrt(aa+cc).
This can also fail to normalize, in which case normalize instead
Unless the aim vector is null, this will succeed. If the aim vector has
not been normalized and its magnitude is m = sqrt(aa+bb+cc), substitute
1->m. That is, use t = r+m and use m+c.

More generally, to rotate unit vector U1 directly to unit vector U2, the
non-unit quaternion will be
q = [U1xU2,1+U1.U2]

Why? If U is a unit vector, then it is normal to a plane through the
origin with equation U.P = 0. Reflection in that plane is given by
reversing the U component of P.
reflect(P,U) = P ^Ö 2(U.P)U
The quaternion product of U1 and U2 is [U1xU2,-U1.U2], so
-2 (U.P) = PU + UP
Noting UU = -1, this gives a quaternion reflection formula.
reflect(P,U) = P + (PU+UP)U
= P ^Ö P + UPU
Reflecting with U1 then U2, by U2(U1 P U1)U2, rotates by twice the angle
between the planes, with axis perpendicular to both normals. Noting U1U2
is the conjugate of U2U1, and -q rotates like q, the rotation quaternion
q = -U2U1 = -[U2xU1,-U2.U1] = [U1xU2,U1.U2]
This q fails to aim U1 at U2 by rotating twice as much as needed, but
its square root succeeds. One square root of unit q is 1+q normalized,
geometrically the bisection of the great arc from the identity to q.
There is an inevitable singularity when U2 is the opposite of U1,
because any perpendicular axis gives an equally direct 180 degree

[These quaternion methods were provided by Ken Shoemake.]

Subject 5.27: How do I transform normals?

In 3D, the orientation of a plane in space can be given by a
vector perpendicular to the plane, a "normal vector" or "normal"
for short. Often it is convenient to keep that vector of unit
length, or "normalized"; be careful of the different meanings
of "normality". A smooth surface has a plane tangent to each
point, and by extension a normal to that plane is called a
"surface normal". Graphics code also cheats by associating
artificial normal vectors with the vertices of polygonal models
to simulate the reflection properties of curved surfaces;
these are called "vertex normals".

The "orientation" of a plane has two meanings, both of which
usually apply. Aside from the rotational tilting and turning
meaning, there is also the sense of "side". A closed convex
surface made of polygons has two sides, an inside and an
outside, and normals can be assigned to the polygons in such
a way that they all consistently point outside. This is
often desirable for shading and culling.

When a model is defined in one coordinate system and used in
another, as is commonly done, it may be necessary to transform
normals also. If the change of point coordinates is effected
by means of a rotation plus a translation, one simply rotates
the normals as well (with no translation). Other coordinate
changes need more care.

Although it is possible to use projective transformations to
map a model into world coordinates, ordinarily they are used
only for viewing. It is usually a mistake to apply perspective
to a normal, as shading and culling are best done in world
coordinates for correct results. Also perspective may be
computed using degenerate matrices which are not invertible,
though that need not be the case. For the importance of this,
see below.

The combination of a linear transformation and a translation
is called an affine transformation, and is performed as a
matrix multiplication plus a vector addition:
p' = A(p) = Lp + t.
When the model-to-world point transform is affine, the proper
way to transform normals is with the transpose of the inverse
of L.
n' = (L^{-1})^T n
However that is not enough.

If L includes scaling effects, a unit normal in model space
will usually transform to a non-unit normal, which can cause
problems for shaders and other code. This may need correcting
after the normal is transformed.

If L includes reflection, the inside-outside orientation of
the normal is reversed. This, too, can cause problems, and
may need correcting. The determinant of L will be negative
in this case.

When a complicated distortion is used, it must be approximated
differently at each point in space by a linear transform made
up of partial derivatives, the Jacobian. The matrix for the
Jacobian replaces L in the equation for transforming normals.

Why use the transposed inverse?
Write the dot product of column vectors n and v as a matrix
product n^T v. Write vector v as a difference of points, q-p.
Let p, q, and thus v lie in the desired plane, and let n be
normal to it. Vectors at right angles have zero dot product.
n^T v = 0
The transform v' of v is
v' = (Lq+t)-(Lp+t)
= (Lq-Lp)+(t-t)
= L(q-p)
= Lv
The transform n' of n will remain normal if it satisfies
n'^T v' = n^T v
Let n' = Mn for some M. Then the requirement is
n'^T v' = (Mn)^T (Lv)
= n^T (M^T L) v
= n^T v
This holds if
M^T L = I
where I is the identity. Right multiplying by the inverse
L^{-1} and transposing both sides gives, as claimed,
M = (L^{-1})^T
When L is a rotation, L^{-1} = L^T, so M = L. When L has no
inverse it will still have an "adjoint" to substitute for
for orthogonality purposes, differing only by a scale factor.

Section 6. Geometric Structures and Mathematics
Subject 6.01: Where can I get source for Voronoi/Delaunay triangulation?

For 2-d Delaunay triangulation, try Shewchuk's triangle program. It
includes options for constrained triangulation and quality mesh
generation. It uses exact arithmetic.

The Delaunay triangulation is equivalent to computing the convex hull
of the points lifted to a paraboloid. For n-d Delaunay triangulation
try Clarkson's hull program (exact arithmetic) or Barber and Huhdanpaa's
Qhull program (floating point arithmetic). The hull program also
computes Voronoi volumes and alpha shapes. The Qhull program also
computes 2-d Voronoi diagrams and n-d Voronoi vertices. The output of
both programs may be visualized with Geomview.

There are many other codes for Delaunay triangulation and Voronoi
diagrams. See Amenta's list of computational geometry software.
Especially noteworthy is the CGAL code: Subject 0.07 and

The Delaunay triangulation satisfies the following property: the
circumcircle of each triangle is empty. The Voronoi diagram is the
closest-point map, i.e., each Voronoi cell identifies the points that
are closest to an input site. The Voronoi diagram is the dual of
the Delaunay triangulation. Both structures are defined for general
dimension. Delaunay triangulation is an important part of mesh

There is a FAQ of polyhedral computation explaining how to compute
n-d Delaunay triangulation and n-d Voronoi diagram using a convex hull
code, and how to use the linear programming technique to
determine the Voronoi cells adjacent to a given Voronoi cell
efficiently for large scale or higher dimensional cases.

Avis' lrs code uses the same file formats as cdd. It
uses exact arithmetic and is useful
for problems with very large output size, since it does not
require storing the output.

On a closely related topic, see
for computation of the 3D medial axis from the Voronoi diagram.




Barber &



Polyhedral Computation FAQ:


[O' Rourke (C)] pp. 168-204

[Preparata & Shamos] pp. 204-225

Chew, L. P. 1987. "Constrained Delaunay Triangulations," Proc. Third
Annual ACM Symposium on Computational Geometry.

Chew, L. P. 1989. "Constrained Delaunay Triangulations," Algorithmica
4:97-108. (UPDATED VERSION OF CHEW 1987.)

Fang, T-P. and L. A. Piegl. 1994. "Algorithm for Constrained Delaunay
Triangulation," The Visual Computer 10:255-265. (RECOMMENDED!)

Frey, W. H. 1987. "Selective Refinement: A New Strategy for Automatic
Node Placement in Graded Triangular Meshes," International Journal for
Numerical Methods in Engineering 24:2183-2200.

Guibas, L. and J. Stolfi. 1985. "Primitives for the Manipulation of
General Subdivisions and the Computation of Voronoi Diagrams," ACM
Transactions on Graphics 4(2):74-123.

Karasick, M., D. Lieber, and L. R. Nackman. 1991. "Efficient Delaunay
Triangulation Using Rational Arithmetic," ACM Transactions on Graphics

Lischinski, D. 1994. "Incremental Delaunay Triangulation," Graphics
Gems IV, P. S. Heckbert, Ed. Cambridge, MA: Academic Press Professional.


Schuierer, S. 1989. "Delaunay Triangulation and the Radiosity
Approach," Proc. Eurographics '89, W. Hansmann, F. R. A. Hopgood, and
W. Strasser, Eds. Elsevier Science Publishers, 345-353.

Subramanian, G., V. V. S. Raveendra, and M. G. Kamath. 1994. "Robust
Boundary Triangulation and Delaunay Triangulation of Arbitrary
Triangular Domains," International Journal for Numerical Methods in
Engineering 37(10):1779-1789.

Watson, D. F. and G. M. Philip. 1984. "Survey: Systematic
Triangulations," Computer Vision, Graphics, and Image Processing

Subject 6.02: Where do I get source for convex hull?

For n-d convex hulls, try Clarkson's hull program (exact arithmetic)
or Barber and Huhdanpaa's Qhull program (floating point arithmetic).
Qhull 3.1 includes triangulated output and improved
handling of difficult inputs. Qhull computes convex hulls,
Delaunay triangulations, Voronoi diagrams, and halfspace
intersections about a point.

Qhull handles numeric precision problems by merging facets. The output
of both programs may be visualized with Geomview.

In higher dimensions, the number of facets may be much smaller than
the number of lower-dimensional features. When this is the case,
Fukuda's cdd program is much faster than Qhull or hull.

There are many other codes for convex hulls. See Amenta's list of
computational geometry software.



Barber &




[O' Rourke (C)] pp. 70-167
C code for Graham's algorithm on p.80-96.
Three-dimensional convex hull discussed in Chapter 4 (p.113-67).
C code for the incremental algorithm on p.130-60.

[Preparata & Shamos] pp. 95-184

Subject 6.03: Where do I get source for halfspace intersection?

For n-d halfspace intersection, try Fukuda's cdd program or Barber
and Huhdanpaa's Qhull program. Both use floating point arithmetic.
Fukuda includes code for exact arithmetic. Qhull handles numeric
precision problems by merging facets.

Qhull computes halfspace intersection by computing a convex hull.
The intersection of halfspaces about the origin is equivalent to the
convex hull of the halfspace coefficients divided by their offsets.
See Subject 6.02 for more information.


Barber &


[Preparata & Shamos] pp. 315-320

Subject 6.04: What are barycentric coordinates?

Let p1, p2, p3 be the three vertices (corners) of a closed
triangle T (in 2D or 3D or any dimension). Let t1, t2, t3 be real
numbers that sum to 1, with each between 0 and 1: t1 + t2 + t3 = 1,
0 <= ti <= 1. Then the point p = t1*p1 + t2*p2 + t3*p3 lies in
the plane of T and is inside T. The numbers (t1,t2,t3) are called the
barycentric coordinates of p with respect to T. They uniquely identify
p. They can be viewed as masses placed at the vertices whose
center of gravity is p.
For example, let p1=(0,0), p2=(1,0), p3=(3,2). The
barycentric coordinates (1/2,0,1/2) specify the point
p = (0,0)/2 + 0*(1,0) + (3,2)/2 = (3/2,1), the midpoint of the p1-p3
If p is joined to the three vertices, T is partitioned
into three triangles. Call them T1, T2, T3, with Ti not incident
to pi. The areas of these triangles Ti are proportional to the
barycentric coordinates ti of p.

[Coxeter, Intro. to Geometry, p.217].

Subject 6.05: How can I generate a random point inside a triangle?

Use barycentric coordinates (see item 53) . Let A, B, C be the
three vertices of your triangle. Any point P inside can be expressed
uniquely as P = aA + bB + cC, where a+b+c=1 and a,b,c are each >= 0.
Knowing a and b permits you to calculate c=1-a-b. So if you can
generate two random numbers a and b, each in [0,1], such that
their sum <=1, you've got a random point in your triangle.
Generate random a and b independently and uniformly in [0,1]
(just divide the standard C rand() by its max value to get such a
random number.) If a+b>1, replace a by 1-a, b by 1-b. Let c=1-a-b.
Then aA + bB + cC is uniformly distributed in triangle ABC: the reflection
step a=1-a; b=1-b gives a point (a,b) uniformly distributed in the triangle
(0,0)(1,0)(0,1), which is then mapped affinely to ABC.
Now you have barycentric coordinates a,b,c. Compute your point
P = aA + bB + cC.

Reference: [Gems I], Turk, pp. 24-28, contains a similar but different
method which requires a square root.

Subject 6.06: How do I evenly distribute N points on (tesselate) a sphere?

"Evenly distributed" doesn't have a single definition. There is a
sense in which only the five Platonic solids achieve regular
tesselations, as they are the only ones whose faces are regular
and equal, with each vertex incident to the same number of faces.
But generally "even distribution" focusses not so much on the
induced tesselation, as it does on the distances and arrangement
of the points/vertices. For example, eight points can be placed
on the sphere further away from one another than is achieved by
the vertices of an inscribed cube: start with an inscribed cube,
twist the top face 45 degrees about the north pole, and then
move the top and bottom points along the surface towards the equator
a bit.

The various definitions of "evenly distributed" lead into moderately
complex mathematics. This topic happens to be a FAQ on sci.math as well
as on; a much more extensive and rigorous
discussion written by Dave Rusin can be found at:

A simple method of tesselating the sphere is repeated subdivision. An
example starts with a unit octahedron. At each stage, every triangle in
the tesselation is divided into 4 smaller triangles by creating 3 new
vertices halfway along the edges. The new vertices are normalized,
"pushing" them out to lie on the sphere. After N steps of subdivision,
there will be 8 * 4^N triangles in the tesselation.

A simple example of tesselation by subdivision is available at

One frequently used definition of "evenness" is a distribution that
minimizes a 1/r potential energy function of all the points, so that in
a global sense points are as "far away" from each other as possible.
Starting from an arbitrary distribution, we can either use numerical
minimization algorithms or point repulsion, in which all the points
are considered to repel each other according to a 1/r^2 force law and
dynamics are simulated. The algorithm is run until some convergence
criterion is satisfied, and the resulting distribution is our approximation.

Jon Leech developed code to do just this. It is available at
See his README file for more information. His distribution includes
sample output files for various n < 300, which may be used off-the-shelf
if that is all you need.

Another method that is simpler than the above, but attains less
uniformity, is based on spacing the points along a spiral that
encircles the sphere.
Code available from links at

Subject 6.07: What are coordinates for the vertices of an icosohedron?

Data on various polyhedra is available at, or, or
For the icosahedron, the twelve vertices are:

(+- 1, 0, +-t), (0, +-t, +-1), and (+-t, +-1, 0)

where t = (sqrt(5)-1)/2, the golden ratio.
Reference: "Beyond the Third Dimension" by Banchoff, p.168.

Subject 6.08: How do I generate random points on the surface of a sphere?

There are several methods. Perhaps the easiest to understand is
the "rejection method": generate random points in an origin-
centered cube with opposite corners (r,r,r) and (-r,-r,-r).
Reject any point p that falls outside of the sphere of radius r.
Scale the vector to lie on the surface. Because the cube to sphere
volume ratio is pi/6, the average number of iterations before an
acceptable vector is found is 6/pi = 1.90986. This essentially
doubles the effort, and makes this method slower than the "trig
method." A timing comparison conducted by Ken Sloan showed that
the trig method runs in about 2/3's the time of the rejection method.
He found that methods based on the use of normal distributions are
twice as slow as the rejection method.

The trig method. This method works only in 3-space, but it is
very fast. It depends on the slightly counterintuitive fact (see
proof below) that each of the three coordinates is uniformly
distributed on [-1,1] (but the three are not independent,
obviously). Therefore, it suffices to choose one axis (Z, say)
and generate a uniformly distributed value on that axis. This
constrains the chosen point to lie on a circle parallel to the
X-Y plane, and the obvious trig method may be used to obtain the
remaining coordinates.

(a) Choose z uniformly distributed in [-1,1].
(b) Choose t uniformly distributed on [0, 2*pi).
(c) Let r = sqrt(1-z^2).
(d) Let x = r * cos(t).
(e) Let y = r * sin(t).

This method uses uniform deviates (faster to generate than normal
deviates), and no set of coordinates is ever rejected.

Here is a proof of correctness for the fact that the z-coordinate is
uniformly distributed. The proof also works for the x- and y-
coordinates, but note that this works only in 3-space.

The area of a surface of revolution in 3-space is given by

A = 2 * pi * int_a^b f(x) * sqrt(1 + [f'(x}]^2) dx

Consider a zone of a sphere of radius R. Since we are integrating in
the z direction, we have

f(z) = sqrt(R^2 - z^2)
f'(z) = -z / sqrt(R^2-z^2)

1 + [f'(z)]^2 = r^2 / (R^2-z^2)

A = 2 * pi * int_a^b sqrt(R^2-z^2) * R/sqrt(R^2-z^2) dz
= 2 * pi * R int_a^b dz
= 2 * pi * R * (b-a)
= 2 * pi * R * h,

where h = b-a is the vertical height of the zone. Notice how the integrand
reduces to a constant. The density is therefore uniform.

Here is simple C code implementing the trig method:

void SpherePoints(int n, double X[], double Y[], double Z[])
int i;
double x, y, z, w, t;

for( i=0; i< n; i++ ) {
z = 2.0 * drand48() - 1.0;
t = 2.0 * M_PI * drand48();
w = sqrt( 1 - z*z );
x = w * cos( t );
y = w * sin( t );
printf("i=%d: x,y,z=\t%10.5lf\t%10.5lf\t%10.5lf\n", i, x,y,z);
X[i] = x; Y[i] = y; Z[i] = z;

A complete package is available at (4K),
reachable from .

If one wants to generate the random points in terms of longitude
and latitude in degrees, these equations suffice:
Longitude = random * 360 - 180
Latitude = [arccos( random * 2 - 1 )/pi ] * 180 - 90

[Knuth, vol. 2]
[Graphics Gems IV] "Uniform Random Rotations"

Subject 6.09: What are Plucker coordinates?

A common convention is to write umlauted u as "ue", so you'll also see
"Pluecker". Lines in 3D can easily be given by listing the coordinates of
two distinct points, for a total of six numbers. Or, they can be given as
the coordinates of two distinct planes, eight numbers. What's wrong with
these? Nothing; but we can do better. Pluecker coordinates are, in a sense,
halfway between these extremes, and can trivially generate either. Neither
extreme is as efficient as Pluecker coordinates for computations.

When Pluecker coordinates generalize to Grassmann coordinates, as laid
out beautifully in [Hodge], Chapter VII, the determinant definition
is clearly the one to use. But 3D lines can use a simpler definition.
Take two distinct points on a line, say

P = (Px,Py,Pz)
Q = (Qx,Qy,Qz)

Think of these as vectors from the origin, if you like. The Pluecker
coordinates for the line are essentially

U = P - Q
V = P x Q

Except for a scale factor, which we ignore, U and V do not depend on the
specific P and Q! Cross products are perpendicular to their factors, so we
always have U.V = 0. In [Stolfi] lines have orientation, so are the same
only if their Pluecker coordinates are related by a positive scale factor.

As determinants of homogeneous coordinates, begin with the 4x2 matrix

[ Px Qx ] row x
[ Py Qy ] row y
[ Pz Qz ] row z
[ 1 1 ] row w

Define Pluecker coordinate Gij as the determinant of rows i and j, in that
order. Notice that Giw = Pi - Qi, which is Ui. Now let (i,j,k) be a cyclic
permutation of (x,y,z), namely (x,y,z) or (y,z,x) or (z,x,y), and notice
that Gij = Vk. Determinants are anti-symmetric in the rows, so Gij = -Gji.
Thus all possible Pluecker line coordinates are either zero (if i=j) or
components of U or V, perhaps with a sign change. Taking the w component
of a vector as 0, the determinant form will operate just as well on a
point P and vector U as on two points. We can also begin with a 2x4 matrix
whose rows are the coefficients of homogeneous plane equations, E.P=0,
from which come dual coordinates G'ij. Now if (h,i,j,k) is an even
permutation of (x,y,z,w), then Ghi = G'jk. (Just swap U and V!)

Got Pluecker, want points? No problem. At least one component of U is
non-zero, say Ui, which is Giw. Create homogeneous points Pj = Gjw + Gij,
and Qj = Gij. (Don't expect the P and Q that we started with, and don't
expect w=1.) Want plane equations? Let (i,j,k,w) be an even permutation of
(x,y,z,w), so G'jk = Giw. Then create Eh = G'hk, and Fh = G'jh.

Example: Begin with P = (2,4,8) and Q = (2,3,5). Then U = (0,1,3) and
V = (-4,6,-2). The direct determinant forms are Gxw=0, Gyw=1, Gzw=3,
Gyz=-4, Gzx=6, Gxy=-2, and the dual forms are G'yz=0, G'zx=1, G'xy=3,
G'xw=-4, G'yw=6, G'zw=-2. Take Uz = Gzw = G'xy = 3 as a suitable non-zero
element. Then two planes meeting in the line are

(G'xy G'yy G'zy G'wy).P = 0
(G'xx G'xy G'xz G'xw).P = 0

That is, a point P is on the line if it satisfies both these equations:

3 Px + 0 Py + 0 Pz - 6 Pw = 0
0 Px + 3 Py - 1 Pz - 4 Pw = 0

We can also easily determine if two lines meet, or if not, how they pass.
If U1 and V1 are the coordinates of line 1, U2 and V2, of line 2, we look
at the sign of U1.V2 + V1.U2. If it's zero, they meet. The determinant form
reveals even permutations of (x,y,z,w):
G1xw G2yz + G1yw G2zx + G1zw G2xy + G1yz G2xw + G1zx p2yw + G1xy p2zw

Two oriented lines L1 and L2 can interact in three different ways:
L1 might intersect L2, L1 might go clockwise around L2, or L1 might go
counterclockwise around L2. Here are some examples:

| L2 | L2 | L2
L1 | L1 | L1 |
-----+-----> -----------> -----|----->
| | |
intersect counterclockwise clockwise
| L2 | L2 | L2
L1 | L1 | L1 |
<----+----- <----|------ <-----------
| | |

The first and second rows are just different views of the same lines,
once from the "front" and once from the "back." Here's what they might
look like if you look straight down line L2 (shown here as a dot).

L1 ---------->
-----o----> L1 o L1 o
intersect counterclockwise clockwise

The Pluecker coordinates of L1 and L2 give you a quick way to test
which of the three it is.

cw: U1.V2 + V1.U2 < 0
ccw: U1.V2 + V1.U2 > 0
thru: U1.V2 + V1.U2 = 0

So why is this useful? Suppose you want to test if a ray intersects
a triangle in 3-space. One way to do this is to represent the ray and
the edges of the triangle with Pluecker coordinates. The ray hits the
triangle if and only if it hits one of the triangle's edges, or it's
"clockwise" from all three edges, or it's "counterclockwise" from all
three edges. For example...

o _
| |\ this picture, the ray
| \ is oriented counterclockwise
------ \ --> from all three edges, so it
| \ must intersect the triangle.
v \
o-----> o

Using Pluecker coordinates, ray shooting tests like this take only
a few lines of code.

Grassmann coordinates allow similar methods to be used for points,
lines, planes, and so on, and in a space of any dimension. In the
case of lines in 2D, only three coordinates are required:

Gxw = Px-Qx = -G'y
Gyw = Py-Qy = G'x
Gxy = PxQy-PyQx = G'w

Since P and Q are distinct, Giw is non-zero for i = x or y. Then

(Gix,Giy,Giw) is a point P1 on L
(Gxw,Gyw,Gww)+P1 is a point P2 on L
(G'x,G'y,G'w).P = 0 is an equation for L

Two lines in a 2D perspective plane always meet, perhaps in an
ideal point (meaning they're parallel in ordinary 2D). Calling
their homogeneous point of intersection P, the computation from
Grassmann coordinates nicely illustrates the convenience. (See
Subj 1.03, "How do I find intersections of 2 2D line segments?")
For i = x,y,w, set

Pi = G'j H'k ^Ö G'k H'j, where (i,j,k) is even

See [Hodge] for a thorough discussion of the theory, [Stolfi] for
a little theory with a concise implementation for low dimensions
(see Subj. 0.04), and these articles for further discussion:
J. Erickson, Pluecker Coordinates, Ray Tracing News 10(3) 1997,

Ken Shoemake, Pluecker Coordinate Tutorial,
Ray Tracing News 11(1) 1998,

Section 7. Contributors
Subject 7.01: How can you contribute to this FAQ?

Send email to with your suggestions, possible
topics, corrections, or pointers to information.

Subject 7.02: Contributors. Who made this all possible.

[All email addresses now removed to protect the authors from spam.]
Jens Alfke
Nina Amenta
Leen Ammeraal
Scott Anguish
Ian Ashdown
Brad Barber
James Beech
David Bouman
Paul Bourke
Lars Brinkhoff
Andrew Bromage
Brent Burley
R. Kevin Burton
Gene Caldwell
Ken Clarkson
Robert Day
Tamal Dey
Martin Dillon
Thomas Djafari
Dave Eberly
John Eickemeyer
John E (Edward) Ellis
Jeff Erickson
Ata Etemadi
Hugh Fisher
David N. Fogel
Arne K. Frick
Olexandr Frantchuk
Robert W. Fuentes
Komei Fukuda
William Gibbons
Normand Grégoire
Eric Haines
Jeff Hameluck
Sandy Harris
Luiz Henrique de Figueiredo
Steve Hollasch
Bill Jones
Richard Kinch
Craig Kolb
Uffe Kousgaard
Stefan Krause
Piyush Kumar
Steve Lamont
Ben Landon
Erik Larsen
Jon Leech
Michael V. Leonov
Sum Lin
Alan J. Livingston
Sebastien Loisel
Fritz Lott
Jacob Marner
Marc Christopher Martin
John McNamara
Samuel Murphy
Alan Murta
S. N. Muthukrishnan
Andrew Myers
David Nixon
Aaron Orenstein
Joseph O'Rourke
Samuel S. Paik
Leonidas Palios
Amitha Perera
Brian Peters
Lavoie Philippe
Christopher Phillips
Tom Plunket
Aaron Quigley
Rudi Bjxrn Rasmussen
Greg Roelofs
Christian von Roques
Dave Seaman
Jonathan R. Shewchuk
Rainer Michael Schmid
Klamer Schutte
Andrzej Serafin
ZhengYu Shan
James Sharman
Ken Shoemake
Jeff Somers
Jon Stone
Dan Sunday
Seth Teller
Saurabh Tendulkar
Yael "YoeL" Touboul
Anson Tsao
Bob van Manen
Remco Veltkamp
Jim Ward
Jason Weiler
Karsten Weiss
Stefan Wolfrum
Daniel S. Zwick

Previous Editors:
Jon Stone
Anson Tsao

Reply all
Reply to author
0 new messages