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From: orourke@cs.smith.edu (Joseph O'Rourke)
Newsgroups: comp.graphics.algorithms,news.answers,comp.answers
Subject: comp.graphics.algorithms Frequently Asked Questions
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Welcome to the FAQ for comp.graphics.algorithms!
Thanks to all who have contributed. Corrections and contributions
(to orourke@cs.smith.edu) always welcome.
----------------------------------------------------------------------
This article is Copyright 1997 by Joseph O'Rourke. It may be freely
redistributed in its entirety provided that this copyright notice is
not removed.
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Changed items this posting (|): 0.03
New items this posting (+): 1.05
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Table of Contents
----------------------------------------------------------------------
0. General Information
0.01: Charter of comp.graphics.algorithms
0.02: Are the postings to comp.graphics.algorithms archived?
| 0.03: How can I get this FAQ?
0.04: What are some must-have books on graphics algorithms?
0.05: Are there any online references?
0.06: Are there other graphics related FAQs?
0.07: Where is all the source?
1. 2D Computations: Points, Segments, Circles, Etc.
1.01: How do I rotate a 2D point?
1.02: How do I find the distance from a point to a line?
1.03: How do I find intersections of 2 2D line segments?
1.04: How do I generate a circle through three points?
| 1.05: How can the smallest circle enclosing a set of points be found?
1.06: Where can I find graph layout algorithms?
2. 2D Polygon Computations
2.01: How do I find the area of a polygon?
2.02: How can the centroid of a polygon be computed?
2.03: How do I find if a point lies within a polygon?
2.04: How do I find the intersection of two convex polygons?
2.05: How do I do a hidden surface test (backface culling) with 2d points?
2.06: How do I find a single point inside a simple polygon?
2.07: How do I find the orientation of a simple polygon?
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)?
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 polygon?
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 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?
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?
7. Contributors
7.01: How can you contribute to this FAQ?
7.02: Contributors. Who made this all possible.
Search e.g. for "Section 6" to find that section.
Search e.g. for "Subject 6.04" to find that item.
----------------------------------------------------------------------
Section 0. General Information
----------------------------------------------------------------------
Subject 0.01: Charter of comp.graphics.algorithms
comp.graphics.algorithms is an unmoderated newsgroup intended as a forum
for the discussion of the algorithms used in the process of generating
computer graphics. These algorithms may be recently proposed in
published journals or papers, old or previously known algorithms, or
hacks used incidental to the process of computer graphics. The scope of
these algorithms may range from an efficient way to multiply matrices,
all the way to a global illumination method incorporating raytracing,
radiosity, infinite spectrum modeling, and perhaps even mirrored balls
and lime jello.
It is hoped that this group will serve as a forum for programmers and
researchers to exchange ideas and ask questions on recent papers or
current research related to computer graphics.
comp.graphics.algorithms is not:
- for requests for gifs, or other pictures
- for requests for image translator or processing software; see
alt.binaries.pictures* FAQ
alt.binaries.pictures.utilities (picture source code)
alt.graphics.pixutils (image format translation)
comp.sources.misc (image viewing source code)
sci.image.processing
comp.graphics.apps.softimage
fj.comp.image
- for requests for compression software; for these try:
alt.comp.compression
comp.compression
comp.compression.research
----------------------------------------------------------------------
Subject 0.02: Are the postings to comp.graphics.algorithms archived?
Archives may be found at:
ftp://wuarchive.wustl.edu/graphics/graphics/mail-lists/comp.graphics.algorithms
It is archived in the same manner that all other newsgroups are
being archived there, namely there is an Index file with all the
subjects, and all the articles are being kept in a hierarchy based
on the year and month they are posted.
----------------------------------------------------------------------
|Subject 0.03: How can I get this FAQ?
The FAQ is posted on the 1st and 15th of every month. The easiest
way to get it is to search back in your news reader for the most
recent posting, with Subject:
comp.graphics.algorithms Frequently Asked Questions
It is posted to comp.graphics.algorithms, and cross-posted to
news.answers and comp.answers.
If you can't find it on your newsreader,
you can look at the latest HTML version at either of these two sites:
| http://www.exaflop.org/docs/cgafaq
http://www.cis.ohio-state.edu/hypertext/faq/usenet/graphics/algorithms-faq/faq.html
The exaflop version should be up-to-date and is nicely converted;
the ohio-state site is sometimes out of date.
Finally, you can ftp the FAQ from several sites, including:
ftp://rtfm.mit.edu/pub/faqs/graphics/algorithms-faq
ftp://ftp.seas.gwu.edu/pub/rtfm/comp/graphics/algorithms/comp.graphics.algorithms_Frequently_Asked_Questions
The (busy) rtfm.mit.edu site lists many alternative "mirror" sites.
Also can reach the FAQ from http://www.geom.umn.edu/software/cglist/,
which is worth visiting in its own right.
----------------------------------------------------------------------
Subject 0.04: What are some must-have books on graphics algorithms?
The keywords in brackets are used to refer to the books in later
questions. They generally refer to the first author except where
it is necessary to resolve ambiguity or in the case of the Gems.
Basic computer graphics, rendering algorithms,
----------------------------------------------
[Foley]
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.
[Rogers:Procedural]
Procedural Elements for Computer Graphics,
David F. Rogers, McGraw Hill 1985, ISBN 0-07-053534-5
[Rogers:Mathematical]
Mathematical Elements for Computer Graphics 2nd Ed.,
David F. Rogers and J. Alan Adams, McGraw Hill 1990, ISBN
0-07-053530-2
[Watt:3D]
_3D Computer Graphics, 2nd Edition_,
Alan Watt, Addison-Wesley 1993, ISBN 0-201-63186-5
[Glassner:RayTracing]
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
[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)
[Watt:Animation]
Advanced Animation and Rendering Techniques,
Alan Watt, Mark Watt, Addison-Wesley 1992, ISBN 0-201-54412-1
[Bartels]
An Introduction to Splines for Use in Computer Graphics and
Geometric Modeling,
Richard H. Bartels, John C. Beatty, Brian A. Barsky, 1987, ISBN
0-934613-27-3
[Farin]
Curves and Surfaces for Computer Aided Geometric Design:
A Practical Guide, 3rd Edition, Gerald E. Farin, Academic Press
1993. ISBN 0-12-249052-5
[Prusinkiewicz]
The Algorithmic Beauty of Plants,
Przemyslaw W. Prusinkiewicz, Aristid Lindenmayer, Springer-Verlag,
1990, ISBN 0-387-97297-8, ISBN 3-540-97297-8
[Oliver]
Tricks of the Graphics Gurus,
Dick Oliver, et al. (2) 3.5 PC disks included, $39.95 SAMS Publishing
[Hearn]
Introduction to computer graphics,
Hearn & Baker
[Cohen]
Radiosity and Realistic Imange Sythesis,
Michael F. Cohen, John R. Wallace, Academic Press Professional
1993, ISBN 0-12-178270-0
[Ebert]
Texturing and Modeling - A Procedural Approach
David S. Ebert (ed.), F. Kenton Musgrave, Darwyn Peachey, Ken Perlin,
Setven Worley, Academic Press 1994, ISBN 0-12-228760-6,
ISBN 0-12-2278761-4 (IBM disk)
[Schroeder]
Visualization Toolkit, The: An Object-Oriented Approach to 3-D
Graphics (Bk/CD) (Professional Description)
William J. Schroeder, Kenneth Martin and Bill Lorensen,
Prentice-Hall 1996, ISBN: 0-13-199837-4 (Published: 02/02/96)
See Subject 0.07 for source.
[Anderson]
PC Graphics Unleashed
Scott Anderson. SAMS Publishing, ISBN 0-672-30570-4
Ammeraal, L. (1992) Programming Principles in Computer Graphics,
2nd Edition, Chichester: John Wiley, ISBN 0 471 93128 4.
For image processing,
---------------------
[Barnsley]
Fractal Image Compression,
Michael F. Barnsley and Lyman P. Hurd, AK Peters, Ltd, 1993 ISBN
1-56881-000-8
[Jain]
Fundamentals of Image Processing,
Anil K. Jain, Prentice-Hall 1989, ISBN 0-13-336165-9
[Castleman]
Digital Image Processing,
Kenneth R. Castleman, Prentice-Hall 1996, ISBN(Cloth): 0-13-211467-4
(Description and errata at: "http://www.phoenix.net/~castlman")
[Pratt]
Digital Image Processing, Second Edition,
William K. Pratt, Wiley-Interscience 1991, ISBN 0-471-85766-1
[Gonzalez]
Digital Image Processing (3rd Ed.),
Rafael C. Gonzalez, Paul Wintz, Addison-Wesley 1992, ISBN
0-201-50803-6
[Russ]
The Image Processing Handbook,
John C. Russ, CRC Press 1992, ISBN 0-8493-4233-3
[Wolberg]
Digital Image Warping,
George Wolberg, IEEE Computer Society Press Monograph 1990, ISBN
0-8186-8944-7
Computational geometry,
----------------------
[Bowyer]
A Programmer's Geometry,
Adrian Bowyer, John Woodwark, Butterworths 1983,
ISBN 0-408-01242-0 Pbk
[O' Rourke]
Computational Geometry in C,
Joseph O'Rourke, Cambridge University Press 1994,
ISBN 0-521-44592-2 Pbk, ISBN 0-521-44034-3 Hdbk
[Goodman & O'Rourke]
Handbook of Discrete and Computational Geometry
J. E. Goodman and J. O'Rourke, editors.
CRC Press LLC, July 1997.
ISBN:0-8493-8524-5
[Samet:Application]
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.
[Mortenson]
Geometric Modeling,
Michael E. Mortenson, Wiley 1985, ISBN 0-471-88279-8
[Preparata]
Computational Geometry: An Introduction,
Franco P. Preparata, Michael Ian Shamos, Springer-Verlag 1985,
ISBN 0-387-96131-3
[Okabe]
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
A. Okabe and B. Boots and K. Sugihara,
John Wiley, Chichester, England, 1992.
Solid Modelling
---------------
[Mantyla]
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 February 1996, it contained 7,154 bib-tex
entries. It can be retrieved from
ftp://ftp.cs.usask.ca/pub/geometry/geombib.tar.Z - bibliography proper
ftp://ftp.cs.usask.ca/pub/geometry/geom.ps.Z - PostScript format
ftp://ftp.cs.usask.ca/pub/geometry/o-cgc19.ps.Z - overview published
in '93 in SIGACT News and the Internat. J. Comput. Geom. Appl.
ftp://ftp.cs.usask.ca/pub/geometry/ftp-hints - detailed retrieval info
The ACM SIGGRAPH Online Bibliography Project, by
Stephen Spencer (biblio@siggraph.org).
The database is available for anonymous FTP from the
ftp://siggraph.org/publications/bibliography directory. Please
download and examine the file READ_ME in that directory for more
specific information concerning the database.
'netlib' is a useful source for algorithms, member inquiries for
SIAM, and bibliographic searches. For information, send mail to
netlib@ornl.gov, with "send index" in the body of the mail
message.
You can also find free sources for numerical computation in C via
ftp://usc.edu/pub/C-numanal. In particular, grab
numcomp-free-c.gz in that directory.
Check out Nick Fotis's computer graphics resources FAQ -- it's
packed with pointers to all sorts of great computer graphics
stuff. This FAQ is posted biweekly to comp.graphics.
This WWW page contains links to a large number
of computer graphic related pages:
http://www.dataspace.com:84/vlib/comp-graphics.html
There's a Computer Science Bibliography Server at:
http://glimpse.cs.arizona.edu:1994/bib/
with Computer Graphics, Vision and Radiosity sections
A comprehensive bibliography of color quantization papers and articles is
available at ftp://hobbes.lbl.gov/pub/doc/cquant95.txt
Modelling physically based systems for animation:
http://www.cc.gatech.edu/gvu/animation/Animation.html
The University of Manchester NURBS Library:
ftp://unix.hensa.ac.uk/pub/misc/unix/nurbs/
For an implementation of Seidel's algorithm for fast trapezoidation
and triangulation of polygons. You can get the code from:
ftp://ftp.cs.unc.edu/pub/users/narkhede/triangulation.tar.gz
Ray tracing bibliography:
ftp://ftp.eye.com/pub/graphics/papers/rtbib95.tar.Z
ftp://ftp.eye.com/pub/graphics/papers/rtbib95.zip
Quaternions and other comp sci curiosities:
ftp://ftp.netcom.com/pub/hb/hbaker/hakmem/hakmem.html
Directory of Computational Geometry Software,
collected by Nina Amenta (nina@geom.umn.edu)
Nina Amenta is maintaining a WWW directory to computational
geometry software. The directory lives at The Geometry Center.
It has pointers to lots of convex hull and voronoi diagram programs,
triangulations, collision detection, polygon intersection, smallest
enclosing ball of a point set and other stuff.
http://www.geom.umn.edu/software/cglist/lowdvod.html
A compact reference for real-time 3d computer graphics programming:
http://www.cs.mcgill.ca/~zed
----------------------------------------------------------------------
Subject 0.06: Are there other graphics related FAQs?
BSP Tree FAQ by Bretton Wade
http://reality.sgi.com/bspfaq/
Gamma and Color FAQs by Charles A. Poynton has
ftp://ftp.inforamp.net/pub/users/poynton/doc/colour/
http://www.inforamp.net/~poynton/
The documents are mirrored to space provided by Fraunhofer Computer
Graphics in Rhode Island, U.S.A. at
ftp://elaine.crcg.edu/pub/doc/colour/
in Darmstadt, Germany at
ftp://ftp.igd.fhg.de/pub/doc/colour/
----------------------------------------------------------------------
Subject 0.07: Where is all the source?
Graphics Gems source code.
http://www.acm.org/tog/GraphicsGems/
This site is now the offical distribution site for Graphics Gems code.
Master list of Computational Geometry software:
http://www.geom.umn.edu/software/cglist
Described in [Goodman & O'Rourke], Chap. 52.
Jeff Erikson's software list:
http://www.cs.duke.edu/~jeffe/compgeom/code.html
General 'stuff'
ftp://wuarchive.wustl.edu/graphics/graphics
There are a number of interesting items in
http://theory.lcs.mit.edu/~seth including:
- Code for 2D Voronoi, Delaunay, and Convex hull
- Mike Hoymeyer's implementation of Raimund Seidel's
O( d! n ) time linear programming algorithm for
n constraints in d dimensions
- geometric models of UC Berkeley's new computer science
building
You can find useful overviews of a number of computer graphic
topics in http://archpropplan.auckland.ac.nz/People/Paul/Paul.html
including:
- area/orientation of polygons
- finding if a point lies within a polygon
- generating a circle through 3 points
- description and psuedo-code for Delaunay triangulation
- basic viewing in 3D
etc.
Sources to "Computational Geometry in C", by J. O'Rourke
can be found at ftp://grendel.csc.smith.edu/pub/compgeom
Greg Ferrar has uploaded his heavily commented C++ 3D rendering
library at ftp://ftp.math.ohio-state.edu/pub/users/gregt
TAGL is a portable and extensible library that provides a subset
of Open-GL functionalities.
ftp://sunsite.unc.edu/pub/packages/programming/graphics/tagl21.tgz
Try ftp://x2ftp.oulu.fi for /pub/msdos/programming/docs/graphpro.lzh by
Michael Abrash. His XSharp package has an implementation of Xiaoulin
Wu's anti-aliasing algorithm (in C).
Example sources for BSP tree algorithms can be found in
ftp://ftp.qualia.com/pub/bspfaq/
Mel Slater (mel@dcs.qmw.ac.uk) also made some implementations of
BSP trees and shadows for static scenes using shadow volumes
code available
http://www.dcs.qmw.ac.uk/~mel/BSP.html
ftp://ftp.dcs.qmw.ac.uk/people/mel/BSP
The Visualization Toolkit (A visualization textbook, C++ library
and Tcl-based interpreter) (see [Schroeder]):
http://iuinfo.tuwien.ac.at:8000/htdocs/vtk/
See also 5.17:
Where can I get the spline description of the famous teapot etc.?
----------------------------------------------------------------------
Section 1. 2D Computations: Points, Segments, Circles, Etc.
----------------------------------------------------------------------
Subject 1.01: How do I rotate a 2D point?
In 2-D, the 2x2 matrix 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 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 2-D
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 (XC,YC) and the line be AB (XA,YA) to (XB,YB).
The length of the line segment AB is L:
L=((XB-XA)**2+(YB-YA)**2)**0.5
and
(YA-YC)(YA-YB)-(XA-XC)(XB-XA)
r = -----------------------------
L**2
(YA-YC)(XB-XA)-(XA-XC)(YB-YA)
s = -----------------------------
L**2
Let I be the point of perpendicular projection of C onto AB, the
XI=XA+r(XB-XA)
YI=YA+r(YB-YA)
Distance from A to I = r*L
Distance from C to I = s*L
If r<0 I is on backward extension of AB
If r>1 I is on ahead extension of AB
If 0<=r<=1 I is on AB
If s<0 C is left of AB (you can just check the numerator)
If s>0 C is right of AB
If s=0 C is on AB
----------------------------------------------------------------------
Subject 1.03: How do I find intersections of 2 2D line segments?
This problem can be extremely easy or extremely difficult depends
on your applications. 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
XA+r(XB-XA)=XC+s(XD-XC)
YA+r(YB-YA)=YC+s(YD-YC) for some r,s in [0,1]
Solving the above for r and s yields
(YA-YC)(XD-XC)-(XA-XC)(YD-YC)
r = ----------------------------- (eqn 1)
(XB-XA)(YD-YC)-(YB-YA)(XD-XC)
(YA-YC)(XB-XA)-(XA-XC)(YB-YA)
s = ----------------------------- (eqn 2)
(XB-XA)(YD-YC)-(YB-YA)(XD-XC)
Let I be the position vector of the intersection point, then
I=A+r(B-A) or
XI=XA+r(XB-XA)
YI=YA+r(YB-YA)
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 coincident
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, I is located on extension of AB
If r<0, I is located on extension of BA
If s>1, I is located on extension of CD
If s<0, I is located on extension of DC
Also note that the denominators of eqn 1 & 2 are identical.
References:
[O'Rourke] 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
Reference:
[O' Rourke] 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, p.195ff]. For a direct algorithm, see:
| S. Skyum, "A simple algorithm for computing the smallest enclosing circle"
| Inform. Process. Lett. 37 (1991) 121--125.
----------------------------------------------------------------------
Subject 1.06: Where can I find graph layout algorithms?
See the paper by Eades and Tamassia, "Algorithms for Drawing
Graphs: An Annotated Bibliography," Tech Rep CS-89-09, Dept. of
CS, Brown Univ, Feb. 1989.
A revised version of the annotated bibliography on graph drawing
algorithms by Giuseppe Di Battista, Peter Eades, and Roberto
Tamassia is now available from
ftp://wilma.cs.brown.edu/pub/papers/compgeo/gdbiblio.tex.gz and
ftp://wilma.cs.brown.edu/pub/papers/compgeo/gdbiblio.ps.gz
----------------------------------------------------------------------
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.
References: [O' Rourke] pp. 18-27; [Gems II] pp. 5-6:
"The Area of a Simple Polygon", Jon Rokne.
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 one fixed vertex
v0 of the polygon, and the two endpoints of consecutive edges of
the polygon: (v1,v2), (v2,v3), etc. The area of a triangle with
vertices a, b, c is half of this expression:
(b[X] - a[X]) * (c[Y] - a[Y]) -
(c[X] - a[X]) * (b[Y] - a[Y]);
Code available at ftp://grendel.csc.smith.edu/pub/code/centroid.c (3K).
Reference: [Gems IV] pp.3-6; also includes code.
----------------------------------------------------------------------
Subject 2.03: How do I find if a point lies within a polygon?
The definitive reference is "Point in Polyon Strategies" by
Eric Haines [Gems IV] pp. 24-46.
The code in the Sedgewick book Algorithms (2nd Edition, p.354) is
incorrect.
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 <wrf@ecse.rpi.edu>
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 the references below for more detail.
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.
References:
[Gems IV] pp. 24-46
[O'Rourke] pp. 233-238
[Glassner:RayTracing]
----------------------------------------------------------------------
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], pp. 242-252.
This walks around the boundaries of P and Q in concert, intersecting
edges. An implementation is included in [O'Rourke].
----------------------------------------------------------------------
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, 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, 35-39], and is part
of many other software packages. See Subject 0.07: Where is all the
source?
----------------------------------------------------------------------
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. One can find the lowest, rightmost
point of the polygon, 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 ftp://grendel.csc.smith.edu/pub/code/polyorient.C (2K).
The reason that the lowest, rightmost 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).
----------------------------------------------------------------------
Section 3. 2D Image/Pixel Computations
----------------------------------------------------------------------
Subject 3.01: How do I rotate a bitmap?
The easiest way, according to the comp.graphics 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
matrix:
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)
where
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.
Reference:
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)
"An Introductory Course in Computer Graphics" by Richard Kingslake,
(2nd edition) published by Chartwell-Bratt ISBN 0-86238-284-X
[Gems I]
[Foley]
[Hearn]
----------------------------------------------------------------------
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.
The Laplace is the kernal:
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
ftp://ftp.cs.unc.edu/pub/users/eberly/magic.
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. Many
times bitmaps have another limitation imposed: The values canot be
negative. The resolution limitation is perhaps most important.
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]
+4.0*(A[i-1][j]+A[i][j-1]+A[i][j+1]+A[i+1][j])
-20.0*A[i][j])/6.0
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
+A[i][j-1]+A[i][j+1]-A[i][j-3]-A[i][j+3])/16
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
-A[i][j-1]-A[i][j+1]+A[i][j-3]+A[i][j+3])/16
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?
For the general solution to the problem get Ata Etemadi's software
package and associated papers from one of the addresses below.
It's very fast and detects polygons too.
The Object Recognition Tookit:
http://www2.team17.com/~aetemadi/archive.html
----------------------------------------------------------------------
Subject 3.08: Where do I get source to display (raster font format)?
ftp://oak.oakland.edu/SimTel/msdos/graphics
----------------------------------------------------------------------
Subject 3.09: What is morphing/how is it done?
See [Anderson], Chapter 3, page 59 - 90.
----------------------------------------------------------------------
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::/ftp.uwp.edu/pub/msdos/demos/programming/source
ftp::/ftp.luth.se/pub/msdos/demos (Sweden)
ftp::/NCTUCCCA.edu.tw:/PC/uwp/demos
http://www.wit.com:/mirrors/uwp/pub/msdos/demos
ftp::/ftp.cdrom.com:/demos
----------------------------------------------------------------------
Subject 3.11: 3D Noise functions and turbulence in Solid texturing.
See:
ftp://gondwana.ecr.mu.oz.au/pub/siggraph92_C23.shar
ftp://ftp.cis.ohio-state.edu/siggraph92/siggraph92_C23.shar
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:
ftp://avalon.chinalake.navy.mil/utils/SPD/SPD33f4.tar.Z
ftp://avalon.chinalake.navy.mil/utils/SPD/spd33f4.zip.
Now moved to http://www.viewpoint.com/
References:
[Ebert]
Noise, Hypertexture, Antialiasing and Gesture, (Ken Perlin) in
Chapter 6, (p.193-), The disk accompanying the book is available
from
ftp://archive.cs.umbc.edu/pub/texture.
For more info on this text/code see:
http://www.cs.umbc.edu/~ebert/book/book.html
For examples from a current course based on this book, see:
http://www.seas.gwu.edu/graphics/ProcTexCourse/
[Watt:Animation]
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:
ftp://avalon.chinalake.navy.mil/utils/SPD/SPD33f4.tar.Z
ftp://avalon.chinalake.navy.mil/utils/SPD/spd33f4.zip.
Now moved to http://www.viewpoint.com/
Reaction-Diffusion Algorithms:
For an illustartion of the parameter space of a reaction
diffusion system, check out the Xmorphia page at
http://www.ccsf.caltech.edu/ismap/image.html
References:
[Ebert]
Entire book devoted to this subject, with RenderMan(TM) and C code.
[Watt:Animation]
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)
http://www.cs.unc.edu:80/~turk/reaction_diffusion/reaction_diffusion.html
A list of procedural texture synthesis related web pages
http://www.threedgraphics.com/pixelloom/tex_synth.html
----------------------------------------------------------------------
Subject 3.13: How do I convert between color models (RGB, HLS, CMYK, CIE etc)?
References:
[Watt:3D] pp. 313-354
[Foley] pp. 563-603
----------------------------------------------------------------------
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.
----------------------------------------------------------------------
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 2-D or 3-D. 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 3-D. 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,
f111.
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.
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).
----------------------------------------------------------------------
Section 5. 3D computations
----------------------------------------------------------------------
Subject 5.01: How do I rotate a 3D point?
Assuming 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 3-D, 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 3-D vector pointing along the axis. The concept
of an axis is unique to 3-D. 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
* definitions of FPOINT, ANGLE, VECTOR, QUAT, MATRIX, MUL, HALF,
* TWICE, COS, SIN, ONE, and ZERO.
* The following is an example of floating point definitions.
#define FPOINT double
#define ANGLE FPOINT
#define VECTOR QUAT
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)
{
QUAT q;
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:
#define POINT VECTOR
*/
QUAT MatrixFromAnyAxisAngle(POINT o, VECTOR axis, ANGLE theta, MATRIX m)
{
QUAT q;
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)
{
VECTOR u;
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. */
FPOINT scl = RECIPROCAL(SQRT(norm));
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)
{
QUAT q;
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 3-D 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
ftp::/ftp.cis.upenn.edu/pub/graphics.
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.
----------------------------------------------------------------------
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. Any of the references listed in
the FAQ should have it. 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.)
----------------------------------------------------------------------
Subject 5.04: How do I clip a polygon against another polygon?
Klamer Schutte, klamer@ph.tn.tudelft.nl has developed and implemented
some code in C++ to perform clipping of two possibly concave 2-D
polygons. A description can be found at:
http://www.ph.tn.tudelft.nl:/People/klamer/clippoly_entry.html
To compile the source code you will need a C++ compiler with templates,
such as g++. The source code is available at:
ftp://ftp.ph.tn.tudelft.nl/pub/klamer/clippoly.tar.gz
References:
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
with:
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 polygon
First find the intersection between the ray and the plane in which
the polygon is situated. Then see if the point of intersection is
in the polygon.
Reference:
[Glassner:RayTracing]
----------------------------------------------------------------------
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
where:
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.
Reference:
[Glassner:RayTracing]
----------------------------------------------------------------------
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.
Reference:
[Glassner:RayTracing]
----------------------------------------------------------------------
Subject 5.09: How do I ray trace caustics?
There is a discussion in [Watt:Animation], but I don't know how
good it is.
It's hard to get right.
One right (but incredibly expensive) answer is:
@InProceedings{mitchell-1992-illumination,
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:
@Article{inakage-1986-caustics,
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:
@Article{yuan-1988-gemstone,
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
surface.
- ----- + - ----- - - ----- + - ----- +
|:::' | |:::::::| |:::: | | ':::|
|:' | |:::::::| |:::: | |. ':|
| | | | |:::: | |::. |
+ ----- + + ----- + - ----- + + ----- -
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.
References:
"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
[Schroeder]
----------------------------------------------------------------------
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
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
----------------------------------------------------------------------
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))+
(y3*((x1*z2)-(z1*x2))+
(z3*((y1*x2)-(x1*y2))+
x1,y1,z1, x2,y2,z2, x3,y3,z3 = 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 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.
ftp://ftp.cs.purdue.edu/pub/pse/PROX/prox9.1.tar.Z
For information about the I_COLLIDE 3D collision detection system
http://www.cs.unc.edu/~geom/I_COLLIDE.html
"Fast Collision Detection of Moving Convex Polyhedra", Rich Rabbitz,
Graphics Gems IV, pages 83-109, includes source in C.
----------------------------------------------------------------------
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
points 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).
I have left out details of how you define and perform the rotation
of the viewplane, but I'm sure someone else will be happy to
supply those if there is demand. It requires knowing how to
describe a plane, and how to rotate vectors to line up. Neither is
difficult, but this is already using a lot of net space. One
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!) But
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 a 3D polygon mesh
References:
"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 ObjectsRendered 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.
----------------------------------------------------------------------
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
then:
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.
References:
Bresenham 3D:
- http://www.sct.gu.edu.au/~anthony/info/graphics/bresenham.procs
- [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
http://www.acm.org/tog/resources/SPD/
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?
The following is robust C code from Seth Teller that computes the
"points of closest approach" on two 3D lines. It also classifies
the lines as parallel, intersecting, or (the generic case) skew.
// 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)
int
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
else
return 2; // distinct: lines skew
}
----------------------------------------------------------------------
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, 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).
----------------------------------------------------------------------
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.
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
generation.
References:
Amenta: http://www.geom.umn.edu/software/cglist
Barber & http://www.geom.umn.edu/locate/qhull
Huhdanpaa ftp://geom.umn.edu/pub/software/qhull.tar.Z
Clarkson: http://cm.bell-labs.com/netlib/voronoi/hull.html
ftp://cm.bell-labs.com/netlib/voronoi/hull.shar.Z
Geomview: http://www.geom.umn.edu/locate/geomview
ftp://geom.umn.edu/pub/software/geomview/
Shewchuk: http://www.cs.cmu.edu/~quake/triangle.html
ftp://cm.bell-labs.com/netlib/voronoi/triangle.shar.Z
[O' Rourke] 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
10(1):71-91.
Lischinski, D. 1994. "Incremental Delaunay Triangulation," Graphics
Gems IV, P. S. Heckbert, Ed. Cambridge, MA: Academic Press Professional.
(INCLUDES C++ SOURCE CODE -- THANK YOU, DANI!)
[Okabe]
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
26:217-223.
----------------------------------------------------------------------
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 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.
References:
Amenta: http://www.geom.umn.edu/software/cglist/
Barber & http://www.geom.umn.edu/locate/qhull
Huhdanpaa ftp://geom.umn.edu/pub/software/qhull.tar.Z
Clarkson: http://cm.bell-labs.com/netlib/voronoi/hull.html
ftp://cm.bell-labs.com/netlib/voronoi/hull.shar.Z
Geomview: http://www.geom.umn.edu/locate/geomview
ftp://geom.umn.edu/pub/software/geomview/
Fukuda: ftp://ifor13.ethz.ch/pub/fukuda/cdd/
[O' Rourke] 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.
References:
Barber & http://www.geom.umn.edu/locate/qhull
Huhdanpaa ftp://geom.umn.edu/pub/software/qhull.tar.Z
Fukuda: ftp://ifor13.ethz.ch/pub/fukuda/cdd/
[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
edge.
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.
Reference:
[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.
----------------------------------------------------------------------
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 comp.graphics.algorithms; a much more extensive and rigorous
discussion written by Dave Rusin can be found at:
http://www.math.niu.edu/~rusin/papers/known-math/spheres/sphere.faq
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
ftp://ftp.cs.unc.edu/pub/users/leech/sphere.c
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
ftp://ftp.cs.unc.edu/pub/users/leech/points.tar.gz.
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 at ftp://grendel.csc.smith.edu/pub/code/sphere.tar.gz (4K)
----------------------------------------------------------------------
Subject 6.07: What are coordinates for the vertices of an icosohedron?
Data on various polyhedra is available at
http://cm.bell-labs.com/netlib/polyhedra/index.html, or
http://netlib.bell-labs.com/netlib/polyhedra/index.html, or
http://www.netlib.org/polyhedra/index.html
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?
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.
Code available at ftp://grendel.csc.smith.edu/pub/code/sphere.tar.gz (4K)
----------------------------------------------------------------------
Section 7. Contributors
----------------------------------------------------------------------
Subject 7.01: How can you contribute to this FAQ?
Send email to orourke@cs.smith.edu with your suggestions, possible
topics, corrections, or pointers to information.
----------------------------------------------------------------------
Subject 7.02: Contributors. Who made this all possible.
Jens Alfke <jens_alfke@powertalk.apple.com>
Leen Ammeraal <ammeraal@ibk.fnt.hvu.nl>
Scott Anguish <sanguish@digifix.com>
Brad Barber <barber@geom.umn.edu, barber@tiac.net>
Paul Bourke <pdbourke@ccu1.auckland.ac.nz>
Andrew Bromage <bromage@mundil.cs.mu.OZ.AU>
Brent Burley <brent@aimla.com>
R. Kevin Burton <engr@waun.tdsnet.com>
Ken Clarkson <clarkson@research.att.com>
Stephen Darnell <Stephen.Darnell@isltd.insignia.com>
Martin Dillon <martind@mv.pi.csiro.au>
Thomas Djafari <frogger@micronet.fr>
Dave Eberly <eberly@cs.unc.edu>
John Eickemeyer <johne@iti.gov.sg>
John E (Edward) Ellis <je_ellis@ccmail.pnl.gov>
Ata Etemadi <Ata.Etemadi@team17.com>
Eric Haines <erich@eye.com>
Luiz Henrique de Figueiredo <lhf@visgraf.impa.br><lhf@csgrs6k4.uwaterloo.ca>
Andrew Fitzgibbon <andrewfg@aifh.ed.ac.uk>
Robert W. Fuentes <fuentesrw@lmtas.lmco.com>
David N. Fogel <fogel@geog.ucsb.edu>
Sandy Harris <sandy@storm.ca>
Steve Hollasch <hollasch@kgc.com>
Craig Kolb <cek@Princeton.EDU>
Steve Lamont <spl@szechuan.ucsd.edu>
Jon Leech <leech@cs.unc.edu>
Sum Lin <slin@esri.com>
Sebastien Loisel <zed@sgi.com>
Fritz Lott <fritz@riverside.MR.Net>
Marc Christopher Martin <mcm@post5.tele.dk>
John McNamara <jmn.ac@fusilla.abanet.it>
Samuel Murphy <sammy@icarus.smds.com>
Aaron Orenstein <aorenste@atitech.com>
Joseph O'Rourke <orourke@cs.smith.edu>
Samuel S. Paik <paik@mlo.dec.com>
Christopher Phillips <christopher@rflect.demon.co.uk>
Tom Plunket <plunket@eidetic.com>
Christian von Roques <roques@pond.sub.org>
Dave Seaman <ags@seaman.cc.purdue.edu>
Klamer Schutte <klamer@ph.tn.tudelft.nl>
ZhengYu Shan <shan@pace10.wh.lucent.com>
James Sharman <James@exaflop.demon.co.uk>
Ken Shoemake <shoemake@graphics.cis.upenn.edu>
Jeff Somers <jsomers@tiac.net>
Jon Stone <jdstone@ingr.com>
Seth Teller <seth@larch.lcs.mit.edu>
Yael "YoeL" Touboul <Yael.Touboul@ifp.fr>
Anson Tsao <ansont@hookup.net>
Jim Ward <jfw@radix.net>
Jason Weiler <weilej@rpi.edu>
Karsten Weiss <karsten@addx.stgt.sub.org>
Previous Editors:
Jon Stone <jdstone@ingr.com>
Anson Tsao <ansont@hookup.net>
Internal Links
Parent Article: Quad Tree Search Project
