Xref: info.physics.utoronto.ca comp.answers:7731 comp.graphics.algorithms:9057 news.answers:30585 Path: destin!jdstone From: jdstone@destin.dazixco.ingr.com (Jon Stone) Newsgroups: comp.graphics.algorithms,comp.answers,news.answers Subject: comp.graphics.algorithms Frequently Asked Questions (FAQ) Supersedes:

Followup-To: comp.graphics.algorithms Date: 9 Oct 1994 10:00:06 GMT Organization: Intergraph Corp., Boulder CO Lines: 1369 Approved: news-answers-request@MIT.EDU Expires: 22 Nov 1994 10:00:03 GMT Message-ID: Reply-To: jdstone@ingr.com NNTP-Posting-Host: destin.dazixco.ingr.com Summary: This posting contains a list of Frequently Asked Questions (and their answers) about computer graphics algorithms. It should be read by anyone who wishes to post to the comp.graphics.algorithms newsgroup. Archive-name: graphics/algorithms-faq Version: 1.14 Last-Modified: September 29, 1994 Posting-Frequency: monthly Welcome to the FAQ for comp.graphics.algorithms! Thanks to all who have contributed. Corrections and contributions always welcome. Changed items are marked with a |. New items are marked with a +. Items needing input are marked with a ?. All ftp references are of the form ftp://node/path All Mosaic references are of the form http://node/path ---------------------------------------------------------------------- Table of Contents ---------------------------------------------------------------------- 0) Charter of comp.graphics.algorithms | 1) Are the postings to comp.graphics.algorithms archived? 2) What are some must have books on graphics algorithms? 3) Are there any online references? 4) Where is all the source? 5) How do I rotate a 2D point? 6) How do I rotate a 3D point? 7) How do I find the distance from a point to a line? 8) How do I find intersections of 2 2D line segments? 9) How do I find the intersection of a line and a plane? 10) How do I rotate a bitmap? 11) How do I display a 24 bit image in 8 bits? 12) How do I fill the area an arbitrary shape? 13) How do I find the 'edges' in a bitmap? ? 14) How do I enlarge/sharpen/fuzz a bitmap? 15) How do I map a texture on to a shape? 16) How do I find the area/orientation of a polygon? 17) How do I find if a point lies within a polygon? ? 18) How do I find the intersection of two convex polygons? 19) How do I detect a 'corner' in a collection of points? 20) How do I generate a circle through three points? 21) How do I generate a bezier curve that is parallel to another bezier? 22) How do I split a bezier at a specific value for t? 23) How do I find a t value at a specific point on a bezier? 24) How do I fit a bezier curve to a circle? 25) What is ARCBALL? 26) Where can I find ARCBALL source? 27) How do I clip a polygon against a rectangle? 28) How do I clip a polygon against another polygon? ? 29) Where can I get source for Weiler/Atherton clipping? ? 30) How do I generate a spline to approximate (insert curve here)? ? 31) Where do I get source to display (raster font format)? ? 32) What is morphing/how is it done? ? 33) How do I draw an anti-aliased line/polygon/ellipse? 34) How do I determine the intersection between a ray and a polygon? 35) How do I determine the intersection between a ray and a sphere? 36) How do I determine the intersection between a ray and a bezier surface? 37) How do I ray trace caustics? 38) How do I quickly draw a filled triangle? 39) Where can I get source for Voronoi/Delaunay triangulation? 40) Where do I get source for convex hull? 41) What is the marching cubes algorithm? ? 42) What is the status of the patent on the "marching cubes" algorithm? 43) How do I do a hidden surface test (backface culling) with 3d points? 44) How do I do a hidden surface test (backface culling) with 2d points? 45) Where can I find graph layout algorithms? ? 46) Where can I find algorithms for 2D collision detection? 47) Where can I find algorithms for 3D collision detection? 48) 3D Noise functions and turbulence in Solid texturing. 49) How do I perform basic viewing in 3d? ? 50) How can you contribute to this FAQ? 51) Contributors. Who made this all possible. ---------------------------------------------------------------------- Subject: 0) 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 ray tracing, 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 software (i.e. gif <--> jpg) ---------------------------------------------------------------------- Subject: 1) Are the postings to comp.graphics.algorithms archived? | Yes. The "official" archive is stored at: | | http://www.cis.ohio-state/hypertext/faq/usenet/graphics/algorithms-faq/faq.html | ftp://rtfm.mit.edu/pub/usenet-by-group/news.answers/graphics/algorithms-faq | | Also available 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. | FYI, all usenet FAQ's are available with Mosaic via | | http://www.cis.ohio-state/hypertext/faq/usenet/top.html ---------------------------------------------------------------------- Subject: 2) 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 [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) [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) [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 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 1979, ISBN 0-13-212365-7 [Pratt] Digital Image Processing, Second Edition, William K. Pratt, Wiley-Interscience 1991, ISBN 0-471-85766-1 [Gonzalez] Digital Image Processing (2nd Ed.), Rafael C. Gonzalez, Paul Wintz, Addison-Wesley 1987, ISBN 0-201-11026-1 [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 [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 ---------------------------------------------------------------------- Subject: 3) 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 September 1993, it contained 5356 bib-tex entries. It can be retrieved from ftp://cs.usask.ca/pub/geometry/geombib.tar.Z - bibliography proper ftp://cs.usask.ca/pub/geometry/geom.ps.Z - PostScript format 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://cs.usask.ca/pub/geometry/ftp-hints - detailed retrieval info Announcing 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. ---------------------------------------------------------------------- Subject: 4) Where is all the source? Graphics Gems source code. ftp://princeton.edu:pub/Graphics/GraphicsGems General 'stuff' ftp://wuarchive.wustl.edu/graphics/graphics ---------------------------------------------------------------------- Subject: 5) 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: 6) 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. ---------------------------------------------------------------------- Subject: 7) 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: 8) 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: 9) 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) If the denominator is zero, then the vector (a,b,c) and the vector (i,j,k) are perpendicular. Note that (a,b,c) is the normal to the plane and (i,j,k) is the direction of the line. It follows that the line is either parallel to the plane or contained in the plane. In either case there is no unique intersection point. ---------------------------------------------------------------------- Subject: 10) 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: 11) 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: 12) 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: 13) 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". ---------------------------------------------------------------------- Subject: 14) How do I enlarge/sharpen/fuzz a bitmap? ---------------------------------------------------------------------- Subject: 15) 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: 16) How do I find the area/orientation of a polygon? Compute the signed area. The orientation is counter-clockwise if this area is positive. There's a Gem on computing signed areas. 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. 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). The key formula is this: If the coordinates of vertex v_i are x_i and y_i, twice the 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}). Reference: [O' Rourke] pp. 18-27. ---------------------------------------------------------------------- Subject: 17) How do I find if a point lies within a polygon? A quick comment - the code in the Sedgewick book Algorithms is wrong. The short answer, for the FAQ, could be: 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 (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: 50) How can you contribute to this FAQ? Send email to jdstone@ingr.com with your suggestions, possible topics, corrections, or pointers to information. Remember, I am not an expert on many of these topics. I'm the editor. Here are some possible topics that may interest many of our readers. Clipping... Splines Nurbs Image Warping/Transformation/Filtering Anti-aliasing Volume Rendering Morphing (synonymous with generalized Warping) MPEG JPEG Z-buffer/A-buffer/etc. interpolation (linear, spline, fft, etc.) Modeling tricks (fractal mountains, trees, seashells) Surfaces Ray Tracing Reflection/Refraction 1) Computing the minimum bounding boxes of various geometric elements such as circular arcs, parabolas, clothoids, splines, etc. What is the most efficient way to do them for the following cases: i) The boxes are all orthogonal to the XY plane; ii) The boxes is oriented such that the smallest area rectangular bounding boxes are computed. 2) What is the most efficient way to tell if a polygon crosses itself? i.e. without intersecting every edge with every edge. ---------------------------------------------------------------------- Subject: 51) Contributors. Who made this all possible. andrewfg@aifh.ed.ac.uk (Andrew Fitzgibbon) atae@spva.ph.ic.ac.uk (Ata Etemadi) atsao@tkk.win.net (Anson Tsao) barber@geom.umn.edu (Brad Barber) bromage@mundil.cs.mu.OZ.AU (Andrew James Bromage) cek@Princeton.EDU (Craig Kolb) fritz@riverside.MR.Net (Fritz Lott) hollasch@kgc.com (Steve Hollasch) jens_alfke@powertalk.apple.com (Jens Alfke) karsten@addx.stgt.sub.org (Karsten Weiss) lhf@visgraf.impa.br (Luiz Henrique de Figueiredo) orourke@cs.smith.edu (Joseph O'Rourke) paik@mlo.dec.com (Samuel S. Paik) sammy@icarus.smds.com (Samuel Murphy) sanguish@digifix.com (Scott Anguish) shoemake@graphics.cis.upenn.edu (Ken Shoemake) slin@esri.com (Sum Lin) spl@szechuan.ucsd.edu (Steve Lamont) weilej@rpi.edu (Jason Weiler) $Id: algorithms,v 1.14 1994/09/29 00:54:31 jdstone Exp $ -- ---------------------------------------------------------------------- Jon Stone Intergraph Corporation jdstone@ingr.com Boulder, CO ----------------------------------------------------------------------

The views and opinions stated within this web page are those of the author or authors which wrote them and may not reflect the views and opinions of the ISP or account user which hosts the web page. The opinions may or may not be those of the Chairman of The Skeptic Tank.

Return to The Skeptic Tank's main Index page.

E-Mail Fredric L. Rice / The Skeptic Tank