"Light Makes Right"
December 2, 1997
Volume 10, Number 3
All contents are copyright (c) 1997, all rights reserved by the individual authors
Archive locations: text version at
HTML at http://www.acm.org/tog/resources/RTNews/html/
You may also want to check out the Ray Tracing News issue guide and the Mother of all Ray Tracing Pages.
As the on-line editor of ACM TOG I finally made a personal web page, http://www.acm.org/tog/editors/erich/, as much for myself as for anyone else - it has links to a number of my papers online, links which I usually have to dig up. At the bottom of my page are some different rendering experiments which might be of interest. Holly Rushmeier, Editor-in-Chief of ACM TOG, also now has a web page, http://www.acm.org/tog/editors/hollyr/, with links to her papers available online. Her pages caused me to burn a good hour surfing some of her co-authors' personal pages, which had some fascinating links.
back to contents
Andrei Sherstyuk Department of Computer Science Monash University Clayton, Victoria 3168 Australia [email protected] http://www.cs.monash.edu.au/~ash
My research interests are: general ray-tracing, modeling/rendering implicit surfaces, convolution surfaces and new methods in ray/surface interactions.
I have written a ray-tracer that is faster than POV-ray and almost as fast as Rayshade. It allows modeling, rendering and animation of implicit surfaces based on skeleton design, ala Jules Bloomenthal's convolution surfaces. Samples of my work, including papers, pictures and animations may be found at my home page. I am a Ph.D. student, finishing some time 1998.
back to contents
Graphics Interface '97 Papers
Electronic copies of all the papers from GI '97 are available from the conference's web page:
Hardcopies of the proceedings may be purchased from Morgan-Kaufmann Publishers. Their web page's URL is the following:
- A. T. Campbell ([email protected])
Fast, Minimum Storage Ray-Triangle Intersection
This paper by Tomas Möller and Ben Trumbore was recently published in the "journal of graphics tools" http://jgt.akpeters.com/. The abstract, code and statistics (but not the algorithm description) is available online at:
I think this is an important paper. The intersection method is comparable to other more traditional methods, with the important advantage that the triangle's normal does not have to be stored in order to efficiently compute the intersection.
Near Real Time RT on a Pentium
A fast (1-2 fps on a Pentium 90) ray trace demo with reflections and shadows on curved surfaces can be found at:
It was faster than expected.
- Eric Haines
Quadric Java Raytracer
I have a neat applet for newbies to Java and / or raytracing. I've written a Java applet that allows web users to interactively render all of the Quadric Surfaces - and my home planet - from any perspective, set by using a custom built, mouse & keyboard responsive, integrated cylindrical and spherical coordinate system graphical user interface:
"Imaging the Imagined: Raytracing tips from da Vinci and me." Raytracing and computer imaging: the math, theory and practice (10+ pages):
[Notable as the first use of a supermodel to help explain ray tracing. -EAH]
- Paul Flavin ([email protected])
65535 Dimensional Ray Tracer
I've written some ray-tracing code and am looking for some feedback. I haven't written any sort of interface, so I can't post an .exe yet, but please let me know what you think of the concept.
Basically, I've written the most gimmick-laden ray-tracer I have ever seen. Gimmicks include:
At the moment, it doesn't do color or textures and only polynomial objects are allowed (stuff like cylinders don't generalize to 65535 dimensions too well).
Basically, I was wondering if anyone would be interested in the finished result. First indications are that it will be *very* tricky to use, and let's face it, pretty much useless. The idea was so cool that I had to do it anyway.
- Mike Ferenduros ([email protected])
There's a lot of software out there on FTP sites which, because it's FTP, does not get caught by web search engines. There is a search engine specifically for FTP:
(There are U.S. sites which also run this engine, but with annoying ad banners.)
UNC Polygon Manipulation Packages
RAPID - Robust and Accurate Polygon Interference Detection: this is a free C++ package available for non-commercial use, available from:
There is also a paper on OBB-Trees, the data structure on which the software is based. Related to this package is V_COLLIDE; see:
for more information. Key features of the package include application to all polygonal models, support of dynamic addition and deletion of objects from an environment and fast overlap tests and incremental computation.
There is also a package for automatic simplification of polygonal models:
and also an improved fast polygon triangulator based on Seidel's algorithm:
This triangulator appeared in Graphics Gems V; since then the authors have improved the code so that it handles holes - a significant addition. (I do not know if the code is robust enough to handle coincident edges).
Mesh decimation, Delaunay triangulation, and similar software is available at http://miles.cnuce.cnr.it/cg/swOnTheWeb.html
CGAL, the Computational Geometry Algorithms Library, is written in C++ and is available free of charge under mild conditions. It is the result of an ESPRIT project conducted by seven leading research groups in computational geometry. It is at:
- Remco Veltkamp ([email protected])
[This library contains a large number of standard computational geometry algorithms, such as convex hull, Delaunay triangulation, etc. - EAH]
Flying Text rendered as GIF animations by POV-Ray while you wait, for free:
- Richard C. Church
[animated GIFs are to 1997 as textured backgrounds are to 1996 and large image maps are to 1995 - too slow for us modem-bound mortals, used too much, and generally not a good idea. That said, this site's a cute idea and I enjoyed it. Make your own eye-candy easily and free. -EAH]
MAC POV-Ray Resources by Shawn Fumo ([email protected])
Here are some places that you can find more information:
Macintosh POV-Ray Resources http://home.pages.de/~mac-pov-resources/
This page has a good amount of information, but unfortunately some of it is
outdated (such as Mondfarilo being out now, etc).
POV-Ray Mac OS Information http://ourworld.compuserve.com/homepages/povraymac/
This is an official page which is done by one of the POV-Ray Team members. A
lot of good information here.
Shawn Fumo's POV-Ray Utilities List: http://www.the-spa.com/shawn.fumo/povutils/
This one is my listing. It is IMHO one of the more complete listings of utilities for any platform. They are sorted by topic, and not platform, but if you do a search for MAC, you should be able to find the entries. It can also usually be found posted in this very group.
Another thing to remember is that all of the ones listed as Include Files can
be used on any platform. There may not be a whole lot of binary utilities for
the MAC, but the include files are your ticket into utility heaven. =)
the PILE http://user.baden-online.de/~mbachman/pile/pile_eng.htm
No discussion of Include Files can be made without mentioning the PILE! When
it comes to include files, this site surpasses all others (including my own!).
Gautam N. Lad's POV-Ray Page http://www.interlog.com/~gautam/povpage.html
You may also be able to find something of interest here. He has a comprehensive listing of a lot of different utilities, general links, and include files.
Thanks to David C. Browne ([email protected]) a WIN32 port of rayshade is now available for ftp at:
- Christoph Kukulies ([email protected])
A reflectance model visualizer is available in SGI executable form and C++ source code form at
The program should be portable to other machines that support the Xforms user interface library and X windows without too much difficulty.
This program is a fairly general viewer for bidirectional reflectance distribution functions (BRDF's). It was written by some students and myself in a graduate course in global illumination that I taught in Fall 1996.
The program implements the following reflectance models:
I demonstrated the program at the Eurographics Workshop on Rendering in St. Etienne, France, and there was enough interest that I decided to release it more generally.
- Paul Heckbert ([email protected])
[There is another BRDF viewer at http://www-graphics.stanford.edu/~smr/brdf/bv/ which works on SGI/IRIX and i386/Linux, and GNU & Mesa source code. -EAH]
The RenderPark renderer currently offers:
It is known to compile and run fine on a SUN Sparcstation (with LEO graphics hardware and the Nth Portable GL), under Linux (with Mesa, a free OpenGL like graphics library) and on a number of SGI machines (with OpenGL).
You can retrieve the source code and example scene files (MGF format) from the URL:
- Philippe Bekaert ([email protected])
There is all sorts of information there about using quaternions.
- Pat Fleckenstein ([email protected])
There are extensive statistics for comparisons using SPD scenes for the GX/GENERIC, FTPOV, and POV ray tracers at:
[A hybrid algorithm of Hybrid VistaBuffer/BSP/LightBuffer wins out as the fastest scheme of those tested, beating ABVH and BSP. -EAH]
There is also an article here about generating near-optimal BSP trees for ray tracing, reminiscent of some of K.R. Subramanian's work. Finally, there is a draft of a paper on the work at:
- Juhu ([email protected])
Errata for "Principles of Digital Image Synthesis" is available at:
Terrain Forge, at:
is a heightfield generator written for Windows 95. It exports to POV-Ray and other formats.
There is a walking motion library by Theresa Willis for POV-Ray at:
(as well as some other tidbits). Theresa's site is:
Guy Rauscher is working on the next generation of Theresa's program, called the Character Animation System. More information can be found at the following site:
[there is also a fun little flag-waving include file for POV-Ray].
- Shawn Fumo ([email protected])
Real Time Stuff
Here's yet another free Wintel 3D engine:
by Chong Jiayi. I haven't a clue if it's any good, but the page is certainly nice enough.
And here are some little fire and water demos:
by Kimmo Roimela.
Parallel Graphics Library
The initial public release of the PGL graphics library is now available for downloading at (http://www.icase.edu/~tom/PGL/).
PGL is a parallel polygon rendering package designed for use in SPMD-style parallel applications running on distributed memory architectures. It exploits the power of the parallel platform to perform rendering operations in place, enabling applications to generate live visual output without having to move large data sets across the network for subsequent post-processing. PGL includes a core set of 3D graphics routines which provide modeling, rendering, and display operations, and auxiliary components which support higher-level graphics and visualization functions, image transport, and user interaction.
- Tom Crockett ([email protected])
Genesis 3D is available at:
It is a surface modeler that does lathe objects, sweeps along paths, and booleans. For Windows 95, and it's free.
The FLIRT (Faster than LIght Ray Tracer) by Wolfgang Stuerzlinger is available at:
for those of you collecting ray tracers. Documentation's in German. This site also has some other graphics code - discontinuity mesher, constrained Delaunay triangulator, and other tidbits.
"An Empirical Comparison of Radiosity Algorithms" is available at:
Some of our conclusions:
- Paul Heckbert ([email protected])
3D BSP Trees
I have made a 3D BSP compiler and there are some pages about algorithms for 3D BSP:
The best FAQ about BSP is:
- Andrea Marchini ([email protected])
[and don't forget the BSP software in Graphics Gems V by Norman Chin - I've never used it, but heard it works well. -EAH]
Mathematic URL Grab-Bag
[passed on by Paul Flavin ([email protected]) - EAH]
http://forum.swarthmore.edu/~steve/steve/mathgraphics.html - a ton of links mathematics and graphics related resources.
http://euclid.math.fsu.edu/Science/Software.html - mathematical software index.
http://www-sci.lib.uci.edu/HSG/RefCalculators.html - 5000+ different calculator programs, on just an amazing array of areas. Aztec date calculator, Honeymoon calculator, direction to Mecca calculator, you name it. (Sadly, gone; try here or here, for example.)
back to contents
For some interesting variants on potential functions for implicit surfaces, see Alfonso Hermida's article in this issue. Also see the "New People" section for someone doing implicit surface ray tracing work - I enjoyed the "implicit seafood" images at Andrei's web site.
The second book is David Rogers' "Procedural Elements for Computer Graphics, 2nd edition", WCB/McGraw-Hill. This volume is almost twice as long as the first edition. There are many minor additions and clarifications, and major new sections on ray tracing, color quantization, radiosity, texturing, and color reproduction. This book is worth having if you need practical advice and pseudocode for common graphics algorithms, be it drawing an ellipse or making an octree. This book gives you the tools which you will use again and again (or, like me, reinvent because you did not read the book carefully the first time - more on that later). Like any book with a lot of solid information, there are bound to be a few minor bugs (more on that later, too). But, from my brief skimming of it, this book looks like a well-crafted work.
Another book in the pipeline is "Rendering with Radiance: The Art and Science of Lighting Visualization" accompanied by a CD-ROM, which should be available from Morgan-Kaufmann (http://www.mkp.com) by early 1998. It is authored by Greg Ward Larson and Rob Shakespeare. Radiance (http://radsite.lbl.gov/radiance/HOME.html) is an excellent free physically based rendering system which has been developed for more than a decade now. The book will cover the uses of Radiance and include individual application chapters by a number of users of the system.
back to contents
I finished my thesis and had some interesting results:
The interesting thing is, Glassner's technique could use these things as well.
So, is my method better than Glassner's? We can't really tell yet.
In regards to my testing scenes, I came up with a few, that had varying characteristics; you can see these anims (in .fli format) at:
My actual thesis is at:
The source code is at:
back to contents
The "diffuse cosine power factor": you can accurately render the moon, if you like. :-) When you start playing with it, you'll quickly notice, that LOTS of things look less plastic when you start fooling with that parameter! It's not just the moon who has this quality, that's for sure. (Wood looks better, for instance).
Non-linear addition of light intensities: That started as a request from one of the two people who ever bought Raytracker while it was commercial.... his company made booths for exhibitions... routinely scattered with TONS of spotlights. His problem - to get light levels right. Either everything was too dark, and the lit areas of the spotlights were okay... or everything was okay, and the spotlight-lit areas was burned out.
The problem is, that if you tested things in real life, with actual spotlights and brought out an actual camera and snapped a photo, then sure, you MIGHT have gotten the same (or similar) results. But f*ck the camera, stand there and use your EYES. The eye is highly nonlinear, and has an AMAZING adaptability to light levels. You don't perceive the dark areas to be as dark as "they really are", or the light areas to be as light as "they really are". You perceive a smaller difference.
So, what I did, is that I created a brightness correction scheme.
Basically, I had a SPLINE that I set up in a cfg file, which defined what the output value should be for a given input value. I think the default spline went like this:
In Out 0.0 0.0 0.7 0.5 1.0 0.8 2.0 0.9 3.0 1.0
That means, that for an input value of 0, you got 0 out. For input of 0.7, you got 0.5 out, etc.
The problem was, where to apply this curve?
In the first generation of the code, it was applied to everything before it went to the screen - which gave the whole thing a much broader "brightness dynamics range", i.e. lighting equations could output values in the range 0 - 3 without burning out the display.
The problem was, applying the curve to RGB individually had bad effects on the saturation of the screen. (Everything became more white).
So I fixed this by, instead of calculating on RGB, summing up RGB to intensity (using factors 0.4, 0.5, 0.1 or somesuch), then I pushed that "brightness" through the equation, and multiplied all RGB with the new brightness divided by the old brightness.
Anyway, the end result looked a bit dull - because of the nonlinearity of the shading. Sometimes it had interesting effects, and it might help the problem.... but sometimes it didn't.
So, I made a different approach.
Instead of applying this color correction at the end of everything, where it just squashed up the luminosity dynamic range of the output - I applied it at the root of the problem - the lights.
So, when I summed up my light, I applied the correction there instead.
Now things get tricky to explain... you see, the Raytracker shading model is a bit strange....
For each light, I calculate two "types" of light. The "light to be multiplied" with the surface color (i.e. the diffuse light, e.g. Ka + Kd) and the "light to be added" to the the surface regardless of its color - i.e. the specular light. (This isn't 100% true, since Raytracker also has a specular-color-affected-by-surface-color-factor, which, when it is 1, the specular reflection is calculated into the "light to be multiplied" (i.e. gets the surfaces color), but when it is zero, the specular light is put in the "light to be added" (i.e. gets the lights color), and when it's 0.5 the specular light is divided fifty-fifty into these two types of light).
Anyway, to get a long story long, once I had calculated these two "kinds" of light, both were affected by my color correction curve. Then, the final intensity at the point was calculated as:
C = (color_at_point * light_to_be_multiplied) + light_to_be_added
where color_at_point is the material's color at that point (e.g. texture color or whatever).
Anyway, this is pure fudge cola all of it - but the end result works, and works well. You can spray many lights on an object, and instead of getting burned out, it just gets "bright". If you set up the spline to end with output value 1.0 (as the default above did), then no matter how much light you spray on a point, it'll never be brighter than 1.0 times its surface color PLUS any specular on top of that... (so it's still possible to get burnouts due to specular reflections - but that just looks cool. Burn outs due to diffuse just looks stupid))
On the fly 256 color palette selection [I asked about this, it worked pretty well, the palette would be biased on the fly to the ray trace being generated. - EAH]: Duh, that's just trivial. It just collects colors that deviates so-and-so much, until the palette is full. When it's full, it increases the "so-and-so" value, applies the same calculation to itself (i.e. colors within the palette closer than so-and-so becomes the same color), remaps the display, and continues.
The advantage of this method is, that there is no stupid histogram made. (I dunno why histograms are even used for color quantization. In what way does the fact that a color is used often or not have to do with the usefulness of it in the palette? Stupid. A single blue pixel in a sea of red and yellow shades would get lost with that method. But with a pure difference method, it won't. Some of the reds and yellows would be set to the same color, but so what? Frankly, if the histogram should be used, it should have the inverse use - the more often a color is used, the less use does it have in the palette.... :-) )
Antialiasing: Another interesting aspect of Raytracker is its antialiasing - it uses a hashing method, so it only antialiases on object boundaries. No extra rays shot for texture differences. (The textures are filtered by calculating the area covered by the pixel, using a simple algorithm of the 'size' of a pixel at the screen multiplied with the Z value (increasing 'into' the screen and being 1.0 at it). This gives us the 'flat' coverage when the texture is hit dead on. Now, divide this by the cosine between the eye ray and the normal, and you have the (approximate) coverage in texture area of that pixel.)
(That's how Raytracker does it. A more detailed method is to calculate the texture's X and Y directions, and instead of dividing the total area by cosine with the surface normal, divide by the sine of the angle from the eye ray to the texture X axis to find out how large area in texture X space to average, and divide by the sine of the angle to the texture Y axis to see how large area in the Y axis to average. That would be more accurate.)
Anyway, there is one problem with the hashing method in Raytracker - and that is rapidly changing surface normals (my method won't antialias under these conditions, since the surface itself doesn't change). So I have invented a much better method, that I haven't implemented.
Think like this: A plain old stupid ray tracer antialiasing just compares the color of this and the neighbor pixels. If they changed by "more than such and such", then supersample.
[This is a fairly important problem: textures are usually best rendered by using mipmaps or summed area tables or similar, better than point sampling with a set of rays. However, traditional "difference between sample colors means shoot more rays" methods ignore this method and foolishly supersample such pixels. A better method is to shade without the texture and do the color comparisons to see if more sampling is needed - this picks up edges and highlights. Also, generally only one texture sample should be generated and shared by the samples in the pixel, IMHO. -EAH]
But - if the texture is already filtered, then what is it that must change "more than such and such" for the pixel to need to be supersampled? The answer is "lighting"!
And since Raytracker already calculates the "light to be added" and "light to be multiplied" as I mentioned above, before actually multiplying with the texture color, well, the 'perfect' antialiasing would be a test if the light levels for the 'to be added' and 'to be multiplied' lights had changed more than such-and-such since last pixels, and if so, supersample!
back to contents
I've played around with ways to speed up the point in polygon test from the comp.graphics.algorithms FAQ and I found another 10x speedup by precalculating the slopes of all the edges and keeping them with the polygon object. Since I'll be reusing this part (x2-x1 / y2-y1) of the equation multiple times, I don't mind the extra space required and since I have a finite set of polygons (sectors), I don't really have to worry about long-term space usage.
previous discussion of point-in-polygon
next discussion of point-in-polygon
back to contents
As far as the new algorithm itself, I was curious about the upper levels of the hierarchy. I was wondering about what sort of number of children there were after creating the bounding volume hierarchy using Goldsmith & Salmon and then merging using your rules. If I understand it correctly, you put grids at all the nodes of the hierarchy. It seems like you could get trees where each node has only a few children at the upper levels, in which case using a grid at such levels might not gain much (i.e. if at a node there are only two bounding box children, just intersect these boxes directly instead of making a 2x2x2 grid at this node).
Good point! Unfortunately, I cannot give you a clear and sure answer - the grid research ended in the summer of 1994 (the sparse sampling scheme predated it by at least a year); things are somewhat rusty in my memory. I believe, the upper levels in some of the test scenes and some other I tried indeed had only a few children. The difference (in the resulting rendering time) between two bounding boxes and a 2x2x2 grid at the top was negligible. The regular grid IS efficient :-) Moreover, I played with both rounding-up and truncating when computing the formulas for the number of voxels. (Truncation would not allow "your" bounding boxes to be converted into grids). The round-up formula, even when used indiscriminately on all the levels of the hierarchy seemed to give better results probably due to the more "liberal" voxelization at the lower, more crowded levels where it is desirable. The difference was not large, however. The entire algorithm appeared to be more sensitive to the way I was merging the bounding boxes. This is part of a more general question: as you know, the entire problem of tracing rays in complex scenes tends to be a matter of statistics. Some improvements may be taken advantage of too seldom, detrimental factors may come into play infrequently, etc. This is why theoretical predictions often fail. For example, my experience was that even with the best known ray-triangle intersector, bounding boxes applied to every triangle in large and completely and fairly uniformly triangulated scenes still some time savings could be made. If my memory serves, 11-23% in the case of scenes I tried. Another example is the 3DDDA: before its implementation, Fujimoto was convinced that the octree would always be faster than the regular grid.
Andrew Woo writes:
When I read your article, the first thought that came to my mind was Frederic, George and Claude's paper in Eurographics 95 (reference 13). Their "hierarchy of uniform grids" is similar to what you call adaptive grids, that you should do a comparison in some fashion.
I've dabbled with heterogeneous grids a bit myself. Got decent results in a GI 92 paper. And your subvoxel grid's ray-bounding box idea (p.46) was very clearly illustrated in Snyder + Barr's paper.
I'm a bit disappointed at your evaluation of Jevans + Wyvill's accelerator - it was treated as a side-thing that might be helpful. His recursive voxels can stand alone as an intersection culler, it might be interesting if you did a comparison of your technique against the Jevans technique. At Alias, we've implemented the Jevans technique + Snyder's ray bounding boxes, and it's pretty fast.
When we get huge databases to render, we find that the Jevans technique may be better or worse than yours:
One final point: comparing your technique against a uniform voxel scheme is disappointing. Yes, it sounds good when you can say it's 50 times faster... When I played with Snyder's bounding boxes, that alone gave me huge magnitude of improvement over uniform voxels. All I'm saying is that you're comparing against a clearly primitive interesection culler. Comparison against something like Jevan's technique might be an interesting start.
Thanks for your comments. All of them are appreciated.
[in response to the mention of Cazals Eurographics paper:]
Although this will sound defensive, a bit of history is needed here. I wish I could call myself a rendering dude, but I am not one anymore and this matters here. My adaptive grid research ended in the summer of 1994. The dissertation was published through UMI in December of the same year and a portion of it was submitted as an article to CG&A soon after that. My contact with the state-of-the-art in ray tracing ended at that point. Although it is embarrassing, I admit, I have not read the Eurographics 95 paper you refer us to and even do not know if that paper is the same we mention in our article at the end of the "Future directions" section. The name seems to be different. Since the dissertation was out there before Eurographics '95, one may regret George did not compare his grids with mine - I am aware of at least two companies that managed to incorporate my algorithm into their products before Eurographics '95.
[in response to the GI 92 references and Snyder & Barr:]
I tried to get as conscientious survey of previous work as possible. Among the 130 references in my thesis there are three relevant papers of yours, but the GI 92 paper is something I simply have not come across or found referenced by others. (My university had only one issue of GI proceedings and to get Jevan's paper I had to resort to an inter-library loan.) Honestly, my assumption was "heterogeneous grids are so simple and obvious that every one is using them, but no one is either brave enough or bothers to make a deal out of it in public."
Regarding Snyder & Barr's paper, it is my recollection and understanding that the authors came up with the subvoxel bounding box, not a subvoxel grid. Our paper introduces the latter and calls the box and other classic Snyder & Barr's improvements "standard." Besides, S & B clearly say that their grid hierarchy was built manually, while we create it automatically - a step forward, I hope. Moreover, our article states up front that it is merely "filling gaps and expanding upon" the work done by the giants on whose shoulders I (Kris) was trying to treacherously stand. I assure you, "Give credit where credit is due" is one of the first things Dr. Sederberg teaches and demands from his students.
[on Jevans & Wyvill's work:]
Certainly it was not our intention to disregard anybody's work, including Jevans'. You are right, his technique is very powerful. You may have noticed I mention near the end of the paper doubling the speed of the algorithm. I have never finished that work, but some aspects of Jevans' concept played a certain role in it. However, at that time, Yagel's Raster Ray Tracer was the last word on the acceleration techniques and this is why I chose it over Jevans' scheme, or any other for that matter.
Also, it may be of some interest to you that the Japanese ray tracing company I worked for earlier evaluated Jevan's algorithm (which they dubbed N-tree). The results were mixed. The algorithm performed really well with some scenes, while with others it would lose to the SEAD-based solution we were using then. The conclusion was that both algorithms were clearly scene-dependent and the company kept looking for alternative solutions.
I read an internal document in 1990 or 1991 - the company systematically evaluated newly published algorithms regularly, "N-tree" was just one of them. The findings were quite often different from what we read in articles. Mainly, I think, because our scenes were usually dramatically different from those presented in the papers. I think Akira Fujimoto, as I am referring to his company, would be the best source of information, although I'm sure he will blast me for divulging any info about the matter. It is relevant, I think, to say that this unusual man gave up the "glory of publishing" his ideas for the sake of his company. He had come up with several solutions years before the world got to know them by the names of others. When I mentioned to Eric "the best known ray-triangle intersector," I meant something that is faster than Badouel's algorithm and existed at least two years earlier in our production code, Fujimoto's ingeniously simple idea. When I said that my sparse sampling was earlier than Kenji Mase's method, I did not brag. Long before Mase's paper appeared, Akira's firm had a - perhaps a little crude, yet fairly efficient - pixel interpolator in its ray tracer. In conclusion, I cannot predict if Akira will be willing to share information he's always considered sensitive, but you may try. ([email protected])
Speaking of comparisons: to the best of my knowledge, even before all of the acceleration techniques were implemented, the same program we present in the paper was tested by Evans & Sutherland against a number of commercial ray tracers. Dr. Sederberg may know better, but it seems to me that Alias and/or Wavefront, as well as the Japanese company I mentioned earlier, were among the contestants because they were referred to as "leading ray tracing companies," or something to that effect. The Car scene shown in the paper was one of the benchmarks. As a result, E&S chose our program. As you can see, the value of any comparisons, even those conducted independently, may be limited.
I would have been more inquisitive concerning the E&S comparisons if I had known what algorithms were in the tested ray tracers. I apologize for my ignorance, but your information about the algorithm Alias|Wavefront is actually using is a sort of revelation to me (In the places I worked before, this was considered a trade secret). I am not qualified to comment on the results of the E&S tests. They surprised me, to say the truth. But - if interested - you should be able to obtain more information from E&S itself or Pro-E, Parametric Technologies, which later bought the program along with E&S's CDRS division. (Which of the two, the program or the division, were more important to the buyer, remains a matter of speculation :-)
Traversal time: originally I intended to use a concept of a "wall street," not to get rich, but to create a modified SEAD, non-hierarchical organization in which rays would travel from wall to wall using Osterhout's corner stitching technique (Corner stitching had actually been used earlier by Arnaldi). The profiling I did on my code showed that traversal should not be the prime target to attack and eventually I gave up the idea.
I do not believe there is "the best (fastest and producing quality pictures) ray tracing algorithm." There is always, however, "the best RT algorithm" for a particular scene. So, I think, you are right: it is conceivable that my algorithm may lose to another algorithm and vice versa. This is why we wrote "we do not consider the new method as the fastest ray-tracing scene decomposition possible." The right answer would be to cleverly choose on the fly from a number of algorithms depending on the complexity of the scene ("Complexity" is of course understood here not as merely the number of objects, primitives, etc.). But so far, all the attempts to do so - at least those I know of - looked good in theory, but failed in practice.
[on comparing to a uniform voxel scheme:]
Looking good was not my objective.
I played with Snyder's bounding boxes (as opposed to subvoxel grids) before I had adaptive grids too, quite a bit. Alas, the improvements were rather small - never more than a few dozen percent, 30+at best. I also tried the dynamic bounding box, that is, one that shrinks after an intersection is found. (Also Snyder's idea.) The improvement with respect to the "static box" was practically none. If my memory serves, the boxes increased memory usage significantly since I created them during preprocessing, but I may be wrong. This is not to say that Snyder's idea is bad. Once more we deal with scene dependence here. I was using a fine subdivision of the regular grid, which was giving me consistently better timings than coarser voxelizations. On the average, the distribution of the primitives in the voxels was more or less even, the primitives were relatively small. Snyder's ray boxes could not do much good; in too many voxels they were too large. I tried both triangulated and non-triangulated scenes. Still the improvement was not impressive, mildly speaking. In the adaptive grid scheme, the matter looks differently. Typically, we begin with much larger voxels and uneven object distribution. This sets a better stage for ray boxes and, consequently, for subvoxel grids.
Why just regular grid? Again, there were more recent developments in the fields at the time and I was eager to pitch my ray tracer against them. Secondly, in my view, the lack of a common basis on which various techniques might be compared is a major problem the field has struggled with. Eric's Standard Procedural Databases was a step in the right direction, but it addressed only one aspect of the issue. This is not an excuse, but I think you will agree that there are many papers that failed to compare their results with anybody's. There are even articles that propose solutions without actually implementing them (palm of victory without work?). Examples on demand.
I chose the regular grid for two reasons. First, because it is one of the best known and easy to implement acceleration techniques, on which many tried to improve. This, I hoped, would make comparisons with numerous papers straightforward.
Second, I wanted to make sure that the implementation of the algorithm against which I was to test my method (or methods - the paper treats only some of them) was best possible. For someone who comes from the school where 3DDDA originated (Akira Fujimoto was my mentor for many years) and the regular grid and its optimization tricks were practically a staple, the choice was quite obvious (Among others, Fujimoto's disciples were required to use "A Proposal for a Hybrid Voxel Traversal Approach" by A. Woo in lieu of our normal "makura's," Japanese wooden pillows, so that even our dreams be properly stimulated).
As to the cost of intersections, I was advised that some companies in Japan dealing in lighting simulation for interior design use lighting equations (computed in physical units) that are so complex that ray-object intersection calculations do not dominate rendering time anymore [People at Pixar appear to agree with this statement. -EAH]. Intersection culling dudes - call me so, please! - may be squabbling over a dead duck these days.
[what follow are some interesting bits from back and forth discussions. -EAH]
Telling you we use the Jevans technique is no big deal - it's a published algorithm after all.
Right, but any information about what you actually have inside your ray tracer (or any product for that matter) gives your competitor an edge. This is how many Japanese companies tend to think at least.
Not so in most companies. In SoftImage's raytracer, they even try and educate users on what 5d subdivision involves and how to optimize for them. Don't need to do much guessing there.
In our raytracer, user parameters include subdivision recursive level, voxel resolutions, etc. Not much of a secret :-)
What is your portion of time spent on traversal vs ray-triangle intersections?
Your interest in the ratio makes me uneasy: I hate thinking I missed an opportunity to make the program even faster.
I'm getting more and more interested in this, as I'm very surprised by this statistic every time I benchmark scenes.
[I agree: when I've benchmarked I generally find traversal takes say 60% of the time and actual intersection testing takes say 20% or less (the rest is shading and bookkeeping). It seems contradictory, so much time spent passing the ray on and so little actually testing objects. But then, the whole point of an efficiency scheme is to avoid testing objects; it's the overall time that is to be minimized. -EAH]
Frederic Cazals joins in a week or two later:
So it goes: Eric forwarded me a couple of emails related to the chat you have been involved into about RT speed-ups, and since I have been interested in this topic in the last 3 years --- with a 16 months break because of my national service, here are my $.02.
Before discussing the main points raised in Kris's email to Andrew let me define the following bibliography:
 Goldsmith and Salmon IEEE CG&A, 7(5), 1987  Jevans and Wyvill GI'89 paper  Filtering, Clustering and Hierarchy construction: a New Solution for Ray-Tracing Complex Scenes, F. Cazals and G. Drettakis and C. Puech, Eurographics'95, at http://pauillac.inria.fr/algo/cazals/xfc_research.html  Kris's recent paper in IEEE CG&A  Bucket-like space partitioning data structures with applications to ray-tracing, F. Cazals, C. Puech, 13th ACM Symposium on Computational Geometry, 1997 at http://pauillac.inria.fr/algo/cazals/xfc_research.html  F. Cazals, Combinatorial properties of one-dimensional arrangements Experimental Mathematics, 6(1), 1997 at http://pauillac.inria.fr/algo/cazals/xfc_research.html
 and  were clearly developed independently and I think it was almost impossible for each party to be aware of the other one early 95. The goal pursued is exactly the same and the resulting hierarchies are slightly different since the grids of  are nested - please correct me if I am wrong - while the ones of  are not so.
The ways the hierarchies are built are very different: -  proceeds by:
-  proceeds by:
It would be very interesting to test [2,4] against , and this can indirectly be done by just comparing  to recursive grids - see point 2. just below. But even more interesting from my point of view would be to get precise statistics describing how the hierarchy of  is organized and works:
All these statistics are precisely defined in  and I think they should provide enough insights to make the solutions even better or to understand why some optimum has been reached.
Also, I think an intuitive explanation of why the 'serendipitous discovery' of page 44 in  works, is given in section 5 of . The long version of  also contains some negative results I got with heterogeneous griddings based on overlap numbers defined in .
a method for determining the optimal resolution and tree depth for a given scene
Jevan's contribution is important, but subdividing a grid containing n objects into alpha*n voxels, with alpha being some positive constant sounds as much important to me - please correct me if I am wrong, but the first time I saw this criterion is in E. Jansen's note in RTNews 5(1), 1992. Jevan's recursive DS [data structure] with this criterion is what we use to call a recursive grid - well, I think!
And so, I believe that the reference DS every new solution should be tested against is a recursive grid and not a uniform one. This is especially easy since once one has a uniform grid, implementing a recursive one is straightforward: as an example, the .h and .C of my recGrid are jut 46 and 102 lines of C++ code!
What is your portion of time spent on traversal vs ray-triangle intersections?
is very interesting. In our implementation which relies on OORT, the polygons are not necessarily triangles or squares - even if more than 95are so for the test models of [2,4]. As a consequence - note that this observation is sufficient but not necessary since one may proceed in the same way for triangles only, the ray-polygon intersection test is as follows:
t1. if the ray and the polygon are coplanar => return else compute the intersection point p between the ray and the polygon plane t2. if p does not belong to the polygon 2D bounding box => return t3. compute the crossing number and conclude
of course test t3 is more expensive than t1 or t2. As pointed out in , the exit point of this algorithm depends of the 'coarseness' of the DS the ray-polygon intersection test is called through, which has an integral geometry explanation.
This is just to say that in our implementation the average call cost of the ray-polygon test varies within a non negligible factor - 7 or so in the worst case, which makes impossible writing something like:
t_r = t_t + n_i * t_0
with t_r the total rendering time t_t the traversal time n_i the number of intersection tests t_0 the cost of an intersection test
because the higher the number of intersections, the lower t_0, and vice-versa.
this is not of major importance, but it should be pointed out that in the theory of one-dimensional search and sort data structures, N-trees refer to the work of Marku Tamminen. see e.g.
M. Tamminen, Analysis of N-Trees, Information Processing Letter, 1983, vol. 16
unlike Jevan's DS, the point here is that the branching factor of a node is not constant.
After looking over this article just before this issue went out, Kris writes:
Regarding the discussion of superiority of one spatial enumeration scheme over another, last summer I had a an interesting conversation with an employee at Evans & Sutherland. He told me that in 1993 or 1994 (I don't remember) they compared 15 best ray tracers they could find. The first, crude version of adaptive grids and Fujimoto's physically accurate commercial ray tracer were among them. It stuck in my memory that it took over 67 hours to render the test image - Car, shown in my paper - using Fujimoto's program. According to my article, Car was ray traced with adaptive grids at standard resolution in 75 secs. From what I know, none of the other programs came even close to adaptive grids. The employee asked me what accounted for such a big difference in speed and practically none in image quality. I could only speculate as I did not know what acceleration techniques the competing packages utilized. I am not suggesting you should put this info into the article, but am mentioning the fact as a yet another indicator that adaptive regular grid approaches (be it mine or Jevans', or Cazals') may be superior to other known methods.
Last, I would like to stress the fact that using the regular grid as a standard against which new space subdivision algorithms are compared resembles the concept of your procedural database. Probably both (the regular grid and database) are not what we deal with in the so-called real world, but being easy to use or implement, they became de facto standards. Similarly, many years after the 1 MFLOP VAX machine became a museum piece, it served the useful purpose of an informally, but commonly, approved basis for measuring the performance of computers that were ages away as far as sophistication, architecture, storage capacity and speed were concerned. While I certainly understand Andrew's concern, I would call for "declaring" the regular grid the "VAX of spatial enumeration techniques."
back to contents
Here are some sample benchmarks. The database used is from Eric Haines' procedural database. Most of this was benchmarked in 1988 on a SUN 3/280 with fpa. If I had data on 20x20x20 (which I don't), that will even be better, as it's the n**(1/3) rule.
Sphere Flakes: Grid Resolution #objects Uniform Ray Bounding Box ------------------------------------------------------------------ 40x40x40 7382 391 mins 163 mins 50x50x50 7382 270 mins 142 mins 60x60x60 7382 220 mins 138 mins 70x70x70 7382 199 mins 133 mins
Note that at a pretty high grid resolution of 40x40x40, we got close to twice the speedup. Also check out the #intersections needed for testing.
Grid Resolution Uniform Ray Bounding Box ------------------------------------------------------- 40x40x40 128566950 3858198 50x50x50 89045372 3662689 60x60x60 68387039 4816035 70x70x70 55492658 4152675
I also tried this against the tetrahedron and rings model. The results showed ray bounding boxes to be slower, because the model was so uniformly distributed. Then I played with determining when ray bounding boxes should be used or not, on a per voxel basis, then again I got good improvements in performance. The heuristic is very simple: if the ray's bounding box is above 75% the volume of the voxel, skip the ray bounding box test, because the ray bounding box is so huge that its culling is likely useless.
back to contents
The entries and results for the first "official" year, May 1996 to April 1997, are now available on a CD-ROM from Hallam Oaks, http://www.aussie.org/products/. Profits from the CD's help fund the continued existence of the IRTC and POV-Ray (http://www.povray.org/) web sites.
The CD-ROM itself is painless to use. Pop it in, and autoplay starts your web browser and you begin on an index page. While most of the disk is filled with contest entries, there are also a few other goodies, like version 3.02 of POV-Ray (binaries for many platforms, source, documentation) and a museum tour movie which runs a few minutes. There are also short (auto-)biographies of some of the competition winners.
The main thrust is, of course, the entries themselves. You can view all entries or just the winners. There are about 1000 images from the competitions and of these, about 500 come with ZIP archives containing the entire scene's data. So not only can you see the images, you can also recreate them, play with them, and learn from them. I found the creators' notes on how they made the scenes the most interesting part of the package. For example, it is almost painful to read that a classroom scene with three detailed (blobby) children was modeled entirely by hand.
The images themselves: me, I've seen a lot of synthetic mirror reflections and too sharp shadows in my life so I'm probably jaded. The winners are generally worthwhile, usually because of the elaborate modeling and materials work and sometimes because of the scene's concept or layout. I would put most of them in the "Ripley's Believe it or Not" category - not great art, not particularly photorealistic, but amazing to contemplate that many of the scenes were done for the love of it in spare time with free software and a lot of talent and a lot more sweat. Amazing and inspiring, seeing all this time and energy and spent cycles in one place.
All in all, this is a pleasant CD-ROM. Most of the content can be found on the web, but for most people, accessing the images off of this CD-ROM will be much faster and more dependable, not to mention more portable. If you do any non-trivial rendering with POV-Ray, or just want to have a collection of fun images to look at and show others, it's worth getting.
back to contents
The best explanation of Plücker coordinates I've seen is in Jorge Stolfi's book "Oriented Projective Geometry" (Academic Press, 1991). I'll try to give a sketch here, but you should really check out Stolfi's book if you want to learn about these things.
Plücker coordinates are an incredibly useful way of specifying lines and rays in 3-dimensional space by six-dimensional vectors. Suppose L is the oriented line passing through two points with homogeneous coordinates [a,b,c,d] and [w,x,y,z] -- or with standard Cartesian coordinates (b/a,c/a,d/a) and (x/w,y/w,z/w) -- in that order. Then the Plücker coordinates of L are
[a*x-b*w, a*y-c*w, b*y-c*x, a*z-d*w, b*z-d*x, c*z-d*y].
Plücker coordinates are homogeneous -- multiplying all six coordinates by any real number gives you new Plücker coordinates for the same line. Also, not every set of six numbers is the Plücker coordinates of a line. Plücker coordinates [L1, L2, L3, L4, L5, L6] always satisfy the following equation:
L1*L6 - L2*L5 + L3*L4 = 0.
Two oriented lines L and M can interact in three different ways: L might intersect M, L might go clockwise around M, or L might go counterclockwise around M. Here are some examples:
|M |M |M L | L | L | -----+-----> -----------> -----|-----> | | | V V V intersect counterclockwise clockwise |M |M |M L | L | L | <----+----- <----|------ <----------- | | | V V V
The first and second rows are just different views of the same lines, once from the "front" and once from the "back". Here's what they might look like if you look straight down line M (shown here as a dot).
L ----------> -----o----> L o L o ----------> intersect counterclockwise clockwise
The Plücker coordinates of L and M give you a quick way to test which of the three it is.
cw: L1*M6 - L2*M5 + L3*M4 + L4*M3 - L5*M2 + L6*M1 < 0 ccw: L1*M6 - L2*M5 + L3*M4 + L4*M3 - L5*M2 + L6*M1 > 0 thru: L1*M6 - L2*M5 + L3*M4 + L4*M3 - L5*M2 + L6*M1 = 0
So why is this useful? Well, suppose you want to test if a ray intersects a triangle in 3-space. One way to do this is to represent the ray and the edges of the triangle with Plücker coordinates. The ray hits the triangle if and only if it hits one of the triangle's edges, or it's "clockwise" from all three edges, or it's "counterclockwise" from all three edges. For example...
/^ ...in this picture, the ray / | is oriented counterclockwise --------- | --> from all three edges, so it / | must intersect the triangle. L-------> (The L is an arrowhead!)
So if you use Plücker coordinates, ray shooting tests like this take only about ten lines of code.
(there's more on Plücker coordinates in the next RT News)
back to contents
I will start with a minor but important error in Dr. Rogers' new "Procedural Elements for Computer Graphics, 2nd edition", WCB/McGraw-Hill. We discussed this topic back and forth, and while I pressed my arguments hard, I did not continue to nag about it and it slipped into the book. The errata is from page 551 on, and has to do with the ray tracing illumination model. The problem is that the distance along reflection and refraction rays is used to attenuate these rays' contributions. Specifically, the intensity of the ray's contribution is divided by the length of the ray - this is incorrect. It is an educated error (physics says that light from a source drops off with the distance squared, for example), but is not a good simulation of what actually happens in the real world. Note that the illumination model presented does not attenuate eye ray contributions, these are left as is.
The error does have a logic to it, in that shadow rays generated from surfaces can get attenuated (if desired) by a function related to the distance to the light source, so why shouldn't reflection rays? [To digress a bit, this reminds me of an interesting erroneous algorithm someone starting out in ray tracing told me: they would not simply ignore the light if the surface was in shadow, but rather would attenuate the light's effect based upon the distance the shadow ray went before it hit the occluder. This idea is interesting! I've never tried it out, and you do have to modify good ray tracers to return the first (not just any) object hit by the shadow ray. But think about it: as an occluder gets further away from the object being occluded, more light appears in the shadow - this is sort of a poor man's radiosity effect, and has a rationale. Surfaces which are close together usually minimize the amount of light which bounces into the area between them. There are cases where this algorithm is nonsense (two occluders, one large and then one small while moving from the light, would give a darker shadow where the smaller occluder casts a shadow), but it's an interesting idea.]
There are also cases in which the contribution from reflections or refractions (or from the eye) are attenuated, e.g. atmospheric effects, absorption due to the glass, etc (and these are usually handled using exponential drop-offs, not divisions by the distance). Nonetheless, always dividing by the distance is incorrect, it does not happen in reality. The simplest counterexample is a mirror. You look directly at something far away. You now turn around and look into a hand mirror at the same scene. It may dim a little bit simply due to the fact that the mirror is not 100% reflective. However, it does not dim due to the distance (not any more than it does looking at the scene directly with the eye). The ultimate case is a reflector telescope (which normally has almost perfect mirrors): the light from the stars (point sources) or moon (area source) reaches the eye from mirror reflections, and the mirror does not dim the perception of the stars just because the stars were reflected and not seen directly. The eye itself works by refraction, and the refracted rays coming in through the lens are not dimmed by the distance just because they are refracted vs. pinhole-camera type eyes where no lens is present.
On to other errors. The error of omission is one of my own, and I came to know about it when I was writing Dr. Rogers about the bug above. I realized that Cyrus-Beck 3D clipping is almost exactly what I was doing in my article in Graphics Gems II about ray/convex-polyhedron intersection testing (which is a variant on Kay/Kajiya slab testing). Kay & Kajiya properly note that their slabs method is similar in spirit to Cyrus-Beck. I never followed up this reference and so did not fully comprehend how closely related these were, and how my own ray intersection method was essentially identical to Cyrus-Beck. What is the quote about those who do not study history are doomed to repeat it? I guess I am in good company: Rogers makes a case in his book how Liang-Barsky 2D clipping (1984) is a special case of the Cyrus-Beck algorithm (1978), showing how the flowcharts and steps of the two algorithms interrelate. Evidently Liang and Barsky did not know of Cyrus-Beck, another case of history repeating itself. Rogers thorough treatment in his book of all the variants of 2D and 3D clipping algorithms leaves no excuse for anyone to reinvent any of these basic algorithms again.
back to contents
Actually, there is a neat way of doing a ray polygon intersection with Cyrus-Beck clipping. A containment test is not required. Just consider the polygon as two opposite pointing faces of a 3-D polyhedron and use the 3-D C&B.
[For example, consider a triangle to be a five sided polyhedron, with two faces (one in each direction) on the triangle's plane and three faces forming the edges, with each face perpendicular (though not required) to the plane. Interesting scheme! I would suspect the traditional "hit the plane, test the point" method of ray tracing a triangle might be faster, but I won't swear to it; this new method gives a tight single loop of code, testing the ray against each plane in turn (with two planes sharing results). Whatever the case, Rogers' method is definitely more elegant. For another elegant recent solution to this problem, see Möller and Trumbore's paper in the "journal of graphics tools"; abstract, code and statistics (but not the algorithm description) at: http://jgt.akpeters.com/papers/MollerTrumbore97/ -EAH]
back to contents
A front surface mirror can be as high as 99reflectance, though I haven't measured any lately. Reflectances this high are also reported by some makers of aluminized plastic reflectors used in light fixtures, and they do indeed achieve this when new. Dust diminishes this value by 5-15, unfortunately.
A typical bathroom mirror has a reflectance of around 90when it's clean, due mostly to internal absorption of the glass itself. The actual RGB reflectance of the mirror in our bathroom is:
0.916 0.938 0.858
back to contents
Eduardo Marreiros ([email protected]) writes:
I'm interested in finding a QUICK and LINEAR way to determine if some 4x4 matrix produces a transformation in which all 3 scale factors are equal. I mean, if some object is to be transformed, it will not be deformed. An object will be grown/shrunk by the same amount in all 3 axes.
Charles Iliya Krempeaux ([email protected]) replies:
One way to do it is...
| a11 a12 a13 a14 | | a21 a22 a23 a24 | | a31 a32 a33 a34 | | 0 0 0 1 |
Check that the upper-left 3x3 sub-matrix matrix-multiplied with its transpose gives you a matrix of the form:
|k 0 0| |0 k 0| |0 0 k|
This makes sure that the upper-left 3x3 sub-matrix is orthogonal - and nicely ignores scaling but not spatial inversions so...
Then make sure that:
| a11 a12 a13| determinant( | a21 a22 a23| ) is positive | a31 a32 a33|
This makes sure that you don't invert your space/coordinate-system.
Peter-Pike Sloan ([email protected]) notes:
If you test that:
(a11)^2 + (a21)^2 + (a31)^2 = (a12)^2 + (a22)^2 + (a32)^2 = (a13)^2 + (a23)^2 + (a33)^2
a11*a12 + a21*a22 + a31*a32 = 0 a11*a13 + a21*a23 + a31*a33 = 0 a12*a13 + a22*a23 + a32*a33 = 0
then the matrix is orthogonal and its axes are all the same length.
[The first set of equations checks the length of the axes (to determine that the scales are all the same), the second set that the axes are mutually perpendicular. These tests are equivalent to doing the transpose multiplication test, without wasting time duplicating dot product tests. The determinant test is still needed to determine whether inversion occurs. So you may be able to get away without doing all these tests, depending on what your system allows. Tolerances are also obviously important. - EAH]
back to contents
Bob Lipman (also of NIST) talked about a newly available database of BRDFs for various common materials, collected by the CIA for use in analyzing satellite images. Recently declassified (and simultaneously defunded), the NEF database (obviously stands for "Nonconventional Exploitation Factors") contains hundreds (?) of exterior finishes and natural materials, characterized by full spectral reflectance models and measurements. It was meticulously created by the National Imaging and Mapping Agency, and represents a huge effort. I strongly recommend to anyone interested in material reflectance data that they go check it out. For more information, see the following web sites:
[The first page has some good links to BRDF related work elsewhere.] The interface to the database currently runs only under Solaris, and Bob is looking to create a simpler interface to access the data under other systems, as well as seeing if the current interface can be ported. (His initial attempts to do so failed for unknown reasons.) [Looking at the pages, also SGI/IRIX and IBM/AIX versions are now available.]
Bob Lipman is at [email protected], (301) 975-3829.
back to contents
I am doing gobs of coordinate transformation stuff, and the "Curse of the Monkey Paw" intrigued me. Your example is really only 2D. Imagine looking down at the top of the cone and projecting everything into a plane. Lo and behold, you have a polar projection of the texture. Is it possible to design a set of triangles such that the transformation you want is achieved? I don't know. But, if you transform the texture first via something simple, it's easy, although you lose a little information. Filter the texture so that it compresses linearly as you travel vertically. That is, convert the texture you have into a triangular texture by compressing the data near the tip to a point, then linearly out towards the base. I'm not sure if this will work perfectly with texture mapping hardware; there is probably some constant or two to deal with, but the general idea of applying a filter seems like it might do the trick. I haven't thought it through completely, and I imagine "linearly" and "triangle" are errors, but the basic idea is to transform the original texture so that the triangles forming the cone map into the texture appropriately.
How general is this? Fairly so, it seems. This will work somewhat for spheres, too, if you cut them in half at the equator. Will it work for all point singularities? Probably not.
The prefiltering actually does make some sort of sense for cones. There's less surface area near the tip (as there is in a polar projection of a plane) than at the base, so there really is less information in a sense in the texture to be applied there. More importantly, consider the reverse problem: you have a texture on a "cone" and want to draw it onto a rectangle. You cut the polar projection radially then stretch out the pole, converting what was really once a triangle into a rectangle. To do the reverse, it seems obvious to convert a rectangle into a triangle. Since you only have locally linear texture lookup, a little bit of tinkering will need to be done, but for small triangles, a linear approximation will probably suffice.
back to contents
[Personally, I would use a sphere and just apply a non-uniform scaling matrix to my equation results; I need a transformation matrix to move the object away from the origin anyway. - EAH]
1) Take an ellipsoid with radii A, B and C in the axes X,Y and Z respectively. The equation for an ellipsoid located at the origin is
X^2 / A^2 + Y^2 / B^2 + Z^2 / C^2 = 1
2) Let the center of the ellipsoid be defined by a 3D vertex named Q, which for simplicity is actually the origin <0,0,0> (we can always use a transformation matrix later to move this point).
3) An arbitrary test point anywhere inside or on the surface of the ellipsoid will be named P. Our goal is to compute the potential at this point.
4) Construct a vector V such that V = P (trivial, but we'll modify this definition later). We can define a ray V * t, e.g. it generates the set of points P' = V * t, where t is a parameter, t >= 0
at t = 0: P' = Q = <0,0,0> at t = 1: P' = P
5) Now, let M be the point where the ray V * t intersects the ellipsoid, at t = tm
so M = V * tm = P * tm
6) The potential of the blob at any point P can then be calculated as
Potential = S * ( 1 - (r/R)^2 )^2
where S = maximum strength of the blob (scalar) at the center r = mag (P) R = mag (M)
7) To solve for the potential of point P we need to find the ratio r/R which in turn depends on the intersection point M:
using X^2 / A^2 + Y^2 / B^2 + Z^2 / C^2 = 1 and M = P * tm
let M have the components <Mx, My, Mz> P have the components <Px, Py, Pz>
so substituting everything into the ellipsoid equation:
(Px * tm)^2 / A^2 + (Py * tm)^2 / B^2 + (Pz * tm)^2 / C^2 = 1
Now solve the previous equation for "tm" and find M using M = P * tm. This gives you the intersection point M.
8) Since you know have the points M, P and the strength value S, solve for "r" and "R" and use equation (6) to find the potential value of point P.
9) If we repeat this for a number of points inside and on the surface of the ellipsoid and use a marching cubes algorithm, we can build the surface by polygonizing it. This entails calculating potential values of points located in a 3D grid in space and selecting points in space where the value of the potential is near some arbitrary threshold value. These points are then used to polygonize the surface.
Modifying the Previous Steps
In the previous steps the origin of the ellipsoid is used as the starting point for the ray V * t. Now, say that the origin for generating our V vector is some defined point E.
Why am I doing this? I want the surface of the ellipsoid to be the "physical" boundary or limit of the implicit surface...that is, when E = Q (center of ellipsoid), then the implicit surface generated with the algorithm will be similar to that of the ellipsoid....BUT, when E is not equal to Q we should have some interesting surface that will be enclosed by the ellipsoid boundary surface but may or may not be similar in shape to it.
Using E as the start of the vector V, let's see how the equations change:
ellipsoid: X^2 / A^2 + Y^2 / B^2 + Z^2 / C^2 = 1 vector V = P - E ray equation: P' = E + V * t
then M = E + V * t = E + (P - E) * tm
Potential = S * (1 - (r/R)^2 )^2
where S is now the maximum strength of the blob at point E (before, it used to be at center of ellipsoid).
r = mag (P - E) R = mag (M - E)
Substituting M = E + (P - E) * tm into the ellipsoid equation:
let D = P - E
(Ex + Dx * tm )^2 / A^2 + (Ey + Dy * tm )^2 / B^2 + (Ez + Dz * tm )^2 / C^2 = 1
(Dx^2/A^2 + Dy^2/B^2 + Dz^2/C^2) * tm^2 + 2 * (Ex*Dx/A^2 + Ey*Dy/B^2 + Ez*Dz/C^2) * tm + Ex^2/A^2 + Ey^2/B^2 + Ez^2/C^2 = 1
This one is a tad more complex than the other one. Now solve the previous equation for "tm" (select the positive value) and find M using:
M = E + (P - E) * tm
Since we know points M, P, E, and the strength value S, solve for "r" and "R" and use equation (6) to find the potential value of test point P.
If this new style ellipsoid is rendered as an implicit surface at a potential of 0 it won't look a bit different than a standard ellipsoid. Where it gets interesting is when the surfaces at potentials greater than 0 are used, and when such ellipsoids are blended with other primitives. The isosurfaces radiate out from point E instead of the geometric center of the shape.
If you try this out you'll see that we have now broken the symmetry of the potential value. If you plot the change of potential across one of the axes, in the standard method, the peak of the potential distribution was located at the origin, now with the new method, the potential has a peak, offset at a distance equal to mag(E - Q).
I gave you a straightforward example of the new method by using a simple object such as an ellipsoid. What about other shapes? Well, how about a cylinder? Researchers such as Jules Bloomenthal have used implicit cylinders to do blobs. One of the approaches is to use the perpendicular distance between a line segment and the test point P to be equal to a value "d". This creates a cylinder with capped (i.e. hemispherical) ends of radius "d" - a cheese log. If we were to break the symmetry we would have to go beyond using a new point E to be the center of the cylinder - we would have to create a new line segment (with the same or different length) that would be translated and rotated relative to the cylinder's original line segment. Now, the method would be to calculate the perpendicular distance between the NEW line segment and the test point P. The point M would be the intersection point of a vector (that is perpendicular to the new line segment and passes thru point P) and the surface of the cylinder (as defined by the original line segment).
We can go now one step further: in the case of the cylinder, instead of creating a new line segment and moving it inside the cylinder's surface use a spline instead. This would make the implicit shape more complex and interesting. The method stays the same but now we have to calculate the distance between the spline and the point P.
Here's yet another idea. So far, I have suggested moving the "center" point of the ellipsoid somewhere else and moving the line segment that defines the cylinder somewhere else also. This modifies the vector's V origin. How about using a point instead of a line segment in the cylinder? that is, the boundary surface is the cylinder, but the vector V is calculated as if the original shape was an ellipsoid....hmmm. Also, it would be interesting to check out weird surface boundaries such as superquadrics with points, line segments or splines to break the symmetry!
If you think this method only applies to blobs, here's another idea. In 1994 I talked a few times over the phone with Paul Borrel at the IBM T.J. Watson Research Center. He had written a paper about a deformation method they named SCODEF. His method used something similar to what I described in detail in the first section of this article, but he uses SCODEF to perform deformations rather than to do implicit surfaces. I was able to modify his equations and implement the eccentricity effects as described in this article with no difficulty.
NOTE: Paul Borrel's paper is "Simple Constrained Deformations for Geometric Modeling and Interactive Design", Paul Borrel and Ari Rappoport, ACM Trans. on Graphics, Vol 13, #2, 1994, pp. 137-155.
back to contents