Ray Tracing News

"Light Makes Right"

May 11, 2004

Volume 17, Number 1

Compiled by Eric Haines, Autodesk, 10 Brown Road, Ithaca, NY 14850 [email protected]. Opinions expressed are mine, not Autodesk's.

All contents are copyright (c) 2002,2003,2004, all rights reserved by the individual authors

Archive locations: text version at http://www.acm.org/tog/resources/RTNews/text/
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 ray tracing FAQ.



So if you don't have any issue out during a year, do you label the next year's issue as N+1 or N+2? I'll go for N+2, and this is the beginning of Volume 17. Yes, it's been way, way too long, even for me. I've been kept too busy reviewing and editing papers and books, and let's not talk about my "Battlefield: 1942" addiction (well, the one cool thing I learnt from that experience was how to set up a wiki; if you don't know what a wiki is, go find out right now). The latest thing getting in the way of me finishing this issue was compiling the results of the Fantasy Graphics League, see http://www.realtimerendering.com/fgl/siggraph2004/index.htm. Oh, and then I had to read 133 sketches for the SIGGRAPH Sketches committee. But now I'm really going to get out an issue...

In book news, Peter Shirley and R. Keith Morley have put out a second edition of the book "Realistic Ray Tracing", adding much C++ code, along with new information on photon mapping and other advanced topics. This currently is the best book on how to make a ray tracer. That said, it is going to have some competition soon. Matt Pharr and Greg Humphreys have coauthored a book, entitled "Physically-Based Rendering: From Theory to Implementation". It should be out by SIGGRAPH from Morgan Kaufmann. It's a pretty impressive work (I was one of the technical editors, so have been able to read it), covering the topic from soup to nuts. The usual areas are covered, such as object intersection, shading, texturing, etc. What's noteworthy is that topics such as tone mapping, sampling patterns, Monte Carlo integration, and many other areas are also covered. Similar to Shirley's book, the authors do not attempt to cover all possible methods but instead focus on what they consider best practices. They explain their own rendering system and along the way discuss why they made the choices they did. A rather amazing thing is that the whole codebase is done using literate programming, making this easily the longest literate program I've ever seen (and I suspect the longest in the world). See Matt Pharr's page at http://graphics.stanford.edu/~mmp/ for a taste of some images and other information. Matt also has some interesting comments in the Roundup notes, below.

Another noteworthy book that came out last SIGGRAPH is "Advanced Global Illumination" by Phil Dutre, Philippe Bekaert, Kavita Bala, from A.K. Peters. This volume is focused on giving a broad overview of theory and techniques. It is a more traditional college-level text, and it does a thorough job of presenting the field. The authors are professors, and much of the material has been honed by being used in classes and presentations at SIGGRAPH. This results in a clear and well-illustrated book on the subject. Recommended.

By the way, July 8th will be a day of celebration, or something: the Unisys LZW (GIF) patent runs its course in the last country in the world, Canada. See http://www.unisys.com/about__unisys/lzw. The assumption nowadays in computer graphics is that if you see a research paper coming from a business, the technique's patented, from a university, it's probably patented. Patents and publication go hand-in-hand, since a patent is a public declaration of invention.

I like to offer up a graphics or geometry related puzzle each issue. Since this issue is so long in coming, I'll offer five: http://www.acm.org/tog/editors/erich/SubtleTools.zip. This is the slideset for a talk I gave at NVIDIA University last year. It was fun to make, and the skeleton is five computer graphics related test questions. Long-time readers of the RTNews should be able to get most of them (since I stole them from these pages).

If you need to use up some more time: my blog (which is really just a place for me to store fun links) is at http://erich666.blogspot.com/, and Andrew Glassner's (which is chewier) is at http://www.glassner.com/OpenSeason/.

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Ray Tracing Roundup

Jack Hsia, one of the authors of the classic 1977 paper on BRDF nomenclature, by Nicodemus et al., has kindly agreed to have the paper optically scanned and placed online. You can find it at:


- Marc Levoy (levoy(at)cs.stanford.edu)


Check out the following--all POV-Ray images rendered with scene description files of size 256 bytes or less.


This one is probably the most impressive, IMHO:


- Matt Pharr (mmp(at)graphics.stanford.edu)

By the way, POV-Ray continues to improve, it's now up to version 3.6 beta 4: http://www.povray.org/. If you haven't looked lately, POV-Ray now includes radiosity and photon mapping extensions. If you want even more extensions, visit MegaPOV, http://megapov.inetart.net/ (though this site is a bit old now; some of these additions have since been folded into POV-Ray itself).

If you'd just like to check out some nice POV-Ray work and wander around looking at resources, see http://www.oyonale.com/oy_en.htm. Some of his images at http://www.oyonale.com/new_en.htm are pretty incredible.


JGT's Quite Alive, Thanks

The ACM sold subscriptions to the "journal of graphics tools" (http://www.acm.org/jgt) through its membership services for many years. They have decided to terminate this arrangement, and some subscribers have not received word about this change. So if you subscribed through the ACM and have not been receiving issues, please contact A.K. Peters, the publisher of the JGT: 1-800-450-2210 or at [email protected]. Also, if your university library receives this journal (but hasn't lately), please do pass this note on to them.

By the way, this journal began in the middle of the year, meaning that issue No. 3 comes out around February of the next year (in other words, we’re on schedule).

- Eric Haines ([email protected])


Tone Operator Code

Regarding tone reproduction operators, here's what I know/have: the Radiance package implements Greg Ward's histogram adjustment technique.

Source code for our photographic tone reproduction operator (Siggraph '02) is freely available online (it implements a newer version which automates two parameters that used to be manual):


I have research code for a number of other operators, but this code is not cleaned up, messy, and comes with no guarantees, and is for those reasons not online. If any one has a need for that code, I'd be happy to share - please send e-mail.

- Erik Reinhard (reinhard(at)cs.ucf.edu)


Have you seen this? I think it's one of the clever-er things I've seen in a while. Mark VandeWettering took the Escher picture of the guy holding a reflective sphere:


And ran it through an unwrapper to make a cubical environment map:


Came out pretty close!

- Matt Pharr (mmp(at)graphics.stanford.edu)

[Here's another Escher-y thing for fun: http://www.mantasoft.co.uk/_stuff/Recursive.htm]


Here are a bunch of short hacks for optimizing various aspects of ray tracing for little interactive ray tracers: http://www.tigrazone.narod.ru/raytracing/. There's also a little Java ray tracer here.


An interesting thing: some competitions to render various data sets. There are pretty nice results, see them at


(poke around in the "quick overview" section of the Cathedral page.)

- submitted by Christian Bauer (christian(at)cgtechniques.com)


Fast, high-quality previewing is becoming in reach for production rendering. Gelato is NVIDIA's rendering product, which uses the GPU to accelerate high quality non-interactive rendering: http://film.nvidia.com/page/gelato.html. The race is on, as CPU-side previewers are also available, e.g. FPrime for Lightwave: http://www.worley.com/fprime.html.


My name's Thomas Ludwig (or lycium, ldwtho001(at)mail.uct.ac.za). I noticed in one issue of RTNews you listed lray as "a ray tracer with caustics, soft shadows, and implicit surfaces".

I currently have two ray tracers online on my page http://lycium.cfxweb.net, lRay and trace3D. trace3D was my experimental renderer for learning about monte carlo methods in ray tracing (soft shadows, global illumination etc), and is really far from realtime. lRay, on the other hand, is my (by recent standards set by the realstorm guys ;) simple real time ray tracer, which supports neither caustics or soft shadows (it does however support implicit surfaces). Actually, even trace3D doesn't really have any support for doing caustics efficiently (via photon mapping).


If you haven't seen it yet, check out recent papers by Vlastimil Havran, in particular V.Havran and J.Bittner: "On Improving KD-Trees for Ray Shooting", Proceedings of WSCG'2002 conference, pp. 209--217, February 2002, http://www.mpi-sb.mpg.de/~havran/ARTICLES/wscg2002.pdf.

Since that review copy of the book went out, we implemented more or less the stuff described in that paper, and it works quite nicely. To the point where we simplified the grid in the book to just be a dumb uniform grid, since all the extra code and discussion of doing it hierarchically wasn't worth the bang for the buck.

The papers on interactive ray-tracing by Wald et al also have some good discussion of building good kd-trees.

There are some nice visualizations of how various acceleration schemes partition space for a variety of (mostly SPD) scenes here--pretty fun/interesting to look through.


[This site really is pretty cool. -EAH]

- Matt Pharr (mmp(at)graphics.stanford.edu)


Go to http://www.realstorm.com/Download.html and grab "Still Sucking Nature" and run it. Pretty cool, it's some more real-time raytracing on the CPU - starts out slow, then gets to some fairly interesting environments. Lots of sampling artifacts, though, such as shimmering reflective pipes, etc, due to undersampling. Still, definitely worth 5 minutes of your time.

An article from the previous RT News about their engine is at http://turing.acm.org/pubs/tog/resources/RTNews/html/rtnv15n1.html#art3.


I just thought I'd drop you a note to let you know that I've finally got around to making available some source code for a ray tracer implementing my hoary old SMART alogorithm for octtree traversal, and taking your SPD models as input.

The source is in plain old ANSI C - no SSE, 3D-Now!, assembler or other such optimizations - and in case you are interested is available at http://www.johnspackman.ukf.net/raytrace/smart.zip with some brief accompanying documentation at http://www.johnspackman.ukf.net/raytrace/index.htm

As a coarse benchmark, the ray tracer is currently rendering a default sized sphere-flake in 3-and-a-bit seconds on a 1GHz Athlon machine; this includes pre-processing and ray tracing time since models are decomposed lazily into an octtree during during ray tracing.

An interesting feature of the ray tracer is that it can be made to output the octtree decomposition generated for a model in NFF format, which may be subsequently ray traced itself for a "cubist" visualisation of the decomposition. I've generated some very large models with this :-)

Happy ray tracing!

- John Spackman (JohnSpackman(at)ukf.net)


My physically-based renderer is available at: http://www.hxa7241.org/perceptuum/


...it *might* be interesting to some - it was engaging to write.

- h.artifex(at)virgin.net


"Interactive Raytraced Caustics", by Chris Wyman. Find this technical report at http://www.cs.utah.edu/~wyman/publications/index.html.


Due to budget constraints, SIGGRAPH 2002 was the last year there were Panel Sessions (happily they're back this year), and the first year I was actually on a panel - coincidence? Anyway, many of the panels' content are all available online: http://online.cs.nps.navy.mil/DistanceEducation/online.siggraph.org/2002/ Panels/. Of particular interest to ray tracers are "When Will Ray Tracing Replace Rasterization?" I'm also happy to find the Demo Scene panel is there, with even little movies of the panelists. In the last half of http://online.cs.nps.navy.mil/DistanceEducation/online.siggraph.org/2002/ Panels/06_DemoScene/haines2.mov I discuss some of the cool tricks of the "Heaven 7" real-time raytracing demo (at http://www.demoscene.hu/%7Epicard/h7/). More fun is Theo Engell-Nielsen's presentation on the demo scene: http://online.cs.nps.navy.mil/DistanceEducation/online.siggraph.org/2002/Panels/06_DemoScene/engell_nielsen.mov. The best is his Pixor spoofs, get them in http://www.scene.org/dog/files/sig02/siggraph02slides.zip.


In the hope that this might be useful for somebody, I've put two bibliographies for adjoints and importance up on the following web page:


"importance_background.bib" contains references to mathematics texts on adjoints in general and articles and books about the use of importance in nuclear physics (where it is an adjoint of neutron density).

"importance_graphics.bib" contains references to articles, dissertations, and books about the use of importance (defined as an adjoint of light) in global illumination and ray tracing. The entries are divided into five categories: 1) theoretical results in rendering, 2) "classic" ray tracing and distribution ray tracing, 3) finite element methods, 4) Monte Carlo methods, 5) volume ray tracing and participating media. The entries are sorted chronologically within each category.

The bibliographies are in BibTeX format and are intended to supplement Ian Ashdown's excellent radbib bibliography. Many thanks to Philippe Bekaert who pointed out some of these references. If you know of any references that should be added, please send me an e-mail.

- Per Christensen (per.christensen(at)acm.org)


Check out Rendermania, http://www.rendermania.com/, for RenderMan and HDR related information.


The 320 file format (!!!) shareware 3D object converter from Hungary now has its own web site. Check it out at http://web.axelero.hu/karpo/


Some of the research Philipp Slusallek, Ingo Wald, and others have been doing on accelerating ray tracing is being sold through a company Philipp started: http://www.inTrace.com. Interesting site, check it out.


Sourceforge (http://3d.foundries.sourceforge.net/ and http://sourceforge.net/softwaremap/trove_list.php?form_cat=100) is, of course, the first place to look for useful open source. In case you missed it, check out http://pygmalion3d.sourceforge.net/, a subdivision surface modeller (mainly for POV-Ray).

For more odd little bits (or large chunks) of code, some graphical in nature, check out Sweetcode: http://www.sweetcode.org/, and the archives: http://sweetcode.org/archives.html.

Code relating to the second edition of _Texturing and Modeling, A Procedural Approach_ by Ebert et al. is available at http://dynamo.ecn.purdue.edu/~ebertd/book2e.html. Admittedly the third edition is out now, but code doesn't rust. There's even a cloud ray tracer in Musgrave's area.


It's made waves as the "unifying equation for computer graphics" and other odd claims. That said, the so-called Superformula http://www.nature.com/nsu/030331/030331-3.html does give a nice range of interesting and evocative organic shapes: http://astronomy.swin.edu.au/~pbourke/curves/supershape/.


In a similar vein, check out the Electric Sheep screensaver: http://electricsheep.org/. Some nice concepts here: each user of the screensaver downloads the latest MPEG movie made, which is used as a screensaver. At the same time, each user contributes compute power while in screensaver mode to make a frame of the next movie everyone will get. If you leave the screensaver by hitting the "up" key (I believe), you'll give it a thumbs up and so request that the pattern used to generate the movie continue to evolve. There will be a sketch at SIGGRAPH about it. Me, I want the version that automagically distributes data that raytraces interesting little screensaver movies.


As a coauthor, I'm definitely guilty of visiting Amazon and checking out the sales rank, etc. I even wrote a Perl program once that sucked ranking data off the site hourly, mostly because I was curious about the semi-random algorithm they use that has your rank jump from 11,635 to 5,119 in an hour ("woohoo! we sold one!"). Now someone has a service that does pretty much this same thing: http://www.junglescan.com/, e.g. our latest book's ranking is at http://1.amazonscan.com/scan/details.php?asin=1568811829 (I recommend you buy a few and see how the graph changes).

While we're on the topic of Amazon, if you haven't heard the word, you can now search inside many of the books sold there. Clever system: it finds relevant pages, lets you look at each and the two pages before and after. If you like what you see, you'll probably get the book. Start searching at http://www.amazon.com/exec/obidos/tg/browse/-/10197021/.


Quote for the day, from Ditherati (http://www.ditherati.net/archive/):


"You can make this stuff. You can make things that are better than what you get out of Hollywood."

- Apple CEO Steve Jobs, on unveiling new desktop movie-editing tools that promise to put his other company, Pixar, out of business, San Jose Mercury News, 7 January 2003


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OpenEXR and Radiance Floating Point Image File Formats, by Simon Bunker (simon(at)rendermania.com), Steve Westin (westin(at)graphics.cornell.edu), and Greg Ward (gward(at)lmi.net)

Neil Gatenby asked about OpenEXR vs. the Radiance file format for storing synthetic images.

Simon Bunker replies:

The OpenEXR image format, developed by ILM, allows higher precision formats to be written and read, including support for the 16-bit floating point "half" format used in NVIDIA's Cg format. It is an extensible format that allows arbitrary buffers of data. The latest version's "exrdisplay" program uses hardware acceleration to implement the display pipeline. Find it at http://www.openexr.org/.

Simon Bunker writes (1/23/03) on globillum, comparing OpenEXR to Greg Ward's Radiance file format:

From a really quick look these formats look very similar. Except ILM's one has more dynamic range information as it uses 16 bits per channel instead of the 8 (I think) the .hdr/.pic format uses (which Greg Ward-Larson created for Radiance). Both use a mantissa for basic colour information, and an exponent - just OpenEXR has more of them! .pic uses 8bits /colour channel to save space. The fourth channel is the common exponent instead of an alpha. OpenEXR can have different exponents for each channel, plus you can mix data types between channels.

Radiance file format information can be found here http://radsite.lbl.gov/radiance/framer.html

.pic has the advantage that it is already supported in several applications - eg. HDRshop, Radiance and Lightwave. OpenEXR has the advantage it comes from ILM and has a cool geek factor.

If you want to store radiance maps in a high dynamic range format and use it in commercial software anytime soon then you'll have to use .pic. You could use OpenEXR if you a) write your own renderer b) are very good friends with someone who writes a renderer or c) are willing to wait for the tools to catch up.


Steve Westin replies:

First choice for compatibility: RADIANCE .pic/.hdr/.rgbe format. For more precision, TIFF either in floating-point format (compatible in the RenderMan world as texture map input, I think) or in LOGLUV format which Greg invented as a more universal sequel to pic/hdr/rgbe. Both are supported by libtiff (the Sam Leffler publicly-available library), but not by Photoshop.


Greg Ward replies:

I have been working with Drew Hess of ILM to get OpenEXR to compile under Mac OS X, and I have the following comments. The 16-bit/primary EXR "half" float format uses what's called an "s-e5-10" bit arrangement, for "sign plus 5 exponent plus 10 mantissa." This arrangement allows for correct sorting by bit pattern, and is the same format designed at SGI for their never-happened- generation of high-end graphics, which eventually became NVidia's cg system. It covers values of 6e-5 to 6.5e4, positive and negative, where the bottom part of the range "denormalized" to a linear form. This follows IEEE float conventions very closely, including a zero bit pattern that corresponds to 0.0 and representations for infinity and NaN, I believe. Heck, let me just make a table showing the different representations I know.

The comparison is here: http://www.anyhere.com/gward/hdrenc/. The summary is that OpenEXR compares favorably with Radiance format, offering 9 orders of magnitude in dynamic range at much better precision (0.1% rather than the 0.8% of Radiance). The down side is that it takes up slightly more space, and the i/o library is C++ and a LOT more complicated. (It's quite nice, though.)

If what you care about is color accuracy, the LogLuv TIFF format is what I recommend. It uses a perceptual color space to get the most relevant information into the fewest bits. There's a 24-bit and a 32-bit/pixel format to choose from (plus a 16-bit/pixel luminance variant). A lot of programmers balk at the CIE XYZ color interface, but it's the best way to be sure of the color you're getting. Most of the other formats just assume you know, without making the color space explicit. Radiance supports an additional tag (as does floating-point TIFF) saying what the primaries are, but since Radiance doesn't allow for negative primary values, your gamut is restricted to the colors within this space. (Not so for EXR or floating-point TIFF.) Also, LogLuv TIFF is supported for reading by such common viewers as ACDSee, because it was incorporated into Sam Leffler's library to convert automatically to 8-bit RGBA. The main advantage I find, though, is that the separation of luminance and chrominance channels allows for very fast global tone-mapping using look-ups. I can load and tone-map a LogLuv TIFF in less time than it takes to decompress a JPEG of the same size!

Speaking of which, anyone who wants to play around with HDR images on the Mac should pick up and try out the Photosphere application I've been working on. It's available from the URL below for a limited time only to beta testers, and catalogs regular TIFFs and JPEGs as well as the various HDRI formats. It also has facilities for building HDRI's from multiple, hand-held exposures. It's the only program that can as far as I know. I also have a command-line tool that works on Linux and under Windows, but I need to recompile it for those platforms.

Photosphere at: http://www.anyhere.com/


Bretton Wade (brettonw(at)microsoft.com) followed up with this question: "Could you also speak to how you see these file formats playing into next generation graphics hardware from nVidia and ATI, and their support for floating point buffers?"

Well, I'm not much of a graphics hardware expert, but the fact that the OpenEXR "half" data type matches that used by nVidia in their new card has to be a plus. (I don't know anything about ATI's format, but I can't imagine it's too different.) Taking an EXR image, you should be able to go straight into a texture map on these cards. Both cards will have the problem of mapping their output to a display, since you presumably put your hardware between your software and your display, not between your software and more software. The study of how to map high dynamic-range imagery to low dynamic- range displays has garnered a lot of attention, recently. There were three papers in last year's [2002] Siggraph, and one at the Eurographics Workshop on Rendering.

[A bit more about this at http://www.nvidia.com/object/IO_7998.html - EAH]

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Raytraced Games, by Greg Alt (galt(at)eskimo.com)

[reprinted from the realtime raytracing email list, realtime_raytracing- [email protected]. -EAH]

I thought I'd share some things I came across and see if anyone has other links. I'm a game programmer professionally, and I'm very interested in real-time raytracing on the side. I'm especially looking forward to the next generation of game consoles, since I think they will be up to the task of allowing the first comercially produced real-time raytraced game.

As far as I know, there are two real-time raytraced games in existence:

RTChess: http://www.newimage.com/~rhk/rtchess/

AntiPlanet: http://www.virtualray.ru/eng/shots.html - a ray-tracer's dream, a world made entirely of spheres.

[More on this one at http://www.digit-life.com/articles/virtualrayengine/index.html?54121, and an in-depth article by one of the programmers of this system on using SSE for raytracing: http://www.digit-life.com/articles/rtraytracing/index.html. -EAH]

And one in the works: http://www.realstorm.com/Screenshots.html

Has anyone seen any others?

Also, one gap that I can see is research on animating characters with non- rigid vertex weighting (meaning a vertex can be influenced by 2 or more bones in the skeleton). Does anyone know of research into appropriate bounding structures for such a character? What I imagine would be necessary would be to also animate on demand -- only characters that are rendered would be animated at all, and only the bones that correspond to the rendered polys would be animated. Unfortunately, I haven't seen any research addressing these issues yet, and it seemms like a difficult (though solvable) problem.

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Randomly Sampling Bounding Volumes, by Philipp Slusallek (slusallek(at)cs.uni-sb.de), Pete Shirley (shirley(at)cs.utah.edu), Fredo Durand (fredo(at)mit.edu), and more

I wanted to do some tests of various ray/object intersection algorithms. To do so, I wanted to generate random rays that would pass through a volume of space. I had some vague recollections on the topic, but couldn't find or think of anything good. I asked on globillum, and thought I'd summarize some of the answers here (so I'll know where it's written down, for next time).


Philipp Slusallek notes a simple unbiased method if the volume is a sphere:

If I remember correctly from the global line people (Sbert and company), they are using two random points on the sphere and connect them. I have not verified this in the paper as I do not have access to the paper right now, so please be careful. It should at least be a good starting point for your search and, if I am not mistaken, also contains a proof (in one of the early papers).

---- Pete Shirley gave a nice summary of a method for any convex volume of space, which Bruce Walter and Greg Ward also outlined:

I am pretty sure the following observations are true using geometric probability for justification (i.e., I read this once but don't have a strong argument, but I think there is one using measure theory that I don't remember!)

* all points on the surface of a convex solid are equally likely to be hit by a random line

* the angle of incidence of random lines near a point follow a cosine distribution with respect to the normal at that point.

So I think the pseudocode is much as Greg Ward suggests:

1. Choose a random face based on area
2. Choose a random point uniformly on that face
3. Choose a random direction with respect to normal N

3 is most easily done by constructing a UVW basis with W = N, and then choosing (u,v,w) as follows:

phi = 2*pi*ra()  // ra is random on [0,1), e.g., drand48()
r = sqrt(ra())
u = r*cos(phi)
v = r*sin(phi)
w = sqrt(1-u*u-v*v)
direction = u*U + v*V + w*W

You could stratify in 4D for all this, but I think that would only start to pay off for billions of samples.


Fredo Durand notes: if you want mathematical justifications, see e.g. Luis Santalo, _Integral Geometry and Geometric Probability_, Addison-Wesley, 1976.

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Ray Tracing's Roots, by Juan Rayces, Andrew Glassner, Jonathan Maxwell, Phil Dutre (Philip.Dutre(at)cs.kuleuven.ac.be), and Chris Trendall (trendall(at)dgp.toronto.edu)

[This letter, from Juan Rayces, a consultant on lens design, was sent to Andrew Glassner; his colleague Jonathan Maxwell contributed his own research later on. I also asked on globillum about what people consider the origins of ray tracing. - EAH]

Dear Dr. Glassner:

I am a lens designer: stupid people used to call us "ray tracers". Like calling barbers "hair cutters".

For years I have browsed "An Introduction to Ray Tracing". I have found it interesting as the approach to ray tracing is somewhat different from methods used by most workers in our profession. The pictures in the book are fascinating.

I suspect that methods proposed in your book are used nowadays in lens design for investigation of stray light and ghost images in optical instruments.

Every time I have your book in my hands, though, it hurts me to see that the Preface does not give much credit to those who have been using numerical ray tracing for centuries for designing optical instruments.

Moreover, in Ray Tracing Bibliography, page 295 you may read:

"5. Appel, A., Some techniques for shading machine rendering of solids. AFIPS 1968 Spring Joint Computer Conf.,32, 37-45, 1968. First ray tracing paper, light ray tracing, b&w pictures on Calcomp plotter."

First ray tracing paper? Oh boy!

I always wanted to write to you about this subject but I could not prove to you that numerical ray tracing was used in lens design very many years before it was used to synthesize images. I myself started "tracing rays" with a mechanical desk calculator 24 years before this "first ray tracing paper"!

A recently published book, "Classical and Evolutionary Algorithms in the Optimization of Optical Systems" by Darko Vasiljevic (Kluwer Academic Publishers, 2002), throws light on the subject of numerical ray tracing history. The book starts on page 1 with the following statement:

"The optical design is one of the oldest branches of physics, with a history going back over three centuries ago" . And after a few lines: "The first ray tracing is done probably by Gascoigne who died in 1641 at the age of 24. His work is not published but it is known to Flamsteed, the first English Astronomer Royal."

It is regrettable that Dr. Vasiljevic does not tell us where he got that information. If this is important to you may be you want to send him an email to "darkov(at)diginaut.com" (obtained from his page at the Internet). I tried contacting him on a different subject and I did not have any luck.

Besides this historical question, may I take exception to this statement of yours (Preface, page ix):

"When designing lenses, physicists traditionally plotted on paper the path taken by rays of light starting at a light source, then passing through the lens and slightly beyond"...

The idea that anyone believes that lens systems were ever designed by graphical ray tracing bends my mind. Even in the days of painful use of log tables photographic objectives were designed using numerical methods for tracing rays on meridional planes. Astigmatism along the principal rays was always computed. Some ambitious people even traced skew rays. Excellent lenses were designed.

Very sincerely yours,

Juan Rayces


Andrew Glassner replies:

Early history? Allowing for some poetic license in the definition of ray tracing, we have the "Mo Jing", a text written circa by Mo Zi in China circa 450 BC. In it are discussions of the camera obscura and the principles of concave mirrors with their focal points (the "central fire.") This work was unfortunately lost eight centuries later, but resurfaced in the eighteenth century.

Moving forward 1,600 years or so (this is ancient China, where such time spans are immaterial), we have Sung dynasty scientist Shen Gua with his "burning mirrors," wherein the focal point is likened to an oar lock. He wrote, "The oarlock constitutes a kind of 'pivot point'. Such opposite motion can also be observed as follows: when one's hand moves upward, the pinhole image moves downward and vice versa."

Another two hundred years or so brings us to the thirteenth century photometrist Zhao Youqin, who with his experimental apparatus developed the inverse square law and the circle of confusion for finite apertures with pinhole cameras. He used "1,000 candles" as his light source, which may or may not have been more poetic license, but only in terms of quantity.

There are also Indian tracts on optics of similar antiquity, but it is somewhat harder to separate observation and theory from the supporting religious philosophy.

Ray tracing? I cannot imagine that the ancient Chinese scientists (in the modern sense of the word) did not develop at least the rudiments of ray tracing, if only to illustrate their treatises.


From Johnathan Maxwell:

Juan: Thank you for mailing me about image synthesis through raytacing. Like you, when I first saw a book on image synthesis using raytracing (about 12 years ago, I suppose). I was bowled over and most impressed that our good old friends rays were being put to such good use.

As for me, alas, I do not know as much about Gascoigne and Flamsteed as you imply I might, Juan, and your mail has sent me into a fluster of "research" [see below -EAH].

While browsing through the books within reach of my desk for information on the history of ray tracing, I gained some amusement in this quotation from the first edition (1975) of "Optics" by Eugene Hecht and Alfred Zajac:

6.2 Analytical Ray Tracing

Ray tracing is unquestionably one of the designer's chief tools. Having formulated an optical system on paper, he can mathematically shine rays through it to evaluate its performance ... Thus while it would probably take 10 or 15 minutes for a skilled person with a desk calculator to evaluate the trajectory of a single skew ray through a single surface, a computer might require roughly a thousandth of a second for the same job and, equally important, it would be ready for the next calculation with undiminished enthusiasm.


These days, of course, you probably need to explain to people what a hand- cranked "desk calculator" was ... a machine designed to induce repetitive stress injury!


On globillum Chris Trendall writes:

Depends a little on what you mean by 'ray-tracing'.

How about Euclid's account of 'backward ray-tracing' (including perspective, reflection, and occlusion) http://www.chass.utoronto.ca/~ajones/cla203/eucarist.pdf http://www.albertson.edu/math/History/edietz/Classical/Optics.htm http://farside.ph.utexas.edu/~rfitzp/teaching/302l/lectures/node107.html http://www.du.edu/~jcalvert/classics/nugreek/lesson24.htm

Followed by AlHazan's work in 'forward ray-tracing' http://www-gap.dcs.st- and.ac.uk/~history/Mathematicians/Al-Haytham.html somewhat disputed in http://www.wordtrade.com/science/optics.htm

Continued by Newton and Kepler, who respectively unified colour and light, and described the function of a lens; Snell/Descartes who mathematically described refraction, all in the geometric optics tradition. As for theories on why certain objects have certain colours, we had to wait until the 20th century for reasonable theories of solids and quantum mechanics...

I guess that if you're just thinking of the geometry of the tracing of rays (including reflective and refractive rules) then Euclid -> Snell/Descartes covers this...


Phil Dutre writes:

There are of course Durer's famous woodcuts (see an example on the cover of the Global illumination compendium -- shameless plug :-) )

Also, Turner (not Whitted, but the British Landscape artist) studied the subject in the early 19th century, but that is of course not the earliest , but it was the first 'course' on perspective and accurate drawing at the Royal Society.

Also very interesting to read is David Hockney's "Secret Knowledge", where he explains how renaissance artists use optics to paint and achieve accurate geometry (basically using a camera lucida).

I guess you can make a point that if some caveman was tracing a silhouette cast by a campfire he was 'tracing rays' ;-)

[Interestingly, in Greek mythology the origin of art is attributed to someone (I forget who) tracing a silhouette on a wall. I forget the details, but believe it's in "A Short History of the Shadow" by Victor Stoichita. - EAH]


Finally, here's a document entitled The Origins of Raytracing? by Jonathan Maxwell, dated January 12, 2003, used with permission:

The question of who was the first raytracer is a fascinating one.

The Conventional History

I think as far as tracing rays in optical instruments is concerned, and taking the law of refraction as opposed to the law of reflection as being the key to it, the readily available standard literature regards Willebrord Snellius (1591-1626) as its inventor in Holland in 1621. Having said this, I accept that discovering the law of refraction is not necessarily the same as tracing rays. Snell's law was not well known until Rene Descartes (1596-1650) published it in 1638, and even then he did not make it clear that he was following Snell, so for some time Descartes was regarded as the originator of Snell's law.

As far as the explicit tracing of rays is concerned, it looks as though William Gascoine (1612-1644) in England was doing this in the 1620's or 1630's, but again this was not widely known until some time later (1692) when William Molyneux, the author of _Dioptrica Nova_, attributed his first clue of how rigorous raytracing might be done to hearing of it from the Astronomer Royal, John Flamsteed, who in turn attributed his knowledge of it to papers by Gascoine in his possession.

Going Back Further

However, the history of raytracing, especially with regard to refraction and also its more generalised application to image synthesis, and the creation of shadows (Sciography) and the like, goes back a lot further; to the Greek philosopher Thales of Miletus (circa 585BC). Thales is thought to have studied the geometry the Egyptians used and thereby came to understand what we now call trigonometry. He not only used this information for astronomy, but he also propounded a law of refraction, which turned out to be wrong.

Moving forward 1500 years, the Arab scientist Alhazan (Abu ibn al-Hasan ibn al-Haitham (965-1039 AD approx) wrote an important treatise on many aspects of optics in 7 books, using the concept of rays with arrows on to explain geometrical optics. These books contained sections on aberrations, and even diffraction, and were published in manuscript form in Arabic around about AD 984, and in Latin in 1269 (preceded by publication of his treatise on rainbows in 1170). Finally they were published in print in Latin in 1572. I do not know if Alhazan knew Snell's law; it wouldn't surprise me if he did, and If he did then it's Alhazan's law, rather than Snell's law.


The historical details of Snell's law and Gascoine's work can be gleened from books such as H.C.King's History of the Telescope (Griffin, London, 1955) and Isaac Asimov's _Eyes on the Universe_ (Imprint Magazine, Bombay 1977, but published before that in the UK I believe)

For the information on Alhazan I am indebted to a close friend Tareq AlBaho, and also _The Oxford History of Science and Technology_.

Robert Temple's _The Crystal Sun_ (Century/Random House, London, 2000) provided the information on Thales, and also further information on Alhazan.

back to contents

Faster Yet Point in Polygon Test? by Stephen Bridgett (s.bridgett(at)qub.ac.uk)

[The CrossingsMultiplyTest() in the online code for Graphics Gems IV, at http://www1.acm.org/pubs/tog/GraphicsGems/gemsiv/ptpoly_haines/, is the fastest general test (without precomputes) for point in polygon testing. It's a bit more efficient than the traditional test found in the comp.graphics.algorithms FAQ (http://www.faqs.org/faqs/graphics/algorithms- faq/), as it avoids division. The idea, from Mark Haigh-Hutchinson (now at Retro Studios), is that you can turn the division into a multiply by keeping track of which direction your test edge is going.

I can't say if the following will help, but knocking a little off this key algorithm can have a noticeable performance effect so it's worth a try if you're doing a lot of ray/triangle testing. - EAH]

I'm writing to say a big thank-you for your article in Graphics Gems IV on point in polygon testing. I'm a student in N. Ireland, and have been writing a small program in my spare time using Borland Delphi, to display models of a few engineering components, and had been looking for an efficient but easily implemented point in polygon test for concave polygons, and the 'CrossingsMultiplyTest()' was just what I needed. So thank you very much.

When converting it to Pascal, I noticed a possible slight improvement, which you've probably thought of already anyway (I hadn't - EAH). I'd read that on newer processors, conditional jumps could cause a slight cache delay. So, I was thinking that the 'if' after the multiply test could be replaced with an 'xor' test, as it has the same truth table, i.e.

Original 'if':
ytest:= (m[p1].z-tz)*(m[p0].y-m[p1].y)
>= (m[p1].y-ty)*(m[p0].z-m[p1].z);
if ytest = yflag1 then Result:= not Result;

(The original C version was:)

       if ( ((vtx1[Y]-ty) * (vtx0[X]-vtx1[X]) >=
             (vtx1[X]-tx) * (vtx0[Y]-vtx1[Y])) == yflag1 ) {
                inside_flag = !inside_flag ;

Using 'xor' test instead:
ytest:= (m[p1].z-tz)*(m[p0].y-m[p1].y)
>= (m[p1].y-ty)*(m[p0].z-m[p1].z);
Result:= (ytest=yflag1) xor Result;

I haven't tested it, and in practice in may not make much actual difference to the processing time, or may even slow it down, but it's a thought anyway.

back to contents

BSP Plane Cost Function Revisited, by Eric Haines

We all know a BSP tree (or octree, or k-d tree) helps in limiting the number of ray/object intersections needed. But how much? MacDonald and Booth [MacDonald89] already analyzed this problem fairly well, though due to space limitations they did not include analysis of what happens when a splitting plane hits one or more objects in a node. Also, their work, which has some fascinating and useful results, appears to be less known than it should; this article covers some of their contributions and goes a bit further.

MacDonald and Booth use the key fact that a child of a node has a chance of being hit proportional to its surface area, one that I will also use extensively (see RTNv10n1 http://turing.acm.org/pubs/tog/resources/RTNews/html/rtnv10n1.html#art4). As a shorthand term, I'll call this the "area expectancy algorithm". MacDonald calls it the "surface area heuristic", but at the time he probably didn't realize there was a solid proof of this result in geometric probability, so "heuristic" is a misnomer. For a formal justification, Fredo Durand recommends _Integral Geometry and Geometric Probability by Luis Santalo, Addison-Wesley, 1976.

What good does splitting a BSP node by a plane do, as far as ray tracing goes? The obvious thing it does is (usually) cut down on the number of objects to be tested against the ray. If a ray hits the node, the splitting plane is compared against the ray. The ray can do one of two types of things: it can progress through just one side of the splitting plane (i.e. one child of the node), or it can progress through one side and then the other (both children). If it progresses through just one side, you then need to test only those primitives on that side. If it goes through both sides, the potential savings are smaller (or nil): you have some chance of hitting a primitive on the first side traversed; if you hit (and the intersection point really is in the first part of the node), then no primitives on the other side need to be intersected. This can be useful for eye/reflection/refraction rays, since you want the closest intersection, but is usually little help in accelerating shadow rays.

Great Expectations

The chance of hitting a bounding box is proportional to its surface area. Say NX, NY, and NZ are the dimensions of the box. The chance of hitting a given child node is:

     node hit expectance == ENH = NX*NY + NX*NZ + NY*NZ

(Actually, there is a factor of 2 we leave out here and elsewhere, which accounts for the area for 6 faces vs. just 3. Since we leave it out everywhere, it is not important in this article. That said, it is important to also then treat polygons as one-sided when computing their expectance.)

This "expectance" is not a percentage chance, but rather a surface area measurement that can be compared with other surface areas to determine the relative chance an object is hit. So, given a random ray, a convex object with twice the surface of another convex object is twice as likely to be hit.

One principle of splitting a node is that you usually want to maximizes the chance that a ray will hit only one of the two children nodes. Say you split the node on the X axis at R, where R = 0 at minimum X and R = 1 at maximum X. The expectance of hitting each child box is then:

    child A hit expectance = ((R*NX)*NY + (R*NX)*NZ + NY*NZ)
    child B hit expectance = (((1-R)*NX)*NY + ((1-R)*NX)*NZ + NY*NZ)

If you divide either of these expectances by the expectance of hitting the node itself, you get the relative chance (a value from 0% to 100%) that a subnode will be traversed if the node is hit. The chance that one child or the other is traversed of course varies with R; lower R, lower chance of being hit. What's interesting is that the chance that both child nodes will be traversed by a ray is independent of R! This chance depends on the ray hitting the plane splitting the node into two children. The ratio of this area (the BSP plane inside the node) vs. the surface area of the node itself is this chance:

    plane hit expectance == EPH = NY*NZ

    plane hit relative chance == CPH = EPH / ENH = NY*NZ / (NX*NY + NX*NZ + NY*NZ)

(Notational note: a variable of the form "E*H" is always the area/2 of an object, which I call its expectance, and a variable of the form "C*H" is a relative chance of hitting something compared to hitting something else, computed by dividing one area by another.)

As an example, say the cube 1x1x1 is split by a plane:

    CPH = 1*1 / ( 1*1 + 1*1 + 1*1 ) = 1/3

So for a cube hit by a ray, an arbitrary plane splitting the cube has one chance in three of being hit, regardless of the split location. Since this axis-aligned splitting plane has the same area as a cube face, this also means that a given cube face has a 1/3 chance of being hit by a ray. This is surprising to me on an intuitive level. I would have thought that a plane inside the middle of the cube would have a higher chance of being hit than a cube face, but that's not the case.

To prove it to yourself, it's helpful to think about the area expectancy principle in terms of orthographically projecting the object on to a plane instead of shooting a ray at it. These are equivalent, in that you can think of shooting a ray through a pixel and testing against an object as the same as computing a pixel's chance of being covered by projecting the object into a z-buffer. So, visualize a box and the splitting plane inside it and how this would project orthographically on a screen. Slide the plane back and forth along one axis of the box and it becomes clear that the screen area of the splitting plane never changes. Therefore it's always equally likely to be hit, regardless of its position in the cube.

The irrelevance of R on the splitting plane's chance of being hit makes it easier to think about how best to split the node. Say the cube's dimension NX is increased slightly, to 1.1. If your plane cuts the X axis, it now has 1 chance in 3.2 (0.325) of getting hit by a ray hitting the box, less than a third. If your plane cuts the Y or Z axis, it now has a 1.1 chance in 3.2 (0.34375) of getting hit, greater than a third. This analysis verifies what we intuitively know, that splitting a box along its longest axis minimizes the chance a ray will hit both nodes and so is more likely to improve efficiency.

To compute a few more examples, say the box was 2x1x1: the values are then a 1/5th chance of both children being hit if a split is made on the X axis vs. 2/5ths for either other axis. For a 10x1x1 it's 1/21 along X vs. 10/21. For a flat, squarish box (like a CD case), say 10x10x1, splitting along X or Y gives 10/120 chance of hitting both boxes, while splitting along Z gives 100/120.

Childcare Costs

We want to learn something about the cost of traversing an unsplit node vs. a split one. The time to test an unsplit node is something like this:

    time for unsplit node = TK1 + M*TP

where TK1 is some constant time spent intersecting the node and accessing its contents, M is the number of primitives in the node, and TP is the time per primitive (which admittedly can vary, but let's ignore that fact for now; we'll come back to it later). Compare this to:

    time for split node = TK1 +
            time to hit both children + time to hit one child

Let's subdivide this equation into the four possible cases that a ray can have when it hits a node: hits child A then B, hits B then A, hits child A only, hits child B only. Call the probabilities of each CabH, CbaH, CaH, CbH. These must sum to 1. We also know that CabH+CbaH = CPH, the probability of hitting the splitting plane. It turns out that CabH==CbaH, i.e. that the chance of a ray going from child A to B is the same as the other direction, regardless of the location of the splitting plane. This is clear if you think about it, since a random ray is as likely to hit the splitting plane coming from one side as the other. So:

    CabH == CbaH == CPH/2. 

We can compute CabH and CbaH given CPH. What we want is the chance that one and only one child is hit. Basically, we want all random rays that pass through a child, minus the rays that hit the BSP plane face. For example, if R is 0, then child A has no thickness and so all rays that hit it will also hit the plane making up its face, so its chance of being the only child hit will be 0. As R increases, rays can enter the sides of the child and possibly not hit the BSP plane. When R is 1, then child A is equivalent to the original node. We know that there is a percentage chance CPH of hitting the BSP plane, so for this case we know that (1-CPH) is the chance of hitting child A but not the plane (i.e. not hitting both children).

At R=0 the relative area of child A is the same as that of the BSP plane, and the chance of hitting only child A was zero, i.e. (EPH-EPH)/ENH=0. At R=1 the relative area of child A is ENH, and the chance of hitting only child A was (ENH-EPH)/ENH=(1-CPH). We also know that at R=0.5 the chance for child A and child B is the same, (1-CPH)/2. The three values we know are on a line, and the function is linear with R. Putting these facts together:

    child A ONLY chance == CaH = (R*NX*NY + R*NX*NZ + NY*NZ) / ENH - CPH
    child B ONLY chance == CbH = ((1-R)*NX*NY + (1-R)*NX*NZ + NY*NZ) / ENH - CPH

As a reality check, we'll test a 1x1x1 node. When R is 0, there is no chance of hitting only child A and a 2/3rd chance of hitting only child B, leaving the CPH chance of hitting the plane of 1/3rd. When R is 0.5, each child has a 1/3rd chance of being the only one hit.

These two functions simplify down to:

    CaH = R*NX*( NY + NZ ) / ENH
    CbH = (1-R)*NX*( NY + NZ ) / ENH

With all this in mind, we can compute the average cost in time of a ray traversing a node with two children leaf nodes:

    node traversal cost = TK1 + TK2 + (1+CPH) * TK3 +
                            CabH * (Pa*TP + Ma*Pb*TP) +
                            CbaH * (Pb*TP + Mb*Pa*TP) +
                            CaH *(Pa*TP) +
                            CbH *(Pb*TP)
                        = TK1 + TK2 + (1+CPH) * TK3 +
                            TP * ( CUH * (Pa + Pb + Ma*Pb + Mb*Pa) +
                                   CaH*Pa + CbH*Pb )
                        = TK1 + TK2 + (1+CPH) * TK3 +
                            TP * ( (CPH/2) * ( (Mb+1)*Pa + (Ma+1)*Pb ) +
                                   CaH*Pa + CbH*Pb )

That's a lot of stuff; the last four terms are the four possible intersection cases, A then B, B then A, A only, B only. TK2 is the cost of testing the ray segment against the splitting plane and other fixed child access costs. TK3 is the cost of accessing a particular child node, multiplied by (1+CPH) since one or two children's lists could be accessed. Pa is the number of primitives inside only child A's box (Pb for child B). Ma is the chance of the ray missing all the primitives in child A and so having to access child B. For example, if there wasn't anything in child A, Ma would be 1.0, as the ray will always reach child B if it were going that way.

Let's give this function a try. Say the TK values are pretty small overall, maybe 0.2 of the time it costs to intersect a primitive, TP. Also assume that the chance that a ray going from one to another child has, say, a 20 percent chance of hitting a primitive in the first child and so not needing to test the second, so Ma = Mb = 0.8. Also say that the node is a cube of length 1x1x1 (though overall scale does not matter), so CPH = 1/3. The formula for these starting conditions are:

    node traversal cost = 0.2*TP + 0.2*TP + (1+1/3) * 0.2*TP +
                            TP * ( (1/6) * ((0.8+1)*Pa + (0.8+1)*Pb) +
                                   CaH*Pa + CbH*Pb )
                        = 0.666*TP +
                            TP * ( 0.3 * (Pa+Pb) +
                                   CaH*Pa + CbH*Pb )
                        = 0.666*TP +
                            TP * ( (CaH + 0.3)*Pa +
                                   (CbH + 0.3)*Pb )
                        = 0.666*TP +
                            TP * ( ((R*NX*( NY + NZ )/ENH + 0.3)*Pa +
                                   ((1-R)*NX*( NY + NZ )/ENH + 0.3)*Pb )
                        = 0.666*TP +
                            TP * ( (2/3)*R     + 0.3)*Pa +
                                   (2/3)*(1-R) + 0.3)*Pb )

So there's some fixed cost for traversing the BSP tree, in this case (2/3)*TP (or whatever, depending how you set the TK values), and the main cost is based on the number of primitives in each child times the chance of hitting only that child, which depends on R, the relative location of the splitting plane along the node's axis.

Let's say we put the splitting plane at R=0.5, the middle of the box, and that there were 12 primitives and they nicely divide evenly into 6 in each child (i.e. none are split by the BSP plane), so Pa = Pb = 6. We get:

    example cost = 0.666*TP + TP * ( (1/3 + 0.3)*6 + (1/3 + 0.3)*6 )
                 = 0.666*TP + 7.6*TP
                 = 8.266*TP

Now, if you had not split the node and intersected all the primitives in it, you'd have intersected 12 objects. So, by subdividing the node into two children, you expect to spend the time equivalent to 8.266 primitives (and on average you'll actually test the ray for intersection against 7.6 primitives, plus traversal overhead). The savings, in this case, are from the fact that there are two bounding boxes, and because there's a small chance that the ray stops in the first child due to intersecting something. So, there's a savings in subdividing a node by a BSP tree, but it doesn't cut the time in half, since a ray potentially can hit both nodes.

If you go through this same analysis and assume the worst case that the ray will always miss everything in the first child tested (i.e. Ma = Mb = 0.0), you'll find that the value 0.3 goes to 1/3 and the final answer is "0.666*TP + 8*TP". If you assume the unlikely event that if the ray enters the first child it won't escape and ever enter the second, the value 0.3 goes to 1/6 and the cost that comes out is "0.666*TP + 6*TP", which is exactly what you would expect, that any ray would test against either the 6 primitives in one child or the other, since the ray is (somehow) guaranteed to not continue and pass between the two children.

One minor idea (that I won't really pursue here) is to compute these occlusion terms based on the areas of the objects inside the node. For example, if we knew each of the six objects in child A has an expectance (surface area) of EOH, and child A's area expectance is EaH, then ((EaH- EOH)/EaH)^6 is a rough estimate of the chance a ray will miss all of these objects. For example, say a 3x3x3 node had 6 1x1x2 object boxes fully inside it somewhere. EaH is 27, EOH is 5 for each box, and the chance of missing them all is (22/27)^6, 0.293, so the occlusion term Ma is 1-0.293=0.707. This assumes the objects in the node are randomly distributed, etc. So it's possible to get a handle on the Ma and Mb occlusion chance terms, but it's work.

The occlusion term does matter, in our example it made the amount of variation go from 6 to 7.6 objects tested per ray. However, this term is meaningless for shadow testing because the shadow ray stops being tested whenever any object is found that blocks it. Testing the contents of child A before child B (or vice versa) does not matter. All that's important is that the BSP tree minimize the number of objects found near the ray - for this purpose, BSP planes that are more parallel to the ray vs. chopping it are much more important (the only exceptions are BSP planes that eliminate testing objects beyond the endpoints of the test shadow ray line segment). Our node traversal equation for shadow testing becomes considerably different, and I won't try to write it out here. It basically depends more on the chance of hitting each primitive in the node, since a hit ends the process.

Analyze This

Is this formula derived reasonable? Let's do an analysis of what happens when we chop our usual 1x1x1 uniform 1000 points of stuff by X, then by Y, then by Z, i.e. chopping it into 8 subnodes total. How much does this save overall? In the first chop each child has 500 objects. We know that there's a 1/3 chance of hitting both nodes, and of hitting only one of the two nodes. So there are 333 objects always tested, and 333 more objects that are tested once (in either one child or the other), for 666 total.

Now we divide each 0.5x1x1 (aka 1x2x2) child into two 1x1x2 subchildren. The average chance of hitting both of these subchildren, for a ray hitting the child, is 0.25, with an 0.375 chance of hitting only one child. For 1000 objects this would give 250+375=625 ray/object tests, on average. Splitting these 1x1x2 subchildren down to 1x1x1, the average chance of hitting both is 0.2, with 0.4 for hitting only one. 200+400 is 600 ray/object tests average when one of these is split and it contains 1000 objects. Note how the performance improves at each level, because a split always does more good the proportionally less surface area (and so less chance that both nodes are traversed) the splitting plane has. The more non-cubic a node is, the better the improvement.

Go down the tree: one way to think about things is that there's an average testing cost of 666 at the first level. Let's instead think of the number of objects tested as being reduced to 0.666 of the total amount. The next level of the tree causes the number of objects to go down to 0.625 of its current level, and the next after that causes a drop by 0.600. Multiply these together: 0.666*0.625*0.600 = 0.250. So we predict that only 250 "point" objects will be tested against a ray when using a BSP tree of three levels.

How does this analysis compare with just looking at the subsubchildren and their average chance of being hit? At the bottom level each child has an average of 125 real objects. Based on surface area, each of the eight subchildren turns out to have one chance in four of a random ray hitting them, so on average two subchildren are hit. So, three subdivisions gives about a 4x speedup, with 250 objects expected to be tested. These predictions match. That said, I believe neither is a perfect analysis, as both ignore the geometric realities of the situation. If a ray hits both children, it cannot also hit both subchildren of these children, it can hit only one additional subchild. In other words, you can draw a straight line through the interiors of only three, not four, of the squares in a 2x2 grid. The next level down again adds only one subchild, i.e. a ray going through a 2x2x2 cube can hit up to a maximum of 4 cells, not all 8. To be honest, I do not know how or whether this matters to the analysis, but at least the exercise shows that two independent analyses match. If anything, because there is an upper limit on how many cells can be visited, this may lower the expected number of tests further. I think this is a place where MacDonald's analysis may be faulty, but am not really sure.

1000 Points of Stuff

Where things get interesting is in analyzing how this function respond to various changes and conditions. One simple scene to analyze is the "1000 points of stuff" dataset. Imagine your 1x1x1 cube has 1000 objects in it, each dimensionless (i.e. each just a point), uniformly distributed. The Ma and Mb values, i.e. the chance of the first child traversed not stopping the ray, are also 1. Furthermore, let's also set the time for testing a primitive TP = 1, and ignore the fixed costs. This then gives us a function that shows how many primitives have to be tested given various settings of the dimensions, of R, and of the way objects in the scene are distributed. This leaves the simple function:

    simple NTC = CPH*(Pa+Pb) + CaH*Pa + CbH*Pb

This function is a fairly reasonable approximation of how the BSP-tree would be used in general. There's a little loss in efficiency from not caring that a ray could be stopped in one child before reaching the other. There's also a little overhead being ignored for traversing the BSP-tree, so this sort of balances out. By using this simpler function, we don't have to guess at what the chance is that a ray will be blocked by a child, and we can set up idealized situations and get "perfect" predictive results.

Expanding and simplifying this function out a bit:
    simple NTC = (Pa+Pb) * CPH +
                   Pa * R*(1-CPH) +
                   Pb * (1-R)*(1-CPH)
               = Pa * CaH(R) + Pb * CbH(R)

As discussed earlier, CaH(R) is the chance that child A is hit at all by a ray that hits the node (i.e. its area divided by the area of the node), CbH(R) is the chance for child B. This is, in fact, the formula that MacDonald uses in his first paper [MacDonald89].

MacDonald gives two interesting related results. He evaluates the cost of the spatial median strategy and the object median strategy. In the spatial median strategy, R is always 0.5, i.e. each node is always split in half. In the object median strategy, the objects are sorted along the axis and the plane inserted at the median of this object list. For example, if there are 1000 points of stuff, the plane is inserted halfway between point 500 and 501 on the sorted list; the corresponding R might be close to 0.5, but might also be very far from it. MacDonald's result: these two strategies will always yield the same cost. Surprising! Here's why.

For the spatial median strategy, CaH(0.5) = CbH(0.5), since the nodes are the same size. The number of objects in the scene is N = Pa+Pb. Also, CaH(R)+CbH(R)=1+CPH. So the cost function is:

    SpatialNTC(0.5) = Pa * CaH(0.5) + Pb * CbH(0.5)
                    = Pa * CaH(0.5) + Pb * CaH(0.5)
                    = (Pa+Pb) * CaH(0.5)
                    = N * CaH(0.5)
                    = (N/2) * (CaH(0.5)+CaH(0.5))
                    = (N/2) * (CaH(0.5)+CbH(0.5))
                    = (N/2) * (1+CPH)

In other words, as we've seen before, the number of primitives intersected is halved by splitting the node, but then CPH raises this value due to rays that hit both nodes. Note how this result is independent of how many primitives are actually in either node.

For the object median strategy, Pa = Pb, and so Pa+Pa=N, so the cost function is:

    ObjectNTC(R) = Pa * CaH(R) + Pb * CbH(R)
                 = Pa * CaH(R) + Pa * CbH(R)
                 = Pa * (CaH(R) + CbH(R))
                 = Pa * (1+CPH)
                 = ((Pa+Pa)/2) * (1+CPH)
                 = (N/2) * (1+CPH)

Again, notice how this cost function has no reference to the distribution of the objects themselves, it is purely a function of the number of objects to be intersected and the relative cross-sectional area of the node. MacDonald notes that this equivalence is useful, in that the optimal splitting plane must lie at or between these two R values (0.5 and whatever the object median R is). This bears repeating and stressing: splitting the node in half (R=0.5) gives the same average number of intersection tests as splitting the node at the median of the sorted list of non-overlapping objects (R is computed by the algorithm). The optimum R value lies between 0.5 and the median R value, and this optimum R value can actually give a cost value better than these two strategies.

Unfortunately, this rule falls apart as soon as objects in the node are pierced by the splitting plane, i.e. as soon as the objects have volume to them and can get split. To skip ahead for a minute, imagine one object spanning the interval [0,0.6] and another from [0.4,0.8]. The spatial median is R=0.5, of course, and the object median is R=(0.3+0.6)/2=0.45, but both of these planes pierce both objects and so have a cost value of 2. At R=0.4 and R=0.6, neither of which are in the interval [0.45,0.5], the cost values go down to 1.733. So, splitting objects throws off this concept.

As we saw with 12 items, splitting the 1x1x1 cube meant that 1/3rd of the rays hit both children and 2/3rds hit just one. With 500 points in each child when R=0.5, this gives an average of 666.66 objects that will be tested by a random ray. With R=0 or 1, the function properly predicts that 1000 objects will have to be tested (since no subdivision is actually taking place). The graph as R goes from 0 to 1 is a parabola centered at R=0.5, pointing upwards.

What if the distribution of points in the node is changed? Say we make it so that the 1000 points are all in the area [0.0,0.2]*NX of the node, with the rest of the [0.2,1.0]*NX volume empty. Do you have any guesses what the best value for R will be? With the object median strategy, splitting the number of objects so that half are in each child, you have R=0.1. You could also simply split the node in half. Another strategy says that there's a large volume of empty space, so the plane should be put right at R=0.2 so that one child is entirely empty and rays hitting only it can quickly be discarded. The object and spatial median strategies must give the same average, as we have seen, and for 1000 objects this value is always 666.66, no matter what the distribution - only the object median's R value varies. So we know the optimum value is between R=0.1 and 0.5. You could find this minimum value of R by expressing Pa and Pb as functions of R, substituting them in, and differentiating this new function and look for where the slope is 0 and at the bottom of a trough. Or you could be lazy like I am and write a Perl program to brute force its way through values of R and find the best one. It turns out the third, "bound the primitives" strategy is the best in this case: R=0.2 gives a minimum value of an average of 466.66 objects that need to be intersected.

Say we do the same experiment, but with the interval [0.0,0.5]*NX filled with our 1000 objects, i.e. half the node is empty. The object median strategy predicts R=0.25, and the "bound the primitives" method says R=0.5 might be good, since it chops off the empty half of space. However, we know that the spatial median strategy picks R=0.5 and will give the same cost as the object median strategy, so the optimal answer lies between these two. Running the analysis, the answer turns out to be that at R=0.375 the average is 625 objects tested against the ray. The split at 0.375 makes child A contain 750 objects and child B contain 250. So there's some point at which it's worthwhile to include some objects in both children, but beyond that point you should always make the splitting plane such that the one child is all empty space.

For the 1x1x1 cube, this point turns out to be at R=0.333 (and at R=0.667, of course), a third. What this means is that if your objects fill up a third or less of the left or right end of a cubic node, then you want to put the splitting plane so as to contain them all in one child. If the objects in the node extend out further than this one third, you'll need to do some work to find the best splitting plane.

Now here is the surprising part: this is true for a node of any dimensions. The node could be 100x20x3, and it will still be true that if all the objects in that node are between say [0,0.2], you'll want to put R=0.2, and this is true up to [0,0.333] for a node of any dimensions along any axis.

Let's look at the cost function if we say that there is a uniform density along the interval [0,Rd], with [Rd,1] devoid of objects. The cost function is then (ignoring the scaling factor of the number of objects):

  F(R,Rd) = [ R*CaH(R) + (Rd-R)*CbH(R)   when R <= Rd
            [ Rd * CaH(R)                when R > Rd

The second part of this function, Rd*CaH(R), is obviously minimized when R=Rd; a higher R just increases the surface area of the child node A. This represents putting the splitting plane to enclose the entire data set in one child.

The first part of this function has a derivative with respect to R that can be used to find the minimum point of the function. This is located where:

    0 = 4R - Rd - 1


    R = ( Rd + 1 ) / 4

You can set Rd=R and solve: indeed, when Rd is 1/3rd, R is also 1/3rd. As Rd increases, R then increases slower (1/4th as fast), so the ideal split point moves from this 1/3rd point to 0.5 as Rd moves from 1/3rd to 1.0. So the 1/3rd answer is verified by the derivative equation, but I can't say I have a good sense of why the value 1/3rd is significant.

Here's another example to ponder. Let's say we put 400 points in the interval [0,0.4] and 600 points in the interval [0.6,1] (so there's nothing in [0.4,0.6]). So, the density of points is 1.5 times as high in the second interval vs. the first. Here's an illustration, showing the node, the relative densities, and the two places to split:

   |  #  #  #  #  #|       |# # # # # # # #|
   | #  #  #  #  # |       | # # # # # # # |
   |#  #  #  #  #  |       |# # # # # # # #|
   |  #  #  #  #  #|       | # # # # # # # |
   | #  #  #  #  # |       |# # # # # # # #|
   |#  #  #  #  #  |       | # # # # # # # |
   |  #  #  #  #  #|       |# # # # # # # #|
   | #  #  #  #  # |       | # # # # # # # |
   |#  #  #  #  #  |       |# # # # # # # #|
   |  #  #  #  #  #|       | # # # # # # # |
   | #  #  #  #  # |       |# # # # # # # #|
   |#  #  #  #  #  |       | # # # # # # # |
   |---------------+-------+---------------+---> X axis
R=0.0             0.4     0.6             1.0

Where is the best place to split this set of points? It will probably be in [0.4,0.6], the empty zone, but where? It turns out that R=0.6 is the best place. The two choices of splitting planes are symmetrical, so the rule appears to be to put the plane closer to the volume with the higher density of objects in it - this makes a certain sense. We could have guessed this from the rule that the optimum is between 0.5 and the object median location (which is at 0.666).

By the way, if the densities were the same, 500 points in each of the two intervals, any plane in the interval [0.4,0.6] has the same minimal cost function, so any would be an acceptable split. Note that there's no preference for R=0.5 by the formula, i.e. the centerpoint is not superior to where either set of data begins.

Stuck in the Middle with You

Up to this point our primitives have all been dimensionless points. The next thing to do is to start looking at the effect of the BSP plane splitting objects. This is where inefficiency comes into the BSP scheme. For example, if your splitting plane intersects all 12 objects in the node, then both children have 12 objects in them, and no efficiency is gained (in fact, it's lost, since a ray has to waste time accessing the useless, no good, dead-beat children). The function above does not show this fact.

Assume mailboxing gets used in the code, so that any primitives in both children get tested only once by a particular ray. Also assume that the cost of checking the mailbox is negligible (we could add it in, but it just confuses things further). This is the formula with it all (well, almost - it's also possible to vary TP per object, which I'll discuss later). From here on in we'll be playing with this function and trying various simplifying assumptions and see what happens. Here it is:

    node traversal cost = TK1 + TK2 + (1+CPH) * TK3 +
                            CUH * (TK3 + (Pa+Pab)*TP + Ma*Pb*TP) +
                            CUH * (TK3 + (Pb+Pab)*TP + Mb*Pa*TP) +
                            CaH *((Pa+Pab)*TP) +
                            CbH *((Pb+Pab)*TP)
                        = TK1 + TK2 + (1+CPH) * TK3 +
                            TP * ( CUH * (Pa+Pb+2*Pab + Ma*Pb+Mb*Pa) +
                                   CaH*(Pa+Pab) + CbH*(Pb+Pab))
                        = TK1 + TK2 + (1+CPH) * TK3 +
                            TP * ( Pab +
                                   CUH * (Pa*(1+Mb) + Pb*(1+Ma)) +
                                   CaH*Pa + CbH*Pb)

where Pa are the primitives exclusively in child A, Pb only in B, and Pab are those shared by both children. The last step of the derivation is something of a jump. What happens is that we know that 2*CUH+CaH+CbH = 1, these are just the probabilities of hitting two children or just one, and so TP*Pab comes out of the function. In other words, any primitive cut by the splitting plane is just as expensive to intersect as if there was no splitting plane at all, since it will always be tested by any ray hitting the node.

This fact brings up an interesting idea for storage: any objects that are split by the BSP plane do not really have to be stored in both children. In other words, a split node could have three lists: objects in child A, in child B, and shared (hit by the split plane). When you test a node for ray intersection, you would first test its shared list, then traverse the two children. This would save a little memory: each object on the shared list is stored in only one list instead of in both children's lists. There's one other noticeable savings: mailbox avoidance. Shared objects are tested only once with this scheme. In the traditional approach, if a ray travelled through both children, each shared object would have to be accessed twice (once in each child's list) and the mailbox test would then avoid the second intersection test. Anyway, this alternate scheme is probably a wash performance-wise, and I'll bet someone's thought of it before I did, but it's one that occurred to me due to analyzing where time goes due to a BSP split.

Hair Splitting

One classic question is, given the choice of splitting a node in half by a plane along the X, Y, or Z axes, which axis should we pick? The functions above address that question in a direct way: choose the one that gives the lowest average number of ray intersections. Let's look at a 2x1x1 node with a uniform distribution of 1000 points. We can all guess that spliting along the center of the X axis is the best plan (so forming two 1x1x1 children), but how much does this help? The answer turns out to be that an X split yields an average of 600 ray/object tests, and a Y or Z split yields 700 ray/object tests, with 1000 being the unsplit value. I thought this was interesting: splitting the obvious axis does save us something, it saves 33% more tests (400 vs. 300 fewer tests). The limit of savings is, of course, 500 ray/object tests, which would happen with splitting something like a 1000x1x1 node, where the splitting plane area is relatively tiny and so extremely few rays hit both children.

What happens if we start paying attention to "Pab", the number of objects chopped by the splitting plane? This changes everything, of course, since you tend to want to minimize Pab.

Say you have a 1x1x1 node filled with 1000 straight strands of hair (or uncooked spaghetti, say) extending along the X axis. It's obviously more worthwhile to chop this node along the Y or Z axis, where Pab=0, because chopping along the X axis will result in each child having the same list of objects, Pab=1000 - no help at all. But what happens if the chance of hitting a strand of spaghetti varies? In other words, say space is filled with strands of spaghetti, but each piece of spaghetti is 0.5 units long.

For an N-length spaghetti strand the set of centers is [-N/2,N/2+1], and the overlap with a plane at say 0.5 is [0.5-N/2,0.5+N/2]. So, if all the X- aligned spaghetti strands were 0.5 units long and randomly overlap a 1x1x1 node, these would have centers from X=-0.25 to X=1.25 and would overlap a plane at say 0.5 from center X=0.25 to X=0.75. So the probability of being split by a plane at 0.5 is (0.75-0.25)/(1.25-(-0.25)) = 0.333.

Taking the difference of the endpoints of the intervals, the general formula is then simply P(N) = N/(N+1). So a strand 10 units long that is known to overlap the node has a 10/11 chance that it will pierce the BSP plane (regardless of where the BSP plane is placed, in fact - the plane's location is irrelevant).

So, say we have spaghetti bits of length 1 throughout space. That will mean that Pab=P(N)*1000=(1/2)*1000=500, and the other 500 1-unit spaghetti bits will divide between the two children, on average 250 each. From our previous analysis, 333+500=833 is the expected number of ray intersections that will have to be performed. In fact, varying the length of the spaghetti bits gives a function going from the "1000 points of stuff" minimum of 667 tests, up to the worst case of 1000 tests, depending on the length of the spaghetti bits. Let's look at the yield for a 1x1x1 node:

Object length    Ray/object tests     Savings
=============    ================     =======
     0.0               667              333
     0.1               697              303
     0.2               722              278
     0.3               744              256
     0.4               762              238
     0.5               778              222
     0.7               804              196
     1.0               833              167
     1.5               867              133
     2.0               889              111
     3.0               917               83
     4.0               933               67
     5.0               944               56
   infinity           1000                0

If the objects we're trying to split up by a BSP plane are 0.1 of the dimension of the 1x1x1 node, then we can save 30.3 percent of tests. If our objects are about the size of the node itself, 1.0, we can save 16.7 percent of tests, continuing to decrease as the object size increases. This implies that as the node size decreases with each BSP cut there will likely be diminishing returns in efficiency, since the relative object size increases compared to the leaf node size.

This is the End, My Only Friend

This section's for the only person to reach this section, whoever he is. We want to create the (nearly) ultimate optimal BSP-tree. We can split on any plane, and we can put the split wherever we want. MacDonald suggests searching between the object and spatial medians, sampling a number of times along this interval. But say we're insane and we want to optimize like crazy, which also means searching outside of this interval.

Here's one way to get the perfect (at least judged locally by the formula) split at any level. The idea is that when a plane is inserted, the only places you need to check is where something is happening, i.e. where an object's bounds in the node is located. MacDonald realized this, but he did not give an ultimate algorithm. So here we go:

For each axis X, Y, Z:
   Pab = Pa = 0
   Pb = number of objects
   Put the interval of each object that overlaps the node into a sorted list.
     If an object's minimum is outside the node,
          increment Pab, decrement Pb.
     If an object's maximum is outside the node, ignore it.
   Evaluate the cost function at every location on this list.
     If it's a start location, evaluate, increment Pab and decrement Pb.
     If it's an end location, decrement Pab, increment Pa, evaluate.

Use the splitting plane and location found to be optimal.

Dropping the constant TP, and including neither the fixed amounts of time spent nor the occlusion of one child by another, the cost function can have a few forms:

    F(R) = Pab + Pa * CaH(R) + Pb * CbH(R)
         = Pab +
           (Pa+Pb) * CPH +
           Pa * R*(1-CPH) +
           Pb * (1-R)*(1-CPH)

These are almost the same as the 1000 points of stuff formula, with the additional cost of objects hit by the split plane included in. The first form is more elegant, but hides the involved computations done by CaH(R) and CbH(R), so we use the second in examples that follow.

So, for example, say you have 3 objects overlapping a 10x10x10 node, going from [0,10] in all directions. Along the X axis object number one goes from [4,7], object two from [-2,8], and object three from [6,12].

object 1      |               |***********|
object 2 -2<************************************|
object 3      |                       |*********************>12
              |-------+-------+-------+-------+-------+---> X axis
             X=0      2       4       6       8       10
            R=0.0    0.2     0.4     0.6     0.8     1.0
When the first object is inserted into the sorted list, the list is (4s,7e), where the "s" means the location is where an object's interval starts, "e" where it ends. Object two's minimum extent, -2, starts outside the cube, so "Pab" increments up to 1 and "Pb" decrements down to 2. The sorted list is now (4s,7e,8e). Object three's maximum extent, 12, is outside the node, so can be ignored. The final sorted list of object starts and ends is (4s,6s,7e,8e).

Now go through the list and evaluate the cost function with each R value. Pab starts at 1 in this case (because of object two). Since the test node is an "s" start node, we know that an object is starting at this distance. So at R=0.4 there are no objects that would be placed in child A, one object is split, and two objects are in child B. So:

    F(0.4) = 1 +
             (0+2) * 1/3 +
             0 * 0.4 * 2/3 +
             2 * 0.6 * 2/3
           = 2.4666

So if we split here, we would expect to test for intersection 2.4666 objects with a random ray. Once we are done computing with the "4s" plane (and any other start plane), we add one to Pab (now 2) and decrement Pb (now 1), since we know that succeeding plane tests will hit two objects (one and two) until we reach the end of one of them. Pa is still 0.

We then test "6s". F(R) is computed first, and turns out to be:

    F(0.6) = 2 +
             (0+1) * 1/3 +
             0 * 0.6 * 2/3 +
             1 * 0.4 * 2/3
           = 2.6

Pab is then incremented to 3 and Pb decrements to 0. Note that in the interval [6,7] all three objects would be penetrated by a plane, so splitting the node would give no benefit.

We then hit "7e", and this signals the end of an object, so we first decrement Pab, to 2, and add one to Pa, to 1, before computing F(R), which turns out to be 2.8. Note that at any interval break, we always evaluate on the "side" where Pab is lower, by evaluating before incrementing Pab and after decrementing Pab.

The last entry on the sorted list is "8e", which first decrements Pab to 1 (only object three is left as being split), increments Pa to 2, and F(R) is 2.7333.

The smallest function value was at R=0.4, 2.4666, so a splitting plane is best put at 0.4. Note that the intervals of the objects should all be grown by some small epsilon, so that when the splitting plane is chosen it won't intersect the object that is close to it. The Y and Z axes can be tested similarly, and the axis plane with the smallest ray/object cost overall is the one that is chosen.

Recall the rule that if objects are near one end or the other the optimal splitting plane chops off the empty volume. So, a quick out on all this testing is to first find the bounding box for the objects. If along an axis the highest end value is less than R=1/3, or the lowest start value is greater than R=2/3, then you can safely put the split plane at this end or start value and be done with evaluating that axis.

Some other implementation notes: if the minimum value for a node comes back as being close to the number of objects (say within 5 or 10 percent - experiments are needed here, and your mileage will definitely vary), this means that inserting a plane is probably not buying much performance and so subdivision can halt. Alternately, you can fold in the fixed costs of traversing children, as shown near the beginning of this article.

You don't really have to track all three of Pab, Pa, and Pb; you can track just two and subtract these from the number of objects in the node to get the third.

As usual, you can also have a maximum level beyond which you won't subdivide. What's nice about the evaluation function is that it's pretty simple: you plug the box dimensions and how many primitives are to the right, left, or pierced by the plane and get an answer. You may also want to experiment with setting the chances a child blocks a ray, Ma and Mb.

The basic algorithm treats all primitives as equally costly to intersect, i.e. TP is a constant. If desired (i.e. you are truly out to massively optimize as much as possible), it's straightforward to factor in the cost of various types of primitives, assuming you have reasonable values for these. Instead of simply incrementing and decrementing the various counts such as Pab in the algorithm, you increment and decrement by the approximate amount of time that the primitive costs to test.

Now the bad news. Recursively doing this split procedure should be fairly near-optimal locally, but it will not necessarily be globally optimal. For a distribution of objects, it may be the case that a sub-optimal split at one level provides a number of perfectly optimal splits to be done at lower levels, giving a better tree overall. MacDonald discusses this problem and how simple BSP-tree algorithms outperformed his plane-searching method under certain circumstances.

Still, for "n" objects to be inserted into an BSP-tree, this algorithm is O(n log n) per level (due to the sort). It gives an interesting mix of the externally imposed octree mixed with splitting planes that try their best to tightly bound at least one object in the node itself.

[MacDonald89] J. David MacDonald and Kellogg S. Booth, "Heuristics for Ray Tracing Using Space Subdivision", Graphics Interface '89, pp. 152--163, 1989.

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RTNews Errata, by Nikos Platis (nplatis(at)di.uoa.gr)

[Included here in case someone's mirroring these pages or printed them out or whatever. I have fixed the web pages in question with this information. - EAH]

I would like to inform you of some mistakes that I believe I found in articles that I used.

In "Ray/Triangle Intersection with Barycentric Coordinates" (http://www.acm.org/tog/resources/RTNews/html/rtnews5b.html#art3), the denominators for s and t must be "(length N)^2". Also, in the picture, s and t (or vertices 2 and 3) must be swapped: s seems to be the barycentric coordinate with respect to vertex 2 and t the barycentric coordinate with respect to vertex 3; the same is implied by the relations given.

In "Pluecker Coordinate Tutorial" (http://www.acm.org/tog/resources/RTNews/html/rtnv11n1.html#art3), in the second relation provided ( L = {P-Q:PxQ} ), I think it should be clarified that the line is directed from Q to P; this is important since Pluecker coordinates describe directed lines, and also in accordance to the next relation, for which it would be U=P-Q.

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Eric Haines / [email protected]