*"Light Makes Right"*

September-December, 1987

Volume 0 (we're C programmers, right?)

Compiled by

All contents are copyright (c) 1987, all rights reserved by the individual authors

Archive locations: anonymous FTP at
ftp://ftp-graphics.stanford.edu/pub/Graphics/RTNews/,

wuarchive.wustl.edu:/graphics/graphics/RTNews, and many others.

You may also want to check out the Ray Tracing News issue guide and the ray tracing FAQ.

This is the first archive collection of "The Ray Tracing News". I hope you will get as much use out of it as I have,

Eric Haines, 3D/Eye Inc, 2359 N. Triphammer Rd, Ithaca, NY 14850 ...!hpfcla!hpfcrs!eye!erich___________________________

I'm presently keeping the list up-to-date. As far as adding new people to this mailing list, I'd personally like to see the list not grow without bounds. Given that the Intro to Ray Tracing course had the highest attendance, there's obviously a lot of people interested in ray-tracing. The group presently consists of researchers and people building ray-tracing systems, and it'd be nice to keep it at this level (and keep all of our long-distance e-mail costs down).

First off, a quick announcement: if you didn't get a copy of the "Intro. to Ray Tracing" course notes at SIGGRAPH 87 and would like a copy (they sold out twice), send me $20 and I'll xerox them. There are only three articles which are reprints - the rest is new stuff and pretty worthwhile.

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Netlib has two addresses. One is:

...!hplabs!hpfcla!research!netlib

(There may be other quicker routes to netlib - you'll have to research that yourself). The second one may be more convenient, as it is an arpa connection:

netlib@anl-mcs.arpa

So after you do "mail [...!]hplabs!hpfcla!research!netlib", the next step is to request what you want, one line per request. For example, to get my databases, simply type on a single line (and don't have anything else on the line):

send haines from graphics

and end the message. Netlib is computerized, and will automatically parse your message and send you the 82K bytes of dox & code for my databases.

The best way to find out about netlib and what it has to offer is to send it a request:

send index

and you'll get sent a listing of all the libraries available. It's mostly numerical analysissy stuff (lots'o'matrix solvers), but there are some things of interest. One particularly yummy database is the "polyhedron" contributions. There are some 142 polyhedra descriptions (vertices, faces, & edge descriptions and more). Some of these descriptions are buggy, but most are good (as netlib says, "Anything free comes with no guarantee").

As far as the question "what do the images look like?" goes, the images will be published in IEEE CG&A in November.

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To kick off discussion of this problem, John Peterson, who is researching this question at the University of Utah, was kind enough to spend some time on a response to me. His reply follows (printed with his permission):

___________________________

RE ray tracing splines.. I've sent a paper on ray tracing splines via polygons to TOG, but since that's going to be stuck in the review process for a while, here's an overview:

Most of the recent published stuff on this have been approaches using numerical methods; which I would avoid like the plague. I recently discovered that Whitted mentions surface subdivision very briefly in his classic paper (CACM '80) and also in Rubin & Whitted (SIGGRAPH '80). The technique they use was to subdivide the surface "on the fly", i.e., an area of surface is split only when rays come near it. Whitted doesn't go into any detail in these papers though, just a couple of paragraphs each.

However, Whitted's motivation for "subdivision on the fly" was lack of memory on his PDP-11 - nowadays I think you're better off just to do all the subdivision initially, then throw the surface away - much less bookkeeping. The polygon/bounding volume database isn't that huge if you use adaptive surface subdivision (more on that in a moment).

In terms of references, I'd highly recommend the "Killer B's" - "An Introduction to the Use of Splines in Computer Graphics" by Bartels, Beatty and Barsky. It appeared as SIGGRAPH tutorial notes in '85 and '86, and I think a book version is coming out from Kaufmann(sp?) in September. Another good reference is, "A Survey of Curve and Surface Methods in CAGD", by Bohm, Farin and Kahmann, in "Computer Aided Geometric Design", July 1984. Both of these give excellent background on all the math you'll need for dealing with splines. If you need to know about rationals, see Tiller's paper "Rational B-Splines for Curve and Surface Representations" in CG&A, September '83.

The subdivision algorithm I used is based on the Oslo Algorithm (Cohen, Lyche & Riesenfeld, Computer Graphics & Image Proc., Oct. 1980). This is a little slower than some of the other subdivision algorithms, but the win is that you're not restricted to specific orders or knot vectors. Since the subdivision time is generally small compared to the ray tracing time (like < 10%) I find it's worth the generality. (By the way, the Killer B's description of the Oslo algorithm is much easier reading than the CG&IP article. Sweeney's paper in the Feb. '86 CG&A also has a good description of it). Other subdivision classics are Ed Catmull's PhD thesis (U. Utah, '75) and Lane, Carpenter, Whitted & Blinn's article in the Jan. '80 CACM.

A couple tricks are noteworthy. First, if you're doing adaptive surface subdivision, you'll need a "flatness criteria" (to determine when to quit splitting the surface). I've found a good approximation is to take the bounding box of the control mesh, and find the point in the middle of it. Then project the size of a pixel onto this point, and use this distance as a flatness criteria.

Other trick: Crack prevention. If you split a surface into two parts, one part may get subdivided more than the other. If this happens, along the seam between the two halves you need to force the polygon vertices in the side with more divisions to lie on the edge of the surface with fewer subdivisions.

_________________________

My reply to John follows:

Thanks much for taking the time to tell me about splines and your findings. You leave me in a quandary, though. I'm interested in the numerical techniques for bicubics, but I also want to take your warnings to heart.

I have to admit, I have a great fear of throwing away the nice compact spline description and blow it up into tons of polygons. From your comments, you say that using adaptive techniques can help avoid this problem. You seem to divide to the pixel level as your criteria - hmmm, I'll have to think about that. Avoiding cracks sounds like a headache. Also, it seems to me that you'll have problems when you generate reflection rays, since for these rays the "flatness criteria" is not necessarily valid. Have you ever noticed practical problems with this (one pathological example I can think of is a lens in front of a spline patch: the lens magnifies the pixel sized patches into much larger entities. However, almost everything has pathological problems of some sort. Have you run into any problems due to your method on a practical level)?

I may try subdividing on the fly to avoid all this. To this end, have you looked at Ron Pulleyblank's routine for calculating bicubic patch intersections (IEEE CG&A March 1987)? What do you think of his "on the fly" subdivision algorithm?

Articles: thanks for the references. Have you seen the article by Levner, Tassinari, and Marini, "A Simple Method for Ray Tracing Bicubic Surfaces," in Computer Graphics 1987, T.L. Kunii editor, Springer-Verlag, Tokyo? Sounded intriguing - someone's hopefully going to get me a copy of it soon if you don't have it and would like a copy. If you do have a copy, is it any good?

__________________________

Now, here's John's response:

RE: Numerical techniques. I guess grim memories of round-off errors and consistently inconsistent results may bias my opinion, but there are some fundamental reasons for the problems with numerical methods. Finding roots of systems of two equations is inherently an unstable process (for a good description of this, see section 9.6 in "Numerical Recipes" by William Press, et. al.). Another way to think about iterative approximations in two variables is the chaotic patterns you see Mandlebrot sets. It's (very roughly) the same idea. Chaos and ray tracing don't mix...

Your comments about the flatness criteria are true, though in practice I've only found one "practical" instance where it really poses a problem. This is when a light source is very close to an object, and casts a shadow on a wall some distance away. The shadow projection magnifies the surface's silhouette onto the wall, and in some cases you see some faceting in the shadow's edge. The work-around is to have a per-surface "resolution factor" attribute. The flatness criteria found by projecting the pixel is multiplied by this factor, so a surface with a "res factor" of 0.5 may generate up to twice as many polygons as it normally would (extra polygons are generated only where the surface is really curved, though).

In order to get a feel for just how much data subdivision generates, I tried the following experiment. I took the code for balls.c (from the procedural database package you posted) and modified it to generate a rational quadratic Bezier surface for each sphere (as well as bounding volumes around each "group" of spheres). I didn't do the formal benchmark case (too lazy), but just choose a view where all the spheres (level 2 == 91 of them) just filled the screen. The results look like this:

Image Size Triangles (pixels) generated 128x128 7800 512x512 30400

The original spline surface definition wasn't small, each sphere has 45 rational (homogeneous) control points + knot vectors. My philosophy at the moment is that the algorithms for handling lots of trivial primatives win over those for handling a few complex ones. Right now the "lots of little primatives" camp has a lot of strong members (Octrees/Voxels, Kay/Kajiya/Caltech, Arvo & Co, etc). If you just want to maximize speed, I think these are difficult to beat, but they do eat lots of memory.

I'd be very interested in seeing a software implementation of Pulleyblank's method. The method seemed very clever, but it was very hardware oriented (lots of integer arithmetic, etc). I guess the question is whether or not their subdivision algorithm is faster than just a database traversal. Something like Kay/Kajiya or Arvo's traversal methods would probably scream if you could get them to run in strictly integer arithmetic (not to mention putting them in hardware...)

Cheers,

jp

__________________________

Anywell, that's the discussion as far as it's gone. We can continue it in one of two ways: (1) everyone writes to everyone on the subject (this is quick, but can get confusing if there are a lot of answers), (2) send replies to me, which I'll then send to all. I opt for (1) for now: if things get confusing we can always shift to (2).

[Actually, we're shifting to (2) around now, though it seems worthwhile to pass on your comments to all, if you're set up to do it. A summary of the comments will (eventually, probably) get put in Andrew's ray-tracing newsletter.]

More responses so far:

>From Jeff Goldsmith:

Re: flatness criterion

The definition that JP gave seems to be based on pixel geometry. That doesn't seem right. Why not subdivide until you reach subpatches that have preset limits in the change in their tangent vector (bounded curvature?) Al Barr and Brian Von Herzen have done some formal studies of that in a paper given this year. (It wasn't applied to ray tracing, but it doesn't matter.) I used that technique for creating polygonal representations of superquadrics with fair to good success. The geometric criterion makes sure that not much happens to the surface within a patch, which is what you really want, anyway.

I, too, by the way, believe in the gobs of polygons vs. one compicated object tradeoff. The two seem to be close in speed, but polygons saves big time in that you never need new code for your renderer. I hate writing (debugging) renderer code because it takes so long. Modeling code is much faster.

--Jeff

>From Tim Kay:

Subject: ray tracing bicubic patches

The discussion about subdivision was interesting. I just want to point out that a paper in this year's proceedings (Snyder and Barr) did just what the discussion suggested. The teapot was modeled with patches, and John hacked them up into very small polygons. He also talked about some of the problems that you run into.

Tim

__________________

>From Brian Barsky:

What numerical techniques is John referring to? He doesn't mean the resolvent work, does he?

____________________________

Response from John Peterson:

I was using a modified version of Sweeney's method. It was extended in two ways; first, a more effective means was used to generate the bounding volumes around the mesh, and it was able to handle surfaces with arbitrary orders and knot vectors. I wrote up the results in a paper that appeared in a very obscure proceedings (ACM Rocky Mnt. Regional Conference, Santa Fe, NM, Nov. 1986)

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My contribution for the week is an in-house memo I just wrote on transforming normals, which is easier that it sounds. Some of you have undoubtedly dealt with this long ago, but hopefully I'll notify some poor soul that all is not simple with normal transforms. Pat Hanrahan mentioned this problem in his talk at the SIGGRAPH '87 Intro to Ray Tracing course, but I didn't really understand why he was saying "don't use the modeling matrix to transform normals!" Now I do, and I thought I'd explain it in a fairly informal way. Any comments and corrections are appreciated!

**Abnormal Normals**

Eric Haines, 3D/Eye Inc.

The problem: given a polygon and its normal(s) and a modeling matrix, how do we correctly transform the polygon from model to world space? We assume that the modeling matrix is affine (i.e. no perspective transformation is going on).

This question turns out to be fraught with peril. The right answer is to transform the vertices using the modeling matrix, and to transform the normals using the transpose of the inverse (also known as the adjunct) of the modeling matrix. However, no one believes this on first glance. Why do all that extra work of taking the inverse and transposing it? So, we'll present the wrong answers (which are commonly used in the graphics community nonetheless, sometimes with good reason), then talk about why the right answer is right.

Wrong answer #1: Transform the normals using the modeling matrix. What this means is multiplying the normal [ x y z 0 ] by the modeling matrix. This actually works just fine if the modeling matrix is formed from translation matrices (which won't affect the normal transformation, since translations multiply the 'w' component of the vector, which is 0 for normals) and rotation matrices. Scaling matrices are also legal, as long as the x, y, and z components are the same (i.e. no "stretching" occurs). Reflection matrices (where the object is flipped through a mirror plane - more about these later) are also legal, as long as there is no stretching. Note that scaling will change the overall length of the vector, but not the direction.

So what's wrong? Well, scaling matrices which stretch the object (i.e. whose scaling factors are not all the same for x, y, and z) ruin this scheme. Imagine you have a plane at a 45 degree tilt, formed by the equation x = y (more formally, x - y = 0). Looking down upon the x-y plane from the z axis, the plane would appear as a line x = y. The plane normal is [ 1 -1 0 ] (for simplicity don't worry about normalizing the vector), which would appear to be a ray where x = -y, x > 0. Now, say we scale the plane by stretching it along the x axis by 2, i.e. the matrix:

[ 2 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ]

This would form a plane in world space where x = 2y. Using the method of multiplying the normal by this modeling matrix gives us a ray where x = -2y, x > 0. The problem with this ray is that it is not perpendicular to our plane. In fact, the normal is now 2x = -y, x > 0. Therefore, using the modeling matrix to transform normals is wrong for the stretching case.

^+Y ^normal ^+Y .../^normal | \ . | \__ .../ | \ . | \__../ | \. | \__ | \ | \_ +---------> X+ +---------------> X+

Figure 1 (a) & (b) - Incorrect Stretching Transformation

Wrong answer #2: Transform the vertices, then calculate the normal. This is a limited response to the wrongness of method #1, solving the stretching problem. It's limited because this method assumes the normal is calculated from the vertices. This is not necessarily the case. The normals could be supplied by the user, given as a normal for the polygon, or on a normal per vertex basis, or both. However, even if the system only allowed normals which were computed from the vertices, there would still be a direction problem.

Say the method used to calculate the normal is to take the cross product of the first two edges of the polygon (This is by far the most common method. Most other methods based on the local geometry of the polygon will suffer from the same problem, or else the problem in method #1). Say the vertices are [ 1 0 0 ], [ 0 0 0 ], and [ 0 -1 0 ]. The edge vectors (i.e. the vector formed from subtracting one vertex on the edge from the other vertex forming that edge) are [ 1 0 0 ] and [ 0 1 0 ], in other words the two edge vectors are parallel to the +x and +y axes. The normal is then [ 0 0 1 ], calculated from the cross product of these vectors.

If we transform the points by the reflection matrix:

[ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 -1 0 ] [ 0 0 0 1 ]

the result is the same: none of the edges actually moved. However, when we use a reflection matrix as a transform it is assumed that we want to reverse the object's appearance. With the above transform the expected result is that the normal will be reversed, thereby reversing which side is thought of as the front face. Our method fails on these reflection transforms because it does not reverse the normal: no points changed location, so the normal will be calculated as staying in the same direction.

The right (?) answer: What (usually) works is to transform the normals with the transpose of the inverse of the modeling matrix. Rather than trying to give a full proof, I'll talk about the three types of matrices which are relevant: rotation, reflection, and scaling (stretching). Translation was already seen to have no effect on normals, so we can ignore it. Other more obscure affine transformations (e.g. shearing) are avoided in the discussion, though the method should also hold for them.

In the case of rotation matrices and reflection matrices, the transpose and the inverse of these transforms are identical. So, the transpose of the inverse is simply the original modeling matrix in this case. As we saw, using the modeling matrix worked fine for these matrices in method #1. The problems occurred with stretching matrices. For these, the inverse is not just a transpose of the matrix, so the transpose of the inverse gives a different kind of matrix. This matrix solves our problems. For example, with the bad stretching case of method #1, the transpose of the inverse of the stretch matrix is simply:

[ 0.5 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ]

(note that the transpose operation is not actually needed in this particular case). Multiplying our normal [ 1 -1 0 ] by this matrix yields [ 0.5 -1 0 ], or the equation 2x = -y, x > 0, which is the correct answer.

The determinant: One problem with taking the inverse is that sometimes it isn't defined for various transforms. For example, casting an object onto a 2D x-y plane:

[ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 0 ] [ 0 0 0 1 ]

does not have an inverse: there's no way to know what the z component should turn back into, given that the above transform matrix will always set the z component to 0. Essentially, information has been irretrievably destroyed by this transform. The determinant of the upper-left 3x3 matrix (the only part of the matrix we really need to invert for the normal transform) is 0, which means that this matrix is not invertable.

An interesting property of the determinant is that it, coupled with method #2, can make that method work. If the determinant of the 3x3 is positive, we have not shifted into the mirror world. If it is negative, then we should reverse the sign of the normal calculated as we have entered the mirror universe).

It would be nice to get a normal for polygons which have gone through this transform. All bets are off, but some interesting observations can be made. The normal must be either [ 0 0 1 ] or its negative [ 0 0 -1 ] for this transformation (or undefined, if all vertices are now on a single line). Choosing which normal is a bit tricky. One OK method is to check the normal before transform against [ 0 0 1 ]: if the dot product of the two is negative, then reverse the normal so that it will point towards the original direction. However, if our points went through the z-reflection matrix we used earlier, then went through the transform above, the normals were reversed, then the object was cast onto the x-y plane. In this case we would want to have the reverse of the normal calculated from the edges. However, this reversal has been lost by our casting transform: concatenating the reflection matrix with the casting matrix yields the same casting matrix. One tricky way of preserving this is to allow 0 and -0 to be separate entities, with the sign of zero telling us whether to reverse the normal or not. This trick is rather bizarre, though - it's probably easier to just do it the simple way and warn whoever's using the system to avoid non-invertable transformations.

THE END:

Well, that's all for now. Do you have any comments? questions? interesting offerings for the group? Either send your bit to everyone on the list, or send a copy on to me and I'll post it to all. I realize this is quite a deluge of info for one message, but all of this has accumulated over a few months. The traffic so far has been quite mild: don't worry about future flooding.

All for now,

Eric Haines
*
more discussion of problem*

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Eric Haines / erich@acm.org