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This month there is only one article, Ray-Tracing with Affine Transforms.
We will be adding the following articles over the next few months. An high
efficiency single phase motor (Patent No. 5,300,870, granted April 5,
1994). The IMint interface (Patent applied for). Contouring in Hyperspace.
Strange Recursion. And many others.

(C)opyright 1995, Seven seas Software Inc., MathVISION Inc.
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                     Ray-Tracing with Affine Transforms
                              Otto J. A. Smith
                               June 13, 1995
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Abstract:

     We derive affine transformations that change the basis of a three
     dimensional world space. The new basis for this space is chosen in
     such a manner that many ray-tracing calculations are simplified,
     particularly when doing z-buffering. We begin with the simple problem
     of determining whether a point is contained in a triangle and extend
     our derivation to three space. We present some algorithm optimizations
     that can make real-world computer programs faster.

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Triangle Containment:

     Let Tri( v1, v2, v3) be a triangle in 2-space. Each v and x are
     2-dimensional row vectors. The parameters p1 and p2 are scalars.
     Figure 1

     Given a point x, determine if x is interior to Tri( v1, v2, v3). In
     order to make this determination use vectors from the triangle to form
     a basis for the space, then any point can be represented by a
     parametric equation:

     (A.) x = v1 + p1( v2 - v1) + p2( v3 - v2)

     Now observe the following:

       1. When p1 = 1 and 0 < p2 < 1 ,then x is on the line segment ( v2 ,
          v3) .
       2. When p2 = 0 and 0 < p1 < 1 , then x is on the line segment ( v1 ,
          v2) .
       3. When 0 < p1 < 1 and 0 < p2 < 1 and p1 = p2 , then x is on the
          line segment ( v1 , v3) .

     We constructed equation (A.) in the manner given here in order to
     insure that the three observations made above were true. Now if p2 <
     p1 then x is on the same side of the line determined by ( v1, v3) as
     v2 . If p2 > p1 then x is on the opposite side of the line. In other
     words, the point x is inside the triangle Tri( v1, v2, v3) when all of
     the following are true:

       1. p2 < p1
       2. 0 < p1 < 1
       3. 0 < p2 < 1

     Given the triangle Tri( v1, v2, v3) and x one can calculate p1 and p2.

     Given the triangle Tri( v1, v2, v3) alone, one can calculate a two by
     two working matrix W such that ( x - v1) W =[ p1, p2 ] where [ p1, p2
     ] is a row vector.

     We construct a matrix M and define W = Inv(M) . The notation Inv(M) is
     used to indicate the matrix that is the inverse of M . That is Inv(M)M
     = I , where I is the identity matrix.

     x = v1 + p1( v2 - v1) + p2( v3 - v2)

     Let M =

             |   v2 -  v1  |

             |   v3 -  v2  |

     Then:
       1. [ p1, p2 ] M = x - v1
       2. [ p1, p2 ] = ( x - v1)Inv(M)
       3. [ p1, p2 ] = ( x - v1) W

     In applications where many points will be checked against a single
     triangle this is a fast algorithm. It is fastest if p1 is calculated
     before p2 and verified against the limits zero and one so the
     algorithm can terminate prematurely if the point x is not contained in
     the triangle. If there are many points, the cost of calculating W =
     Inv( M ) is amortized over many points. W is a feature of the triangle
     and not the point being checked which is used in a multiplication with
     the matrix Inv( M ). When M is singular, W does not exist since v1, v2
     and v3, all lie on a straight line. If this is the case the point is
     contained in the "triangle" ( line segment ) only if it is on the line
     defined by any two verticies from v1 , v2 and v3, and in addition has
     verticies from this set in both a positive and a negative direction.

Discussion of Basic Algorithm:

     I have found no references to the derivation of this technique in the
literature although the technique presented here produces an algorithm that
could also be derived starting from barycentric coordinates. A discussion
of this relationship is included in the appendix for this paper.

Before extending the algorithm to three (and higher dimensional spaces) and
showing how it can simplify Ray-Tracing, particularly when doing
z-buffering, we would like to enumerate some of its advantages and
disadvantages.

One traditional method of determining containment in a triangle, or convex
polygon, is to generate the equations for the bounding lines and then to
determine which side of each bounding line the point falls on.

A second technique is based on the Jorden curve theorem. footnote1

I believe there are several reasons why the advantages of the triangle and
hypertriangle containment algorithms presented here are not commonly used.
One reason is that most computer 3-D graphics geometry has been based on
homogeneous footnote7 system of coordinates and transforms taken from the
theory of projective geometry, rather than affine transformations based in
the theory of linear algebra.

Another reason is that normally a basis for the space would be constructed
from the vectors (v2 - v1 ) and (v3 - v1) ( instead of (v3 - v2) ) so that
hidden advantages of choosing other sets of vectors for the basis of the
space have not been explored. We will call techniques, such as the one
presented above, in which the vectors are not constructed from a common
origin, Free Choice techniques. We will call techniques in which a common
origin is used, Common Origin techniques. In common origin techniques, one
point in the triangle, ( or hyper-triangle ) is chosen as the origin. In
order to generate the set of vectors that form the basis for the new
coordinate system, the value of this origin is subtracted from the values
of the remaining points in the triangle.

I posted a common-origin algorithm on the Internet without comment about
its derivation and recieved a reply from Prof. Dr. Heinrich Giesen pointing
out that it is equivalent to the Barycentric coordinate technique, which is
a well known technique. The demonstration of equivalency is included in the
Appendix .

We will not use the common origin technique in this paper because by using
free choice techniques, we can customize the results of our basis change to
give us additional information that is useful and increases the efficiency
of our algorithms.

Multi Dimensional Extensions:

     The techniques presented above are easily extended to
multi-deminsional spaces.

Let Tri( v1, v2, ... , vn) be a hypertriangle in (n-1) -space.

Given a point x, determine if x is interior to Tri( v1, v2, ... , vn) .

In order to make this determination use vectors from the hypertriangle
Tri( v1, v2, ... , vn) to form a basis for the space.

The Free Choice Extension:

     Any point x can be represented by a parametric equation:

     (B.) x = v1 + p1( v2 - v1) + p2( v3 - v2) + ... + pn-1( vn - vn-1)

     When the following conditions are true, the point is contained in the
     hypertriangle.

       1. 0 < pi < 1
       2. Given any i , j such that i < j then pi > = pj

Common Origin Extension:

     In the common origin technique represent an arbitrary point x as
     follows:

     (C.) x = v1 + p1( v2 - v1) + p2( v3 - v1) + ... + pn-1( vn - v1)

     When the following conditions are true, the point is contained in the
     hypertriangle.

       1. 0 < pi < 1
       2. p1 + p2 + ... + pn < 1 .

Given the triangle
Tri( v1, v2, ... , vn) and x we can calculate p1 + ... + pn.

Given the triangle
Tri( v1, v2, ... , vn) alone we can calculate a working matrix
W = Inv( M ) such that ( x - v1) W =[p1, p2, ... , pn] .

In this paper we will most frequently use the free choice technique to
choose vectors with which to calculate the matrix.

Two Perspective Problems:

     There are two perspective problems that are easily solved using
hypertriangle containment.

The two problems for which we disclose new solutions here are,

  1. The ray tracing perspective projection problem.
  2. The standard perspective problem.

The Standard Perspective Problem:

     The standard perspective problem is as follows. Figure 4 We are given
     the following information:

       1. The location of a single point t, in world coordinates in
          three-space.
       2. The location of the focal point f, of an eye or a camera in world
          coordinates in three space.
       3. The location of the projection plane P in world coordinates in
          three space. This location is determined by a set of three
          vectors.
          P is defined by the set { r1, r2, r3 } . footnote2

     We are asked to determine the point at which a line from the focal
     point of the eye f to the single point t intersects the projection
     plane P. If this point of intersection exists, we will call it t'. We
     are asked to determine the coordinates of t' in the rectangular
     coordinate system of the projection plane P. The projection plane
     represents the surface of a video display device, such as a CRT tube
     and the coordinate system in which we want the information returned is
     the two dimensional pixel coordinates of the CRT tube.

The Ray Tracing Perspective Problem:

     The ray tracing perspective problem is as follows, Figure 5 We are
     given the following information:

       1. The location of the focal point, f of an eye or a camera in world
          coordinates in three space.
       2. The location of the projection plane P in world coordinates in
          three space, (this is three points, the origin of the screen, the
          upper left of the screen and the lower right).
       3. A point x on P in world coordinates footnote3
       4. A two dimensional triangle T = Tri( v1, v2, v3 ) in world
          coordinates. This is a two dimensional triangle embedded in three
          space, its verticies are vectors with three coordinates. In the
          real world systems, the triangle is a portion of the surface of
          an object being ray-traced.

     We are asked to determine two different things:

       1. Does the line starting at the focal point f and passing through
          the point x, intersect the triangle T? If the intersect exists
          call it x'.
       2. If the intersection exists, what is a measure of the distance
          from the point x to its intersection on the triangle T at x'.

Ray Tracing Problem Solved:

     Let us use the free choice tetrahedron containment algorithm to
develope a ray tracing perspective technique. If the line starting at the
focal point f and passing through the point x intersects the triangle T,
the first item we are asked to determine, is equivalent to asking if the
point x is contained in the tetrahedron Tri( v1, v2, v3, f) . That is, is x
contained in the tetrahedron defined by the union of the triangle with the
focal point.? ( T U f ). The second item we are asked to determine, is a
measure of the distance from x to its intersection on T.

We use this measure for z-buffering. We get this measure as a side benefit
when we use the free choice technique of detecting whether x is in Tri( v1,
v2, v3, f ) . Let the point x be represented as: footnote4

(D.) x = f + p1( v1 - f) + p2( v2 - v1) + p3( v3 - v2)

Calculate the working matrix W = Inv( M ) that transforms a point into the
new coordinate system represented by the above equation.

The three by three matrix of row vectors vectors M is represented as:

M =

        | v1 -  f  |

        | v2 -  v1  |

        | v3 -  v2  |

from the three by three matrix M calculate W = Inv( M ). Represent W = Inv(
M ) as a matrix of column vectors.

W =

     | c1   c2   c3 |



The algorithm to obtain the information we need is then as follows:

  1. Calculate the dot product p1 = ( x - f) * c1 . If this product is
     greater than one or less than zero the algorithm terminates. Point x
     is not in the tetrahedron.
  2. Calculate the dot product p2 = ( x - f) * c2 . If this product is
     greater than p1 or less than zero the algorithm terminates. Point x is
     not in the tetrahedron.
  3. Calculate the dot product p3 = ( x - f) * c3 . If this product is
     greater than p2 or less than zero the algorithm terminates. Point x is
     not in the tetrahedron.
  4. If the algorithm has not terminated, and we reach this step, We can
     use p1 as a measure of the distance from x to the triangle T since p1
     is simply the ratio of the length of the line segment ( f, x) to the
     length of the line segment ( f, x') where x' is the point at which the
     line intersects T. Since ( f, x) has a fixed length, we can use the
     ratio as a measure.

In an actual computer system several things must happen in order to perform
a ray tracing of a world object. First, areas that are not partially
contained on the screen are clipped. One technique for doing this is
disclosed as part of the algorithm for the standard perspective technique
footnote5, then the real world coordinates for each pixel on the display
screen is generated, then the ray from the focal point through the screen
pixel is compared against triangles. The value of p1, our measure, is used
for z-buffering to determine which points are in front of or behind other
points.

Standard Perspective Problem Solved:

     In the standard perspective problem we are given the following
information:

  1. The location of a single point t in world coordinates in three-space.
  2. The location of the focal point f of an eye or a camera in world
     coordinates in three-space.
  3. The location of the projection plane P in world coordinates in three
     space, (this is three points, the origin of the screen, the upper left
     of the screen and the lower right.).

Let the viewing area of P that we are interested in be represent by three
coordinates in world-space. P = { r1, r2, r3 } .

That is, P is represented by the set of three vectors { r1, r2, r3 } . In
"real world" situations, the vectors ( r2 - r1 ) and ( r3 - r1 ) are
frequently chosen to be orthogonal, simply because the viewing area of the
CRT is rectangular and has right angle corners. We can think of r1 as the
lower left hand corner of the CRT, r2 as the lower right hand corner and r3
as the upper left hand corner. Let t be represented by the equation,

(E.) t = f + p1( v1 - f) + p2( v2 - v1) + p3( v3 - v1)

This equation has been generated differently than any of the above
equations. It has been constructed this way in order to make two questions
easy to answer.

  1. Is t a point that can be projected onto P such that t', the projected
     point is in the rectangle determined by { r1, r2, r3, r2 + ( r3 - r1)
     } ?
  2. What are the coordinates of t' in the two dimensional coordinate
     system determined by P where: footnote6
       1. r1 is chosen to have the coordinates (0,0)
       2. r2 is chosen to have the coordinates (1,0)
       3. r3 is chosen to have the coordinates (0,1)

Now construct W = Inv( M ) from M as determined by equation (E.) above
where we express M as:

M =

        | v1 -  f  |

        | v2 -  v1  |

        | v3 -  v1  |

We calculate W = Inv( M ) from M and let the columns of W be represented
as:

W =

       |  c1  c2  c3 |



Now in order to solve the standard perspective problem we do the following:

  1. Calculate p1 = ( t - f) * c1 . If p1 < 1 then the point t does not
     project onto the rectangle of interest on P and the algorithm
     terminates.
  2. Calculate p2 = ( t - f) * c2 . If p2 < 0 or p2 > 1 then the point t
     does not project onto the rectangle of interest on P and the algorithm
     terminates.
  3. Calculate p3 = ( t - f) * c3. If p3 < 0 or p3 > 1 then the point t
     does not project onto the rectangle of interest on P and the algorithm
     terminates.
  4. If the algorithm has not terminated, then the point t does project
     onto P in the area of interest and t' has the coordinates t' = (
     p2/p1, p3/p1 )

Optimizing the algorithm:

     There are several ways in which the algorithm can be optimized.
Without going into extensive detail we will give several of them here. Some
properties of the triangles are invarient from different viewpoints. In the
ray tracing problem only one of the rows of the three by three matrix M
depends upon the viewpoint when the matrix is generated using the
free-choice technique. Consequently, two rows of the matrix M are invarient
and need be calculated only once for all viewpoints. This has the added
benefit, that if the inversion is done by calculating the determinant by
expansion of minterms, some of the minterms need be calculated only once
for all viewpoints. Optimizations similar to the above can be realized for
triangles that share a common edge.

One can avoid dividing by the determinant when calculating the inverses and
postpone this computation until the comparisons have been made. In this
case comparisons are made against the values of the determinant instead of
one, and the divisions are only made if the containment has been
established and we need to complete the computation.

Any set of four non-coplanar (non-colinear) points in three space can act
as a basis for the space. From these four points we can find an affine
transformation that takes any point in world space into a new coordinate
system defined by these points. We can also find an inverse transformation
that takes points from our new coordinate system back into world space.

The affine transformation shares the advantages of the homogeneous
coordinate system with transforms. In both systems it is possible to
multiply (concatenate) transforms in order to obtain a new transform
equivalent to applying the transforms in sequence. Transforms for scaling,
reflecting, rotating and distorting are easily constructed and combined and
the inverses of these operations are easily found. This fact is not new,
and has been mentioned in the literature before. The main arguments in
favor of using a homogeneous coordinate system have been that, first, it
reduces the complexity of code. With modern object oriented languages the
code is only minimally more complex for the basic operations with a great
increase in simplicity for more complex operations. Secondly, homogeneous
coordinate systems are used because of the ease of explaining there use and
for historical reasons.

This paper is an outgrowth of a larger research project in computer
graphics that began with a 3-D modelling and display program that ran on an
Apple II-E written around 1983. That system did not use a homogeneous
system of coordinates and transformations but used affine transforms
instead to translate points and calculate perspective. The free choice
algorithms are a recent invention derived from that work. Containment is
calculated by generating an affine transformation that takes the the point
we are interested into a space in which simple comparisons can determine
containment and the new coordinate system contains desirable information.

In that 1983 computer system, "cameras", that is coordinate systems
associated with a viewpoint were stored or created from four points in
three space. For orthographic projections the three vectors generated from
these four points were orthogonal. Each camera could be used to generate an
affine transformation that would move a vector from world space to camera
space. Perspective transforms were done as they are traditionally done
rather than as presented herein above. Transforms could be convolved into
new transforms as they are in systems using homogeneous coordinates.
Inverses of transforms could also be calculated to take a vector from
camera space to world space. Transforms could also be converted back to
cameras.

The system provided a simple language in which sequences of transforms and
objects could be generated mathematically.

Some major advantages of the system were.

  1. It was significantly faster than equivalent systems on the same
     machine.
  2. Planes and lines did not need to be expressed in their cartesian form
     since they were easily expressed simply as two points or three points
     respectively.
  3. Projections onto polygons were simplified.
  4. The 30% overhead associated with storage of homogeneous coordinate
     system matricies was avoided.
  5. Transformations from any camera (coordinate system) to any other
     camera in the system was easily and simply achieved.

On a completely biased note, I prefer the kind of system presented here not
only because it is fast, but because the Mathematics of what is happening
appears to me to be simpler and more transparent to a user of the system.

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Appendix:

The Barycentric Coordinate Technique

     Represent an arbitrary point x using the equation:

     (F.) x = v1 + p1( v2 - v1) + p2( v3 - v1)

     Now the point x is contained in the triangle only when all of the
     following conditions hold:

       1. When 0 < p2 and
       2. When 0 < p1 and
       3. When p1 + p2 < 1 .

     We call this technique, the common origin technique. Figure 3 This is
     based upon a traditional convention that when calculating a new basis
     for a space, the origin of the vectors originate from a common point.

     This is equivalent to the using barycentric coordinates. Barycentric
     coordinates are based upon the fact that any point can be represented
     as the sum of the verticies of a triangle times a weight, where the
     sum of the weights equals one.

     (G.)
       1. x = p0v1 + p1v2 + p2v3
       2. p0 + p1 + p2 = 1
     Let p0 = 1 - p1 - p2, then (G.)-1 above becomes

     x = (1 - p1 - p2) v1 + p1v2 + p2v3 x = v1 + p1( v2 - v1) + p2( v3 -
     v1)

     This is identical to (F.) above.
     Return to referance to appendix

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                                 Foot Notes

   * footnote 1, Triangle Containment Techniques Compared

     Several of these techniques are compared by Eric Haines in Volume 5,
     Number 3, September 2, 1992, of the Ray Tracing News, an internet
     electronic journal. For triangles, his version of the barycentric
     coordinate system algorithm worked best for triangles, but the
     advantages were rapidly lost for polygons with more than three sides.
     We have some optimizations of the algorithm that drastically speed up
     these calculations.
     Return to place document

   * footnote 2, Coordinates of CRT

     That is, the three points { r1, r2, r3 } determine a rectalinear area
     in three space that we are interested in, possibly the origin, the
     upper left hand corner and the lower right hand corner of a CRT tube.
     Return to place document

   * footnote 3, Translating pixel to world coordinates.

     Translating pixel coordinates to world coordinates is easily
     accomplished. Let w0 be the world space coordinate of the origin of
     the screen at pixel coordinates (0,0) , let w1 be the world space
     coordinates of the top left of the screen at pixel coordinates (0,399)
     , let w2 be the world space coordinates of the bottom right of the
     screen at pixel coordinates (639,0) . .

     The world space coordinates of (X,Y) is then

     w0 + Y( w1 - w0 )/399 + X( w2 - w0 )/639.

     Generate 640 values for x and 400 values for y and access them by
     their indicies to avoid frequent recalculations when generating scan
     lines.
     Return to place document

   * footnote 4, Other free choice vector constructs

     It is also possible to substitute a representation as in equation (E.)
     below for equation (D.) and change some of the checking conditions.
     This may be desirable in some real world programs.
     Return to place document

   * footnote5, Other algorithms for clipping are known.

     There are other more effective ways of doing this clipping when using
     BSD trees and other data structures that partition the world space,
     rather than depending on the algorithms presented here alone.
     Return to place document

   * footnote6, Coordinates for projection plane should match CRT

     This coordinate system is used to make reading this paper easy. In
     "real world" computer applications, the coordinates would be chosen to
     represent pixel coordinates on a CRT tube.

     In otherwords, for a full screen 400 X 640 VGA system appropriate
     coordinates might be r1=(0,0), r2=(640,0), r3=(0,400).
     Return to place document

   * footnote7, Definition of homogeneous

     "homogenous", as used here does NOT mean a homogeneous system of
     equations, but derives from a projective geometry construct.
     Homogeneous coordinates are used in computer graphics as a redundant
     method of storing vectors. That is in 3-space a vector of length 4 is
     used to store a coordinate. The fourth coordinate is used to scale the
     first three coordinates. For example (12, 8, 4, 4), and (9, 6, 3, 3)
     both represent the coordinate (3, 2, 1).

     In a homogenous system of coordinates, the equivalent of a matrix
     multiply, combined with a vector addition, is stored in a four by four
     matrix.
     Return to place document

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