CSC 418/2504: Geometric Transformations 3 -- Tutorial solutions

1. The projection is very similar to an orthographic projection discussed earlier in class, that setup by glOrtho(). The difference is that this projection is oblique in the yz-plane. Thus, the transformation for x and z stay the same; that for y changes.

   The equation of the top plane of the view volume is:

   y = -z tan(alpha) + top-near*tan(alpha)

   The equation of the bottom plane of the view volume is:

   y = -z tan(alpha) + bot-near*tan(alpha)

   Any point on the top plane of the view volume should map to y' = 1. Similarly, any point on the bottom plane of the view volume should map to y' = -1.

   y' can be written as a linear function of y and z:

   y' = a*y + b*z +c

   We'll set c=0 because from the diagram, the point y=0,z=0 should map to y'=0 (the center of the view volume). Note that this also fixes the angle alpha as being arctan( (top+bot)/2, near ).

   To calculate the constants a and b, we'll substitute in the constraints for the mapping of the top-plane and bottom-plane:

   1 = a*(-z tan(alpha) + top-near*tan(alpha)) + b*z
   -1 = a*(-z tan(alpha) + bot-near*tan(alpha)) + b*z

   This can be solved for a and b, yielding:

   a = 2/(top-bot),
   b = (a*top-1)/near

   The final matrix thus looks like:
   

   [ A 0 0 B ]
   [ 0 a b c ]
   [ 0 0 C D ]
   [ 0 0 0 1 ]
   

   where A, B, C, D are the same as in the orthographic case, and a,b,c are as defined above.
   
   
2. Multiplying a homogeneous vector by a 4x4 matrix takes 16x, 12+. The required operations are thus:

   modelview transformation: 16x, 12+
   projection transformation: 16x, 12+
   division: 3/
   viewport: 2x, 2+
   
   Total for transforming 1 point: 34x, 26+, 3/
   Total for transforming 3 points: 102x, 78+, 9/
   Total flops per triangle: 102 + 78 + 5*9 = 225
   10^6/225 = 4444 triangles per second

   Note: For lighting calculations, surface normals also need to be transformed, thus requiring some additional work.
   
   

3. The points can be organized such that every new point specifies a new triangle, as shown below. Many graphics libraries, including OpenGL, accept such special "triangle-mesh" primitives.
   
   

   1
   2
   3   vertices 1,2,3 specify A
   4   vertices 2,3,4 specify B
   5   vertices 3,4,5 specify C
   ...