Tutorial
Geometric Transformations 1

1. Show that the compound transformation rot(z,theta1)rot(z,theta2) is equivalent to rot(z,theta1 + theta2).
   
   
2. For the following scene, find the matrix T which transforms points from the block coordinate system, CS_block, into the fish-right-eye coordinate system, CS_eye. Such a transformation would be used when rendering the cube as seen from the point of view of the fish.
   
   
   
   
3. The compound transformation rot(x,90)rot(y,90)rot(z,90) is equivalent to what single rotation?
   
   
4. Find the series of transformations necessary to rotate the house about the axis q by theta degrees, assuming that the house initially lies in the xy-plane.
   
   
5. Find the inverse of the following transformation matrix:
   trans(a,b,c) rot(z,alpha) rot(y,beta)
   Express the answer in terms of trans and rot operations rather than actual matrices.
   
   
6. Find the inverse of the following matrix, assuming that the vectors A, B, and C are an orthnormal set. (Hint: There is a `clean' solution.)
   

   Ax Bx Cx Tx
   Ay By Cy Ty
   Az Bz Cz Tz
   0  0  0  1