Past Midterm and Exam Questions (PDF) - Student.cs.uwaterloo.ca ...

CS488/688 Introduction to Computer Graphics 63
Let C be the coordinate transformation from F ∞ to F ∈ with matrix representation M:
M =
∣
2 0 0
0 1 0
−2 0 1
Let T be an affine transformation whose matrix representation is M when mapping points with
coordinates relative to F ∞ to points with coordinates relative to F ∞
In the following, assume that P , Q, and ⃗v are represented relative to F ∞ .
1. Give the coordinates of T (P ), T (Q), and T (⃗v) relative to F ∞ .
2. Give the coordinates of T (P ), T (Q), and T (⃗v) relative to F ∈ .
3. Give the coordinates of C(P ), C(Q), and C(⃗v) relative to F ∈ .
4. What are the lengths of ⃗v, T (⃗v), and C(⃗v)?
.
∣
19.12 Inverse Transformations [Last Used: ?]
In general, given a transformation T represented by a matrix M, we must invert M to find the
matrix representation of T . Suppose, however, M is represented as the composition of transformations:
M = M 1 ◦ M 2 ◦ . . . ◦ M n ,
where each M i is a simple translation, rotation about the origin, or scalings about the origin. Find
an expression for M −1 that can be computed without explicitly inverting M.
19.13 Perpendicular Lines, Affine Maps [Last Used: Winter 1993 Final]
Do perpendicular lines map to perpendicular lines under affine transformations? If so, then prove
it. If not, give a counter example.
19.14 Projective Transformations [Last Used: Winter 2000 Midterm]
For the following question, assume that we are viewing the scene by looking down the negative
z-axis of a right handed VCS.
a) If we project an arbitrary point (x, y, z, 1) onto the near plane at z=-n, what are the coordinates
of the projected point?
b) Give a matrix that performs the transformation from part (a), assuming the usual normalization
(division by the last coordinate) will subsequently be applied.
c) Suppose we want to map the z values so that the near plane maps to z=a and the far plane
maps to z=b (instead of all z values mapping to -n) while mapping x and y to the same values
as in (a) and (b). Give a matrix to perform this transformation.