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

CS488/688 Introduction to Computer Graphics 65
1. What are the coordinates of P relative to F w ?
2. What are the coordinates of P relative to F v ?
3. Give the change of coordinates matrix M wv mapping from F w to F v .
4. Give the change of coordinates matrix M vw mapping from F v to F w .
5. What is M wv M vw ?
19.17 Transformations [Last Used: Winter 2007 Midterm]
Suppose you are given a triangle in 2D with vertices at positions (0, 0), (1, 0) and (0, 1). In the
following, assume the positions of these vertices are expressed with homogeneous column vectors
as a = [0, 0, 1] T , b = [1, 0, 1] T , and c = [0, 1, 1] T , using an orthonormal basis in which the first
unit-length basis vector points to the right and the second points up.
1. Give a transformation matrix that translates this triangle by 5 units to the left.
2. Give a transformation matrix that rotates this triangle counterclockwise by 90 degrees.
3. Give a transformation matrix that translates this triangle by 5 units to the left and also
rotates by 90 degrees counter-clockwise about its own origin (that is, about point a in its
original coordinate system).
4. Suppose under some affine transformation T the triangle is mapped to a ′ = [4, 2, 1], b ′ =
[2, 4, 1] and c ′ = [4, 4, 1]. Find the matrix that represents this transformation.
19.18 Transformations [Last Used: Winter 2007 Final]
In the following, assume ∆ is a 2D triangle with vertices at A = [1, 1] T , B = [3, 2] T , and C = [2, 3] T .
Suppose P = αA + βB + γC with α = 1/6, β = 1/3, and γ = 1/2.
1. Is P = αA + βB + γC a valid affine combination? Justify your answer.
2. Is the point P in the interior of the triangle? Justify your answer.
3. Suppose points A, B, and C are transformed by an affine transformation T . Give an expression
with numerical values for the weights that expresses T (P ) as an affine combination of
T (A), T (B), and T (C).
4. Compute the coordinates of P in the same frame as A, B, and C.
5. Compute the coordinates of P relative to the frame formed by the vector B − A and C − A
with origin A.