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

CS488/688 Introduction to Computer Graphics 3
– P 1 =
– P 2 =
– P 3 =
1.4 Euler Angles [Last Used: Winter 2000 Final]
Euler angles are one method of representing orientations, with the orientation represented as a
rotation around each coordinate axis. While common in computer graphics, Euler angles have
several problems. To address these problems, people often use quaternions.
For this question, we will use the following representation of Euler angles: With c a = cos(θ a )
and s a = sin(θ a ), the matrix for Euler angles is
R(θ x , θ y , θ z ) =
⎡
⎢
⎣
c y c z c y s z −s y 0
s x s y c z − c x s z s x s y s z + c x c z s x c y 0
c x s y c z + s x s z c x s y s z − s x c z c x c y 0
0 0 0 1
= R x (θ x )R y (θ y )R z (θ z ),
where R x (θ x ), R y (θ y ) and R z (θ z ) are the standard rotation matrices.
The following Euler angles and quaternion represent (to 3 digits of precision) the same orientation:
θ x = 10, θ y = 90, θ z = 30
q 1 = 0.696 − 0.123⃗i + 0.696⃗j + 0.123 ⃗ k
1. (3 pts) Give a second set of distinct Euler angles β x , β y , β z that represent the same orientation.
Your angles must be in the range [0, 360] degrees.
β x =
β y =
β z =
⎤
⎥
⎦
2. (2 pts) Give the quaternion q 2 corresponding to your Euler angle solution in part (a) of this
question.
q 2 =
1.5 Animation Techniques [Last Used: Winter 2006 Final]
You are working on a scene in a movie that will incorporate ray traced CG characters of a Vice
President shooting (with buckshot) an apologetic lawyer in slow motion. (Buckshot is a type of
“bullet” that consists of many small pellets that spread out as they fly through the air.)
Describe what part of the animation each of the following techniques could be used for, and
suggest and justify the use of a fifth technique that would be useful in creating this animation.
1. Keyframing