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

62 CS488/688 Introduction to Computer Graphics
19.7 Rotation About an Arbitrary Axis [Last Used: Spring 1994 Final]
Describe, in general terms, how you would accomplish a rotation about an arbitrary 3D line through
a point p in direction ⃗v (i.e., l(t) = p + t⃗v). Illustrate with a diagram.
19.8 Stroke Fonts [Last Used: Spring 1997 Midterm]
The Hershey stroke fonts define each character in its own local, rectangular coordinate system.
Suppose we wanted to slant or skew every character to the right, simulating the effect of handwriting.
This could be accomplished by a 2D transformation in the xy plane that maps a rectangle
into a parallelogram as depicted below:
Q’
Q
P
Assuming that the lower left corner of the rectangle is P and the upper right corner of the
rectangle is Q, the transformation is completely determined by observing that P maps into P and
Q maps into Q ′ .
(a) Write the equations to accomplish this transformation.
(b) Write the associated homogeneous transformation matrix.
19.9 Parallel Lines, Affine Maps [Last Used: Winter 1994 Final]
Do parallel lines map to parallel lines under affine transformations? If so, then prove it. If not,
give a counter example.
19.10 Orthographic Projections [Last Used: Spring 1994 Midterm]
The transformation T mapping [x, y, z, w] to [x, y, w] (where w may be either 0 or 1) is an example
of an orthograhic projection. Prove that T is an affine transformation.
19.11 Coordinate Frames [Last Used: Winter 1993 Midterm]
Let F ∞ be a Cartesian frame in a two space. Let F ∈ be a frame given by {[ 1 2
, 0, 0], [0, 1, 0], [1, 0, 1]}
relative to F ∞ . Let P , Q, and ⃗v = Q − P have the following coordinates relative to F ∞ :
P = [1, 1, 1]
Q = [2, 1, 1]
⃗v = [1, 0, 0].