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

58 CS488/688 Introduction to Computer Graphics
(d) Recall that a Bézier curve can be constructed from n + 1 contr ol points P 0 , . . . , P n using the
equation
( )
n∑
n
P (t) = Bi n (t)P i where Bi n (t) = (1 − t) n−i t i .
i
i=0
Assume that points P 0 , P 1 , P 2 and P 3 define a cubic Bézier segment in the plane. Suppose
that we wish to draw a second cubic Bézier segment defined by points Q 0 , Q 1 , Q 2 and Q 3
in such a way that the two segments join with C 2 continuity. Give explicit formulas for the
locations of Q 0 , Q 1 and Q 2 , each in terms of the P s, that guarantee C 2 continuity. (You can
derive the formulas by drawing and examining a diagram of the two segments, or by working
from the mathematical definition directl y.)
(e) Bonus. Is it possible to position control points {P 0 , P 1 , P 2 , P 3 } and {Q 0 , Q 1 , Q 2 , Q 3 } in such
a way that the two cubic Bézier segments together form a simple closed curve that is C 2
everywhere? Either show control points that realize this construction, or prove that it is
impossible.
19 Transforms
19.1 Viewport Transformation [Last Used: Winter 2004 Midterm]
I have a picture that I’ve expressed in a normalized 2D coordinate system with top left corner
(0, 0) and bottom right corner (1, 1) (y grows downward, as in screen coordinates). I’d like to print
the picture as PostScript, centered on an 8.5” × 11” page with a one-inch margin. One unit in
PostScript is equal to 1
72
”. The picture should be centered on the page, and scaled uniformly to be
as large as possible while still fitting within the margins. The two coordinate systems are shown
below.