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

CS488/688 Introduction to Computer Graphics 57
18.8 Quadratic Bézier Curves [Last Used: Winter 2009 Final]
1. The diagram below shows three control points P 0 , P 1 and P 2 that together define a quadratic
Bézier spline P (t), t ∈ [0, 1]. Use de Casteljau’s algorithm to evaluate the spline at t = 1 3 .
Sketch the intermediate lines and points that are used in the algorithm and clearly label the
point P ( 1 3 ).
2. Based on the formula for an affine combination of two points, use de Casteljau’s algorithm to
derive an explicit equation for P (t) in terms of P 0 , P 1 and P 2 .
3. Suppose that we wish to place three additional control points Q 0 , Q 1 and Q 2 defining a second
Bézier spline segment, parameterized over [1, 2]. Give formulas for whichever of the Qs are
necessary to guarantee that the two curve segments will be joined with C 1 continuity.
4. Suppose now that we have three control points P 0 , P 1 and P 2 where P 1 lies at the midpoint
of the line segment joining P 0 and P 2 . Show that as the parameter t grows at a constant rate
from 0 to 1, a particle tracing out the curve P (t) will move along the line from P 0 to P 2 with
a constant velocity.
18.9 Splines [Last Used: Winter 2011 Final
(a) Draw and label four points P 0 , P 1 , P 2 and P 3 such th at the cubic Bézier curve P (t) derived
from them is a simple, closed curve. Sketch the resulting curve freehand, as accuratel y as
you can. (A curve is closed if it starts and ends at the sam e point. A closed curve is simple
if it never touches itself anyw here else.)
(b) Explain why it would be impossible to construct a simple, closed cubic Bézier curve that has
C 1 continuity everywhere. Incl ude a small diagram if you want.
(c) The diagram below shows three B-Spline control points B 1 , B 2 and B 3 , arranged in a closed
loop. These three control point s give rise to three cubic Bézier segments that meet each
other with C 2 continuity. Draw and label the twelve Bézier control points {P 0 , P 1 , P 2 , P 3 },
{Q 0 , Q 1 , Q 2 , Q 3 }, and {R 0 , R 1 , R 2 , R 3 }. (It doesn’t matter which segment gets the P s, which
gets the Qs, and which gets the Rs, as long as they are labelled consistently.)