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

CS488/688 Introduction to Computer Graphics 55
• In the figure above, draw and label any control points of Q whose placement is forced by
the continuity conditions. List any control points whose placement is not forced by the C 1
conditions here:
• Draw the de Casteljau evaluation of P at t = 1/2 in the figure above. Label P (1/2) on your
drawing.
18.5 Cubic Hermite with Bézier Segments [Last Used: Winter 1998 Final]
In cubic Hermite interpolation, we are given a set of n points and n vectors, and we want to find
a piecewise cubic C 1 curve that interpolates the points, where the first derivative of the curve at
these points should be the specified vectors.
Given the points and vectors below, draw and label the cubic Bézier control points for a piecewise
cubic that interpolates the points with derivatives that interpolate the vectors. Your curve P should
interpolate the ith data value at parameter value i, i.e., P (i) = P i and P ′ (i) = ⃗v i .
Label the ith control point of the jth Bézier segment as P j
i .
v
P 0
0
P 2
v1
P 1 P 3
v
4
P 4
v
2
v
3
18.6 Cubic Bézier Curves [Last Used: Fall 2002]
Below are six curves and their “control points/polygon.” Two of the control polygons are the
Bézier control polygon for the curve drawn with it; the other four are not. Indicate which of the
two control polygons are Bézier control polygons for the corresponding curve and which four are not
Bézier control polygons for the corresponding curve. Justify your answer for the control polygons
that are not Bézier control polygons.
You may assume that none of the control points overlap or are repeated.