Homework 3 : Curves

Assigned: November 1
Due: November 8, 9:30am

Question 1:

In Chapter 10 of Hearn and Baker (the Curves reading in the reader), they give the matrix for a uniform, cyclic, cubic B-Spline to be:

-1	3	-3	1
3	-6	3	0
-3	0	3	0
1	4	1	0

Show that if this matrix is used to make a piecewise cubic curve, the resulting curve is C2 continuous.

Question 2:

Consider specifying a segment of a cubic by: the position of its end points, the position of its parametric center point (e.g. where the curve is at u=.5), and the tangent at the parametric center point.

Derive the matrix for this cubic that maps from these 4 controls to the canonical cubic parameters.

Question 3:

Consider specifying a cubic by the position of 4 points equally spaced in parameter space (e.g. the controls are points placed at u=0, u=1/3, u=2/3, u=1). Derive the matrix that maps from these 4 controls to the canonical cubic parameters.

Question 4:

Consider a quartic curve segment (that is, a polynomial of degree 4). The matrix that maps from the controls to the canonical parameters will be 5x5.

Derive the 5x5 matrix to map from equally spaced points along the curve (e.g. positions specified for u=0, .25, .5, .75, 1) to the canonical parameters of the quartic.

Question 5:

Plot the curve in Question 4 for the control points placed along the X axis at (1,0), (2,0), (3,0), (4,0) (5,0). Now plot the curve again with the first point moved to (1,1).

Based on these plots, and the kinds of numbers you see in your answers to 3 and 4, comment on why we prefer to use piecewise low order polynomials rather than smaller numbers of higher order polynomials.