CS557 Winter 2001

Homework 3

Due Friday, 26 January 5:00p

1. What are the control points of a degree two polynomial Bezier curve that is C^2 with a cubic Bezier curve whose control points are (0,0), (2,4), (4,4), (6,2). The two curves meet at (6,2).

2. A degree two polynomial Bezier curve P(t) is G^2 with the circle x^2+y^2-1=0 at the point (1,0). What are the control points of P(t)?

3. A rational degree one Bezier curve has control points P0=( 1,1), P1=(2,2) and weights w0=1, w1=2. Find the hodograph of this curve. Express the hodograph as a degree two rational Bezier curve.

4. A rational cubic Bezier curve has weights w0=2, w1=4, w2=12, w3=16. Reparametrize this curve so that the control points are all the same, but the weights w0=1 and w3=1.

5. A polynomial cubic bezier curve P(t) has control points (0,1), (0,0), (1,0), (2,0) and a polynomial quadrtaic Bezier curve Q(t) has control points (4,1), (4,2), and (3,2). Find the control points and weights of a rational cubic Bezier curve that is G2 with P(t) at (2,0) and G^2 with Q(t) at (4,1).

6. What is the area between the X-axis and an explicit Bezier curve with control points (0,1), (1,4), (2,1), (3,5), (4,7), (5,1)?

7. For a cubic polynomial, f(1)=3, f(2)=4, f(3)=5, f(4)=4. What is f(5) and f(6)?

8. The main problem with forward differencing is numerical stability. Experiment with this C code to get a feeling for how stable forward differencing is. This code computes the x-coordinate of n points on the Bezier curve with control points (2,5), (11,320), (397,200), (431,326). Experiment with this code to see how large n can be and still have the last x-coordinate computed by the forward difference be between 430.5 and 431.5. Change the "float" type declarations to "double" and try it again.