Solution Found!
Answer: Exercises 13–15 concern the subdivision of a
Chapter 8, Problem 15E(choose chapter or problem)
Exercises 13–15 concern the subdivision of a Bézier curve shown in Figure 7. Let x(t) be the Bézier curve, with control points \(\mathbf{p}_{0}, \ldots, \mathbf{p}_{3}\), and let y(t) and z(t) be the subdividing Bézier curves as in the text, with control points \(\mathbf{q}_{0}, \ldots, \mathbf{q}_{3} \text { and } \mathbf{r}_{0}, \ldots, \mathbf{r}_{3}\), respectively.
Sometimes only one half of a Bézier curve needs further subdividing. For example, subdivision of the “left” side is accomplished with parts (a) and (c) of Exercise 13 and equation (8). When both halves of the curve x(t) are divided, it is possible to organize calculations efficiently to calculate both left and right control points concurrently, without using equation (8) directly.
a. Show that the tangent vectors y’(1) and z’(0) are equal.
b. Use part (a) to show that \(q_3\) (which equals r0/ is the midpoint of the segment from \(q_2\) to \(r_1\).
c. Using part (b) and the results of Exercises 13 and 14, write an algorithm that computes the control points for both y(t) and z(t) in an efficient manner. The only operations needed are sums and division by 2.
Questions & Answers
QUESTION:
Exercises 13–15 concern the subdivision of a Bézier curve shown in Figure 7. Let x(t) be the Bézier curve, with control points \(\mathbf{p}_{0}, \ldots, \mathbf{p}_{3}\), and let y(t) and z(t) be the subdividing Bézier curves as in the text, with control points \(\mathbf{q}_{0}, \ldots, \mathbf{q}_{3} \text { and } \mathbf{r}_{0}, \ldots, \mathbf{r}_{3}\), respectively.
Sometimes only one half of a Bézier curve needs further subdividing. For example, subdivision of the “left” side is accomplished with parts (a) and (c) of Exercise 13 and equation (8). When both halves of the curve x(t) are divided, it is possible to organize calculations efficiently to calculate both left and right control points concurrently, without using equation (8) directly.
a. Show that the tangent vectors y’(1) and z’(0) are equal.
b. Use part (a) to show that \(q_3\) (which equals r0/ is the midpoint of the segment from \(q_2\) to \(r_1\).
c. Using part (b) and the results of Exercises 13 and 14, write an algorithm that computes the control points for both y(t) and z(t) in an efficient manner. The only operations needed are sums and division by 2.
ANSWER:Solution 15E1. At , And at , Therefore, the tangent vector and are equal to . b) Wit