Solution Found!
A B-spline is built out of B-spline segments, described in
Chapter 8, Problem 6E(choose chapter or problem)
A B-spline is built out of B-spline segments, described in Exercise 2. Let \(\mathbf{p}_{0}, \ldots, \mathbf{p}_{4}\) be control points. For \(0 \leq t \leq 1\) let x(t) and y(t) be determined by the geometry matrices \(\left[\begin{array}{llll} \mathbf{p}_{0} & \mathbf{p}_{1} & \mathbf{p}_{2} & \mathbf{p}_{3} \end{array}\right]\) and \(\left[\begin{array}{llll} {\left[\mathbf{p}_{1}\right.} & \mathbf{p}_{2} & \mathbf{p}_{3} & \mathbf{p}_{4} \end{array}\right]\), respectively. Notice how the two segments share three control points. The two segments do not overlap, however—they join at a common endpoint, close to \(\mathbf{p}_{2}\).
a. Show that the combined curve has \(G^0\) continuity—that is, x (1) = y.(0)
b. Show that the curve has \(C^1\) continuity at the join point, x (1). That is, show that x’ (1) = y’ (0).
Questions & Answers
QUESTION:
A B-spline is built out of B-spline segments, described in Exercise 2. Let \(\mathbf{p}_{0}, \ldots, \mathbf{p}_{4}\) be control points. For \(0 \leq t \leq 1\) let x(t) and y(t) be determined by the geometry matrices \(\left[\begin{array}{llll} \mathbf{p}_{0} & \mathbf{p}_{1} & \mathbf{p}_{2} & \mathbf{p}_{3} \end{array}\right]\) and \(\left[\begin{array}{llll} {\left[\mathbf{p}_{1}\right.} & \mathbf{p}_{2} & \mathbf{p}_{3} & \mathbf{p}_{4} \end{array}\right]\), respectively. Notice how the two segments share three control points. The two segments do not overlap, however—they join at a common endpoint, close to \(\mathbf{p}_{2}\).
a. Show that the combined curve has \(G^0\) continuity—that is, x (1) = y.(0)
b. Show that the curve has \(C^1\) continuity at the join point, x (1). That is, show that x’ (1) = y’ (0).
ANSWER:Solution 6E1. And Using the formula for , but with shifted control point for , you have