Solution Found!
The parametric vector form of a B-spline curve was defined
Chapter 8, Problem 2E(choose chapter or problem)
The parametric vector form of a B-spline curve was defined in the Practice Problems as
\(\begin{aligned}\mathbf{x}(t)=& \frac{1}{6}\left[(1-t)^{3} \mathbf{p}_{0}+\left(3 t(1-t)^{2}-3 t+4\right)\mathbf{p}_{1}\right.\\ &\left.+\left(3 t^{2}(1-t)+3 t+1\right) \mathbf{p}_{2}+t^{3} \mathbf{p}_{3}\right] \quad \text { for } 0\leq t \leq 1,\end{aligned}\),
where \(\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}, \text { and } \mathbf{p}_{3}\) are the control points.
a. Show that for \(0 \leq t \leq 1, \mathbf{x}(t)\) is in the convex hull of the control points.
b. Suppose that a B-spline curve x(t) is translated to x(t) + b (as in Exercise 1). Show that this new curve is again a B-spline.
Questions & Answers
QUESTION:
The parametric vector form of a B-spline curve was defined in the Practice Problems as
\(\begin{aligned}\mathbf{x}(t)=& \frac{1}{6}\left[(1-t)^{3} \mathbf{p}_{0}+\left(3 t(1-t)^{2}-3 t+4\right)\mathbf{p}_{1}\right.\\ &\left.+\left(3 t^{2}(1-t)+3 t+1\right) \mathbf{p}_{2}+t^{3} \mathbf{p}_{3}\right] \quad \text { for } 0\leq t \leq 1,\end{aligned}\),
where \(\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}, \text { and } \mathbf{p}_{3}\) are the control points.
a. Show that for \(0 \leq t \leq 1, \mathbf{x}(t)\) is in the convex hull of the control points.
b. Suppose that a B-spline curve x(t) is translated to x(t) + b (as in Exercise 1). Show that this new curve is again a B-spline.
ANSWER:Solution 2E1. This tells that each polynomial weight is non-negative for , since Expanding the equation 15, The 1 here plus the 4 and 1 here in the coefficient of and respectively, sum to