91Ó°ÊÓ

A quartic Bézier curve is determined by five control points, \(\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3},\) and \(\mathbf{p}_{4} :\) $$ \begin{aligned} \mathbf{x}(t)=(1-t)^{4} \mathbf{p}_{0} &+4 t(1-t)^{3} \mathbf{p}_{1}+6 t^{2}(1-t)^{2} \mathbf{p}_{2} \\ &+4 t^{3}(1-t) \mathbf{p}_{3}+t^{4} \mathbf{p}_{4} \quad \text { for } 0 \leq t \leq 1 \end{aligned} $$ Construct the quartic basis matrix \(M_{B}\) for \(\mathbf{x}(t)\)

Find an example of a closed convex set \(S\) in \(\mathbb{R}^{2}\) such that its profile \(P\) is nonempty but conv \(P \neq S\) .

The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed sets \(S_{1}\) and \(S_{2}\) in \(\mathbb{R}^{2}\) such that \(S_{1}\) is affinely dependent and \(S_{2}\) is affinely independent. In each case, the set should contain either one, two, or three nonzero points.

Prove that the convex hull of a bounded set is bounded.