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Suppose a Bézier curve is translated to $\mathbf{x}(t)+\mathbf{b} .$ That is, for$0 \leq t \leq 1,$ the new curve is$$\begin{aligned} \mathbf{x}(t)=(1-t)^{3} \mathbf{p}_{0} &+3 t(1-t)^{2} \mathbf{p}_{1} \\ &+3 t^{2}(1-t) \mathbf{p}_{2}+t^{3} \mathbf{p}_{3}+\mathbf{b} \end{aligned}$$Show that this new curve is again a Bézier curve. [Hint: Where are the new control points?]

Thus, the new curve is again a Bezier curve.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 6

Curves and Surfaces

Vectors

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

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