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Let $\mathbf{g}(t)$ be defined as in Exercise $21 .$ Its graph is called a quadratic Bézier curve, and it is used in some computer graphics designs. The points $\mathbf{p}_{0}, \mathbf{p}_{1},$ and $\mathbf{p}_{2}$ are called the control points for the curve. Compute a formula for $\mathbf{g}(t)$ that involves only $\mathbf{p}_{0}, \mathbf{p}_{1},$ and $\mathbf{p}_{2} .$ Then show that $\mathbf{g}(t)$ is in $\operatorname{conv}\left\{\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}\right\}$ for $0 \leq t \leq 1$

$g ( t ) = ( 1 - t ) f _ { 0 } ( t ) + t f _ { 1 } ( t ) =$$= ( 1 - t ) \left[ ( 1 - t ) p _ { 0 } + t p _ { 1 } \right] + t \left[ ( 1 - t ) p _ { 1 } + t p _ { 2 } \right] =$$= ( 1 - t ) ^ { 2 } p _ { 0 } + 2 t ( 1 - t ) p _ { 1 } + t ^ { 2 } p _ { 2 }$The sum of the coefficients (weights) in the linear combination for $g$ is:$( 1 - t ) ^ { 2 } + 2 t ( 1 - t ) + t ^ { 2 } = 1 - 2 t + t ^ { 2 } + 2 t - 2 t ^ { 2 } + t ^ { 2 } = 1$Each of the coefficient has value between 0 and 1 when $0 \leq t \leq 1 ,$ so$g ( t )$ is in conv $\left\{ p _ { 0 } , p _ { 1 } , p _ { 2 } \right\}$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

Johns Hopkins University

Missouri State University

Harvey Mudd College

Boston College

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and this exercise, we will find a formula for this specific curve. Um, we know that GFT is one minus t times f zero f t plus tee times of one of t. Now if we succeeded here, the expressions for if zero and if one we have one. Ministry times one minus T time Specie road plus teeth times be one past tee times one Ministry times p one pastie them's p two on board. When performing this product, we obtain one Manistee squared times B zero pause to times t times one Manistee times be one plus t square time. Speak to now. Notice that if he's a number in the closed interval 01 then the following product When my this t squared to tee times one minus t on 30 square are positive numbers. Well, no negative. At least Aunt, the some off these three numbers one minus T square plus two time state into administered E plus T square is the same as one minus D plus D squared on one Rennes duplicitous one. So this miss one. Therefore, what we had before was a linear combination where all the coefficients are not negative and they add up to one. That means that GFT is a comics combination. Off p zero p one and p two that is DFD is in the complex whole, off p zero p one and p two. Whatever T is a number in a close in their ball 01

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