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Let $\mathbf{x}(t)$ be the $\mathrm{B}$ -spline in Exercise $2,$ with control points $\mathbf{p}_{0},$$\mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3} .$a. Compute the tangent vector $\mathbf{x}^{\prime}(t)$ and determine how the derivatives $\mathbf{x}^{\prime}(0)$ and $\mathbf{x}^{\prime}(1)$ are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both $\mathbf{x}^{\prime}(0)$ and $\mathbf{x}^{\prime}(1)$ equal $\mathbf{0} .$ Justify your assertions.b. Compute the second derivative $\mathbf{x}^{\prime \prime}(t)$ and determine how $\mathbf{x}^{\prime \prime}(0)$ and $\mathbf{x}^{\prime \prime}(1)$ are related to the control points. Draw a figure based on Figure $10,$ and construct a line segment that points in the direction of $\mathbf{x}^{\prime \prime}(1) .\left[\text { Hint: Use } \mathbf{p}_{2} \text { as the }\right.$origin of the coordinate system.

a. Further $\quad \mathbf{x}^{\prime}(1)=0$ when $\mathbf{p}_{3}-\mathbf{p}_{1}=-\frac{4}{3} \mathbf{p}_{2}$b. Geometrically, $\mathbf{p}_{2}$ is on the line through $\mathbf{p}_{0}, \mathbf{p}_{1}$ and $\mathbf{p}_{3}$ is on the line through $\mathbf{p}_{1}, \mathbf{p}_{2}$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 6

Curves and Surfaces

Vectors

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University of Michigan - Ann Arbor

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So we're giving the Parametric equations. X is equal to t plus one liberty and why sequel through T minus one over tea. We're interested in the point where T is equal to one. So first thing you want to do is find expression for the furtive. So we have d expert ET is equal to one minus one over t squid. Andy, why over DT were people one plus one over t squid. And from these two, we can find anywhere but the X so d by D. X is equal to divide by GT divided by t expert ET and that we're people too. One just one over t squared over one minus one or T squid. So simply find that would give t squared plus one over t squared and he squared minus 1/2 squared which simplifies to t squared plus one over t squared minus one citizen expression for the dream. Give off that car at any point now for part B, you want first of all, start by finding d y by D x. AT T is equal to one and that would people to one plus 1/1 minus one which is undefined, so we know that at that point a Cajun should be vertical, and then we want to find the point. So Artie's equal to one we can find X is equal to one plus 1/1. Which is it to to and why is equal to one minus 1/1, which is equal to zero. So we have our 00.20 and we know that the slope is undefined, which is a vertical tension. So we need to be able to really to sketch the curve and draw the T agent at that 0.20 So here the green curve is the curve defined by this set of parametric equations, and the purple line here is detain Geant at the 0.0, and you can see that it's a vertical tension.

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