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The parametric vector form of a B-spline curve was defined in the Practice Problems as$$\begin{aligned} \mathbf{x}(t)=& \frac{1}{6}\left[(1-t)^{3} \mathbf{p}_{0}+\left(3 t(1-t)^{2}-3 t+4\right) \mathbf{p}_{1}\right.\\ &+\left(3 t^{2}(1-t)+3 t+1\right) \mathbf{p}_{2}+t^{3} \mathbf{p}_{3} ] \quad \text { for } 0 \leq t \leq 1 \end{aligned}$$where $\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2},$ 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 $\mathbf{x}(t)$ is translated to $\mathbf{x}(t)+\mathbf{b}(\text { as in Exercise } 1) .$ Show that this new curve is again a $\mathrm{B}$ -spline.

a. Thus, $\mathbf{x}(t)$ is a convex combination of the control point for $0 \leq t \leq 1$b. Thus, $\mathbf{x}(t)+\mathbf{b}$ may be witten in a similar form, with $\mathbf{p}_{i}$ , replaced by $\mathbf{p}_{i}+\mathbf{b}$ for each $i$This shows that $\mathbf{x}(t)+\mathbf{b}$ is a cubic B-spine with control point $\mathbf{p}_{i}+\mathbf{b}$ for $i=0,1,2,3$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 6

Curves and Surfaces

Vectors

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Harvey Mudd College

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Idaho State University

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So in this question, we've given each of C equals one minus Tse Do you wanna see laws see jitsu off? See less named execution style we're gonna use in the town now from exercise 23 we have That's just see was given to us as one minus c squared p easier plus suits See one minus c p one lost C squared pizza. So in this case, Gypsy Gypsy Gypsy was determined by p zero p one MP too. So now we're also in our question since G one of C is also determined bye p zero p one on pizza. So this implies that rg one off See, he's just said knows g off, See, because it's also determined by p zero p one on pizza. So that's just one minus c squared p zero lost should see one minus C off p one plus c squared off pizza. Now we've given that jitsu since G two off See is determined. Bye p one pizza p three. This would imply that G two off C would be so it's not be one minus c squared instead of p zero off P. Juan lost shoots in one minus C p two lost C squared. I mean three. So glug G one of C on jitsu off. See into equation star. When you do that, we're gonna off each of C. He called So one minus tse mort implied by June off. See, which is one minus C squared P zero close to see one mile. Ist sie auf p one lost C squared off pizza. Ooh! Lost. See more deployed by jitsu Does one minus c squared P one lost shoot. See one minus c off. Pizza lost. C squared off p 30. This will be equal to if you open that. Or the government of C Cube. P zero lost Keith. See? One minus. Tsutsui lost C squared. He won. Lost C squared miners. C cube pizza. Ooh, I'm just playing this all plus C one miner's Tutsi lost C squared P one lost two c squared one minus C pizza. Ooh, Lost C cubed off. In theory, this will be equal. So when you solve these, you going off one minus the Red Sea lost three c squared minus Tse cube off P zero. So you have been if you expire and this woman is still killed. They are lost. Three C minus 60 square lost three c cube off yuan Lost their it C squared minus three C cube off, too. Lost C cube off in three. Now let's look at some of the coefficients quarter. This is all each of C. No, if you some peach of this coefficient, we should get that's equal to warm. So some off the where physicians in its off scene these forced one for pisa is one minus. Charity lost their squared minus C cubed. Lost the C minus 60 squared Lost three c Cubed Lost, three C squared minus There it's cubed. Lost Cute. Execute Consistent. Cute. There it's the cubes, Constance. There it's a cubed. There it is, quite close. Dignity. Squalus 60 squared minus 60 60 squared. So this three council out minus to reconsider, please. So we'll sit out the insides, of course, to one. No, we can see that if we fucked or the coefficients in questions off each of sea, we can rewrite it. Us. We got rights, Oz each off. See, if you go back to this question right, these us, each off equal to be questions off easier is actually one minus C All Cube P zero loss. This one is the ritzy more supply by one minus. C squared off P Juan off, plus the ritzy squared one minus C off, too. Lost C Cube P three. And we see that's with the fact that this is forced to see that's the were physicians are known negatives for zero less than accosted. See living across the wall. So because there's some of his one and they also know negative. This implies that each off, see is in the convents all off control points easier. You want pizza?

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