exercises-13-15-concern-the-subdivision-of-a-bezier-curve-shown-in-figure-7-let-mathbfxt-be-the-be-2

Question

Exercises $13-15$ concern the subdivision of a Bézier curve shown in Figure $7 .$ Let $\mathbf{x}(t)$ be the Bézier curve, with control points $\mathbf{p}_{0}, \ldots, \mathbf{p}_{3},$ and let $\mathbf{y}(t)$ and $\mathbf{z}(t)$ be the subdividing Bézier curves as in the text, with control points $\mathbf{q}_{0}, \ldots, \mathbf{q}_{3}$ and $\mathbf{r}_{0}, \ldots, \mathbf{r}_{3}$ respectively.
a. Justify each equal sign:
$$
3\left(\mathbf{r}_{3}-\mathbf{r}_{2}\right)=\mathbf{z}^{\prime}(1)=.5 \mathbf{x}^{\prime}(1)=\frac{3}{2}\left(\mathbf{p}_{3}-\mathbf{p}_{2}\right)
$$
b. Show that $r_{2}$ is the midpoint of the segment from $\mathbf{p}_{2}$ to $\mathbf{p}_{3}$
c. Justify each equal sign: $3\left(\mathbf{r}_{1}-\mathbf{r}_{0}\right)=\mathbf{z}^{\prime}(0)=.5 \mathbf{x}^{\prime}(.5)$
d. Use part (c) to show that $8 \mathbf{r}_{1}=-\mathbf{p}_{0} \mathbf{p}_{1}+\mathbf{p}_{2}+\mathbf{p}_{3}+$ 8 $\mathbf{r}_{0} .$
e. Use part $(\mathrm{d}),$ equation $(8),$ and part (a) to show that $\mathbf{r}_{1}$ is
the midpoint of the segment from $\mathbf{r}_{2}$ to the midpoint of the segment from $\mathbf{p}_{1}$ to $\mathbf{p}_{2} .$ That is, $\mathbf{r}_{1}=\frac{1 {2}\left[\mathbf{r}_{2}+\frac{1}{2}\left(\mathbf{p}_{1}+\mathbf{p}_{2}\right)\right]$

Eric Mockensturm · Accepted Answer

and this problem want to find the Bessie a curve with the control points 143 12 6 15 and 74 Well, we can pretty much just plug that into the formula that were given in the equation. 89 and we just get that. The curve is given by X equals one plus 60 plus three T squared minus cube and why equals four plus 24 t minus 15 t squared minus 19. Q.

Gideon Idumah · Answer

So in this question, we&#x27;ve given each of C equals one minus Tse Do you wanna see laws see jitsu off? See less named execution style we&#x27;re gonna use in the town now from exercise 23 we have That&#x27;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&#x27;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&#x27;s just said knows g off, See, because it&#x27;s also determined by p zero p one on pizza. So that&#x27;s just one minus c squared p zero lost should see one minus C off p one plus c squared off pizza. Now we&#x27;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&#x27;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&#x27;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&#x27;m just playing this all plus C one miner&#x27;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&#x27;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&#x27;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&#x27;s cubed. Lost Cute. Execute Consistent. Cute. There it&#x27;s the cubes, Constance. There it&#x27;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&#x27;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&#x27;s with the fact that this is forced to see that&#x27;s the were physicians are known negatives for zero less than accosted. See living across the wall. So because there&#x27;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?

Charles Carter · Answer

Okay, So for this problem, we need Teoh just kind of re copy a basic illustration. So we have two focal points. No, this one is that minus C zero. This is a positive C zero and we have a point. P was Exline case of her part a ah d one. Using the distance formula is just the square root of X plus e. So you can imagine I&#x27;ve drawn a line down. That would be X. This would be length C. It&#x27;s our difference in our exes and then just plus y square and said D one squared is just equal to X plus C square plus life square and similarly indeed to de to square. It&#x27;s just gonna be ah c minus X squared, which is the difference between the two. So we can also right. That is X minus C because we&#x27;re squaring it and then plus why square So says party part B has a subtracting. Ah, Do you two squared from D one squared? Okay, so we dio ah di one squared Manistee to square. Basically the only thing we&#x27;ll have left will have Ah, from the 1st 1 will have a negative two x e and from the one square were positive to exceed. But because we&#x27;re subtracting, our wives would cancel out our see squares were cancelled and her X cords will cancel out, leaving us with negative for XY. So what kind of talk? That one out. But if you if you do a binomial expansion, subtract the only thing that&#x27;s gonna stay is the XY term. Okay. In fact, I think I got that one backwards. So d one squared is the positive. And if we subtract a negative, it would be positive for four times X Dempsey, Right? Okay. Okay. So for parts seen, um, we just wants us to explain why d one plus t two is equal to two a and that just comes from the definition. Ah, that the some of the distances from the folk I to any point has to be constant. Um, And so because those distances are constant Ah, if we add a se well, let&#x27;s just think of a point like on the end. So we have the distance between the two folk, I plus the extra piece s we&#x27;re gonna have to see. Plus that extra peas, Plus that extra piece again should be the same is adding this son which would be equal to to a so I mean, that&#x27;s one geometric way to explain it. Um is the total length from one into the other would be the same as but the some of the distances from both focus to one vertex. Okay, so that&#x27;s one way to think about it. Okay, So now, because of that, we can use the difference of squares to say d one plus t two times d one minus t two is equal to four xy, so we&#x27;re gonna replace the someone to a times D one minus t two is equal to four XY and therefore the difference in the two is gonna be equal. Teoh port of it about two is too. So two x times see over a okay and then the next part says, If we add that now said d plus de too is equal to two A. If we add those two together, the details cancel out. So we have two times D one is equal. Teoh into X and then see over a is just equal to e plus to a We can divide everything about two and rearrange a little bit to see that d one is the go to just a plus eight times X And similarly, for the last time, we can subtract, um, or we can use the fact that D one is now equal. Teoh a plus the x Either way, we can so we can say a plus e x buzz d To is equal Teoh to a and subtracting the A plus e x from both sides. We can get d two is equal to a minus e x. Good. And that&#x27;s all of it. Okay, um, thank you very much. I&#x27;m kind of just scroll through slowly with each step just to kind of let you get back and see the different parts. That&#x27;s part a first part B and C. Erin, there&#x27;s D and E. Okay, thank you very much.

Regina Hays · Answer

Okay. What we want to go ahead and do is to find, um, the area of the ellipse that is defined by the parametric equations X equal to be co sign T. Why equal to a sign? T um, on the interval, Um, from 0 to 2 pi. Okay. And so we do know that area is given by the integral from A to B of why t x Okay. And then there is a little diagram of the ellipse. So let&#x27;s go ahead and re create that. And then, um where, um this is a and that is B. Okay. And so, um, this is gonna be, um, the area equal to the integral role of a signed T. And now what I need to do is go ahead and take the drift of the bags with respect to T. And so that is going to give me negative b sign of tea, multiply both sides by D. T. And so this is times negative B sine of t tt. Now I need to figure out, um what my limits. My limits are. And so I noticed that, um, when um X is, um egg zero. Um, when x is equal to zero. Then tea is equal to pi over too, Because I&#x27;m really going from, um zero to be here. So I&#x27;m going from zero to be in the X values. And so if x is equal to be, then t is equal 20 So if I mean, if I&#x27;m keeping it and wise and exes my exile, you would go from zero to be. And I actually have four quadrants, so I have four quadrants. And if I&#x27;m going from zero to be my time that I use go from pi over too. 20 Um, and so this will actually be equal to, um negative four A. B the integral from pi Over 2 to 0 of signs square T t t Uh, And then also, if I switched my upper and lower limits than I change a sign of the interval. So this is gonna be four a. B integral from zero to pi over too. Time sign square T TT Okay. And then we&#x27;re gonna go ahead and use a calculator, um, to approximate, um, this integral. Now, of course, we&#x27;re not gonna be able to put the baby in the calculator. And So once we get the number, we can multiply it by A and B. So let&#x27;s go ahead and do that. So I&#x27;m gonna go to my calculator and it&#x27;s four and the integral from zero to, um, high over too. Lips. Here you go. And Rhys Waas sign tea. And then we are, um, wearing that and taking the river with respect the tea. And so now you notice that&#x27;s 3.1415 and that she look familiar to you. That is pie. And so, if I switch back to this, then this is going to be pi times a B and, um, which is if it was a multiple choice question. So it&#x27;s option, um d

Eric Mockensturm · Answer

We&#x27;re out with another set of busy eight points and we want to find the curve and then plot plot the curve. So the points are 3 to 0 to 54 and 24 So, again, we can pretty much plug in to our values for before Lila, and we just get, um, again. Three one minus D cube 15 p squared one minus t plus two cube. And for why? We just used the y values now and we get two times one minus T cube plus six t times one minus t squared. Um, right. You should be out here and 12 times. T squared, minus one times One is tea plus for you. Okay, so we can we can plot that. Put that into the computer, and we get this nice kind of backwards s shape. Typical over busy a curve

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Exercises $13-15$ concern the subdivision of a Bé…

Problem 14 Easy Difficulty

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

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