the-parametric-vector-form-of-a-b-spline-curve-was-defined-in-the-practice-problems-as-beginaligned-

Question

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

Gideon Idumah · Accepted 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?

Linh Vu · Answer

into question, but permit questions excluding table three minutes One. Why would you ever like they risk one and is between the finest 3 to 3. So let Micah Tryba from the very witching day. Why an X? So let&#x27;s say these Deborah Ministry wants to once quantum 0123 so x a good do. Uh, here we get. Include, you know, uh, minus 28 on this would be the minus nine minus two minus one and 07 And in 26. Fund of wine and ankle agenda on us for Dean. My last night. Um, minus fall one six in heaven on six in So it would touch a planted craft. You know, See you be. So now the firstborn is minus 28 on Manus wanting. So let&#x27;s say I understand it could be here. A management team will be Ah Oh, yeah. So it would be somewhere here on this Fucking So we have enough us born will be here minus nine minus night. So we have a saying minus nine will be. Then just let us know I don&#x27;t have another boned here and managed to minus fall. We have a minus two. Yeah, but it&#x27;s followed me here, Mr. Minus fall apart here minus one. Anyone? Ah, minus one. Yeah, and one will be above now. So we have this bones now, and is there only give a six? You must be somewhere. Here. Respond now. And 711 7 11 will be about a little. So have now. And gender six were under 16. This will be 26 b somewhere and 16 with Philip. Oh, yeah? Do we get this bone now? So and we turn on the bones. Now you say it is when you meet the graph from the, uh, burly Normal, the poorly Nomiyama Decree three here. And this will be detention of the curve here. This will be the starting point. Is women the Indian point now And you&#x27;ll find that yeah. Found us from the second question. Getting the the in cardio. Why? Minus one. Anybody by five. And wouldn&#x27;t, you know, forced the question when the x PICO Judah Why, Manasquan? That this will be Can Why? Minus one hour? Three of ah, 1 25 minus one. So doesn&#x27;t mean that body No, me, I&#x27;m not a great three

Adnan Gill · Answer

So we have a set off Parametric equations excess in quote to t to the power three minus one. Why is equal to five t plus one and in values of t go from negative three to hostility. So we&#x27;re gonna make the table for for for so he take values for tea. Let&#x27;s start with T is equal to negative 3 91 01 and three and for each team value will calculate an X value. So 40 is equal to negative three. That&#x27;s equal to negative 28 negative t to the power trees. Then it 27 minus one is negative. 28 Similarly forties equal to negative. One X is equal to negative to negative one and zero and then are 26. Similarly, we can calculate families for why so they come out to be called negative 14 Negative 41 six and 16. So for part B, we need to craft these points on the coordinate place. So let&#x27;s start with negative 28 minus 14. So negative 28 minus 14 would be right here. Then we got negative too. My negative four. So negative, too negative. Four is here and then we got negative 11 So negative 11 would be here and then we got 06 so that would be here. And then we have 26 16 26 and 16 would be here. So we tried to draw the best possible curve that we can get through these points. So let&#x27;s go. So it looks like this and then, like this saw the direction off increasing T. It starts in the fourth quarter and then it causing the first quadrant passing through the third quarter trick. So that part&#x27;s be now for part C. We have to find a Cartesian equation for this. So we have X is equal to t cute minus one, and from here, we can city cute is equal to X plus one. And that means T is equal to X plus one to the power 1/3. And why is equal to five t plus one? So we have why is equal to five times x plus y to the power one hour, three plus one. So part didn&#x27;t simply says, described the car. So this is the cube root function. So this is a huge brute function

Thane Stiles · Answer

So if we want to find values of X and y AT T equal to zero t equal to pi over two t equals two pi and t equal to two pi. We can start with the equation off X equal to three co sign of zero, which means the re we can have. Why equal to three sign of zero which here is equal to zero. We can say that won t is equal to 1/2 by we have X equal to three co sign of over to which is equal to zero. And then we have why stripped over why equal to three sign of pi over two. Three Again, we can do the same thing here X equals three co signed pie And then why equals three sign apply to receive negative three and zero And then finally, for our last one, we will have zero and negative three moving on from there. If we&#x27;re going to graph this, we know that scroll down a little bit. Here. We know that our plane curve will have a counterclockwise orientation as we know that our initial point could be over at 30 and then 03 and the negative 30 and then zero Negative three, meaning that we go around in this direction. However, if we were going to obtain the same plane with a clockwise orientation, we just interning a interchange co sign of tea and sign of tea in the Parametric equation has reversed amounts. From there, you could see that all of our X and Y values will be switched, meaning that when we graft this, we would start out with point over here, then here, here, here, and have a circle that loops in the other direction. I have them slightly offset that where you can see that they&#x27;re going into

Adnan Gill · Answer

the parametric equations we have. Our X is equal to three plus t and why is equal to one minus t and the violence of T go from 0 to 1. So first we want to change physical cartage equation. So if X is equal to three plus t from their weekend cities equal to X minus three and then why is equal to one minus T? Or we could say wise, you put one minus X minus lee, and that gives you why is equal to one minus X plus three, or why is equal to negative X plus four. So that equation is the Cartesian equation for this second Parametric equations. Now let&#x27;s figure out the end point. So this would this would be a linear would be lying statement. So we want to find the endpoints off that like segment. So for part, the first thing we do is we say, when t is equal to zero, then X is equal to X is equal to three plus zero, which is equal to three, and why is equal to one minus, you know, which is a clue to what? So we have our 10.31 so we mark that point right here, that would be the first point. And then we say when t is equal to one than X is equal to three plus one, which is four. And why is equal to one minus one, which is zero. So you have a 00.40 so that there&#x27;s a 0.40 and therefore the line will actually or the segment will start act 31 and it would end at 40 So the description is it&#x27;s ah line segment from 31 to 40

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