a-quartic-bezier-curve-is-determined-by-five-control-points-mathbfp_0-mathbfp_1-mathbfp_2-mathbfp_3-

Question

A quartic Bézier curve is determined by five control points, $\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3},$ and $\mathbf{p}_{4} :$
$$
\begin{aligned} \mathbf{x}(t)=(1-t)^{4} \mathbf{p}_{0} &+4 t(1-t)^{3} \mathbf{p}_{1}+6 t^{2}(1-t)^{2} \mathbf{p}_{2} \ &+4 t^{3}(1-t) \mathbf{p}_{3}+t^{4} \mathbf{p}_{4} \quad 	ext { for } 0 \leq t \leq 1 \end{aligned}
$$
Construct the quartic basis matrix $M_{B}$ for $\mathbf{x}(t)$

Donald Albin · Accepted Answer

all right, You can actually see the answer to the question over here on my screen. But I&#x27;ve already put the points into my virtual graphing calculator here. If I press stat and one for edit, there are the points and you put them in by doing left and right, left button and the right button and down and up. And you can just enter them it like this. Three enter for enter. If I wanted to get that negative seven, I just go over and up. Negative seven. Enter. Okay, they&#x27;re all in there. So now I go back to stat, I go to calculate by going over, and then I scrolled down until I get two cubic regression six. Enter. And there it is. I think I just demonstrated for you that it is possible to make mistakes. So, um, that is not as far as I can tell. The correct answer. Um oh, I&#x27;m supposed to use quart IQ regression. So now I understand my mistake. Oops. I just want press the stop button over to calculate go down a quart IQ, regression it. Enter, hit, enter. And now I have the correct cancer. So that&#x27;s a demonstration of If you make a mistake, just ah, think again. Go back to the beginning and consider all that you&#x27;ve done. Um, because there is a mistake somewhere, and I believe in you. You confined.

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?

Michael Jacobsen · Answer

for this exercise, we&#x27;re going to be solving for the degree to polynomial p a t such that this polynomial is going to pass through these three given points. Let&#x27;s start by using the function notation as follows. We know that p of one must be equal to 12 p of two. Must be equal to 15 and p evaluate at three must give us back 16. Then, if we look at the left hand side, P of one is just placing a one here and here, for example, so it would become the equation. A not plus a one times one for T plus a two times one for T squared is equal to the right hand side. 12. Let&#x27;s do this again by placing to here and here and setting it equal to 15 all together, we would obtain the second equation that is a not plus two times. A one two is then going to be squared. Soul go plus four times a two, and that&#x27;s going to be equal to 15 likewise, replacing three into pft and we get the equation that a not plus 3 +81 plus 9 +82 must be equal to 16. Now, at this stage we have a system of three linear equations in three unknowns. So if we air consistent, then will be able to determine a solution for Pft. Let&#x27;s next take this to an augmented matrix for this matrix we see here. This portion has been augment has been placed here, and the right hand side has been augmented with the 12 15 and 16 Let&#x27;s begin role reducing to see if we have a solution. So for the first reduction step, I&#x27;m going to take negative one times. Row one added to row to replacing row two. So first we have 111 12 in Row two will become zero one, three and three now for Row three would do the same operation multiply row one by a negative one added to row three and record the result here. The result is going to be zero to eight and four. Now for the next step in solving this system, let&#x27;s pivot off this entry here. We&#x27;re going to take to row operations like before, But first, let me copy that middle row, which is 0133 Now. Our goal is to eliminate this entry above next. So multiply wrote to that. We see here by negative one at it to row one and place the result here we&#x27;ll obtain one zero negative two and a nine. Likewise, to eliminate this entry multiply row to buy negative to And at the result to row three, we&#x27;ll obtain zero 02 and negative too. Now, we&#x27;re just about there for the next operation. Let me take an extra step where we divide this entry by two, thereby obtaining an entry of one or record. The result down here will have 001 negative one. And you can do row operations at this step filling out these entries here. I feel like that&#x27;s a lot of work to do at once. So let me just copy in the remainder of the Matrix. So now we have two objectives like before pivoting off of this entry here. We&#x27;re going to eliminate first this entry so we can say that this is robe equivalents to the following first. Copy that last row 001 negative one. Then multiply Row three by negative three and add the results of Row two will have 01 zero by design and then six here for the next operation. We want to get rid of this negative, too. So if we multiply row to buy positive too, add it to row one will achieved. That result will have 10 zero here, and a seven at this stage were ready to record what we were able to solve for recall that the unknowns are a not a one and a two. So by our form of the system, a not is equal to seven. A one is equal to six kind of missed their six and A to this is a zero here is equal to negative one. So that tells us that the required polynomial is p f. T equals a knot, which is seven plus a one, which is six times t plus a two, which is negative one times t squared. So this is a required polynomial that goes through all three points

Ankit Gupta · Answer

But everyone this wasn&#x27;t useful functions are you find a bill of a little day. Everyone read your ex, not You know that perfects people go everywhere. Radio X is nothing, but it will do. Forex, beware Debate your eggs. No, this time here for us. All this one sent by I think this four packs. Okay, so that&#x27;s what I do. First Alpine domain for Geo Philip of us. Well, for that this Europe, Exxon or video of the goddesses in denominator Macy&#x27;s your things. They&#x27;re not deported, you know? Then by pix minus Joseph Gordon, you know, then my affection Nautical toe It missed. Excellent article Tokyo Bay. Why next year? Last I checked 94 gets a public support. Paid 80 and 1 25 sq minus So far the person replied in this paperwork. And I want a different meats by you The next u minus it I can I pull to before North fight? I had an idea Night by effects. Hold you minus school or for a few minutes. You know that A few of minus vehicle before the weekend day Your mind is me And do you swear less? Maybe. Let&#x27;s this way so I a place for money In what, again die for the people who are being died. I x minus two pinto a spray this on and considering it this is use that means 25. I just read less, baby baby missed this into this. That is their next less beast with this. But in school square this for So this is now if they find capital of West Indian Day before you For women, The writer Bae&#x27;s your Texas. We&#x27;re doing 56 minus. Do you do? What if I practice my most day night? Let&#x27;s board for the right away. Your picks the opuses five X minus two. Now this one day some handsome reporter Why have when the piper just read? That&#x27;s the next place for that. Is it? So finally I hear Then, deputy leftover people do It would be five. What&#x27;s this play this day? Next, Nespola What away? Excellent Nordic or No, Moulmein. Bye, Mrs Doctor. Thank you.

Ankit Gupta · Answer

well, everyone. This was in your group insect problem a lot bigger before then. Therefore, normal is divided by explosion with them interested. So that mama there, Lucas said. So we know that in my enemies, if you d like this, the OPEC&#x27;s buddy realize that the office, then we get portion to your face. Bless me. This amended Otto. So did I follow normal fee or Mexico dough? Then here. Do I did anyone? It&#x27;s plus one, of course, Sundays more or one less demanding outside the one my beautiful pass, a well known their law beginning for So what I can do here this to affect so narrow the video today. So what I do about your vaccine ledger and make your big pulling on that&#x27;s sitting next to let the excess bread the C X let&#x27;s be less than in the amended right before the is given. Four. Now, this year, offensively derided by explosive and ventilated tinted, is completely so You wonder. Descend this more normalised said this pain, but always in front of the New York ABC. Before that, no barbarian for Massenet&#x27;s Dylan. So on the distant Don translate this into practice. One won&#x27;t be offering support to get, but nobody extra here. I&#x27;ll get extra part for Blessed Be X Q. Let&#x27;s it&#x27;s a square. This may explain what I want and I get here The X No place it&#x27;s expired less see its place be Felicity, but very person invited you over people. Great day. The extra for less excuse here. It&#x27;s still here so bad. I had to for Monday What I&#x27;ll get Laissez indo and still Mexico just like in your exits by the end of 200 just like Texas exes. Yet it&#x27;s in here. Are they going? I did be a blessing less day last day. So this is very quiet for long enough. Four maybe See the arterial proposition.

Problem

The "B" in B-spline refers to the fact that a seg…

Problem 9 Easy Difficulty

Answer

$M_{B}=\left[\begin{array}{ccccc}{1} & {-4} & {6} & {-4} & {1} \\ {0} & {4} & {-12} & {12} & {-4} \\ {0} & {0} & {6} & {-12} & {6} \\ {0} & {0} & {0} & {4} & {-4} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right]$

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$M_{B}=\left[\begin{array}{ccccc}{1} & {-4} & {6} & {-4} & {1} \\ {0} & {4} & {-12} & {12} & {-4} \\ {0} & {0} & {6} & {-12} & {6} \\ {0} & {0} & {0} & {4} & {-4} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right]$

Linear Algebra and Its Applications

Heather Z.

Caleb E.

Kristen K.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Heather Z.

Caleb E.

Kristen K.

Joseph L.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications