A quartic Bézier curve is determined by five control points,Po, P1, P2, P3, and p4:x(t) = (1 – t)*Po + 41(1 – t)°P¡ + 6t²(1 – t)²p2+ 41°(1 – t)P3 + t*P4 for 0 < t < 1%3DConstruct the quartic basis matrix MB for x(t).

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Answered: A quartic Bézier curve is determined by…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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). (a) Given the points (1,2), (2,6), and (4,1). Find the equation of the straight line that best fits these points. (b) Find P, the projection matrix on C(A), where A = (c) Verify that P² = P. 1 1 12 1 4
5: Let W be the plane with the equation 6x -y + 5z = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula = A(4"A)\A", P = where the matrix A is constructed using any basis for W as its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e.., enter the values p11, p12- P13.
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
1. (a) Express the polynomial q(t) = 3t² + 5t - 5 as a linear combination of the polynomials P₁ (t) = ² + 2t + 1, P2 (t) = 2t² + 5t + 4, P3(t) = 1² + 3t+ 6. (b) Consider the linear map T: R³ R³ such that T Y Z i. State the matrix of T in standard basis B of R³. ii. Determine the matrix of T in the basis C = = 2 {0)·()·()} - -1 y + z x + z y + x You may assume without proof that C forms a basis. 1 of R³.
5. Find the standard matrix for the linear operator T: R3 → R³ that first transform by the formula T(x, y, z) = (2x – y + z, –x + 2y + z,x – z), then rotate the resulting vector anti clockwise about the y-axis through an angle 0 = 120°, and then project the resulting vector about yz-plane. Hence compute T(1, –1,1).
Define T(f) = f′ from P4 to P4. (a) Prove T is linear. (b) Give the matrix for T. Is it invertible? Why or why not?
(a) Given the camera projection matrix: 500 250 100 300 200 80 1 40 and, the 3D world point: (7, –4, 22) compute the corresponding 2D image projection on the point (to the nearest integer location): X: y: (b) A given camera is calibrated to project between a planar surface and the image plane resulting in the following project matrix: 60 20 50 30 70 90 6. 9. 1 What point on the planar surface projects to the origin of the image plane (i.e. (0, 0))? Select your answer from the following:
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (2x − 2y+z, x − 4y+z, x-y-z) relative to the bases B={u=(5,16,1); v= (2, 1, –4); w=(7,17,2)} and B'={u'=(2,0,1); v' = (1,0,1); w'=(3,2,0)) Then the second column of A is: *) 0 O A. (0 O B. "B) °) SOC. O D. (6)
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (3x−y+4z,y+z−x,x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,−5)} and B'={u'=(2,2,1); v'=(1,0,0); w'=(3,2,0) } Then the first column of A is: OA.-16 O B. 13 1 4 17 17 °C) O D. (2 213 O E. "G) ņ m
#10: Let W be the plane with the equation 3x-y + 62 = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula P = A(ATA)-¹AT, where the matrix A is constructed using any basis for W as its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e., enter the values P11, P12, P13.

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A quartic Bézier curve is determined by five control points, Po, P1, P2, P3, and p4: x(t) = (1 – t)*Po + 41(1 – t)°P¡ + 6t²(1 – t)²p2 + 41°(1 – t)P3 + t*P4 for 0 < t < 1 %3D Construct the quartic basis matrix MB for x(t).

A quartic Bézier curve is determined by five control points, Po, P1, P2, P3, and p4: x(t) = (1 – t)Po + 41(1 – t)°P¡ + 6t²(1 – t)²p2 + 41°(1 – t)P3 + tP4 for 0 < t < 1 %3D Construct the quartic basis matrix MB for x(t).