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# 0 p i 1 3 a 3 2 a 2 a 1 a marks that the patch is to

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Unformatted text preview: 01 a0 P0 B a1 C BC B C = A B P1 C @ a2 A @ P2 A a3 P3 c Given that the spline patch is one of several making up a more complex curve, and the points are deﬁned as ✓◆ ✓◆ ✓◆ ✓◆ a2 , a1 and a0 are vector constants that deﬁne the shape of the curve. 0 P i +1 = 3 a 3 + 2 a 2 + a 1 a Marks: that the patch is to be drawn between two points Pi and Pi+1 and the Given 0 0 gradients at the ends are to be Pi and Pi+1 respectively, write down four 00 equations connecting Pi Pi+1 Pi Pi+1 and a3 a2 a1 a0 . b Solve the above equations to ﬁnd the values in matrix A in the equation: 01 01 Pi = 0a0 a)! a P0 0 BC PiBa1a1 =C B C = A B P1 C @a A @P A Pi+1 = 2a3 + a2 + a12 + a0 a 0 Pi+1 = 33 a3 + 2 a2 P3 a1 + 4 Splines" b)Marks: equations in part a can be written ! The 0 10 1 0 1 1000 a0 P0 B to 1 the C Ba1 C B A in b Solve the above equations 0 ﬁnd 0 0values in matrix P1 C the equation: B CB C=B C @1 1 1 1 A @ a2 A 1 @ P2 A 1 0 0 0 1 a0 3 2 a3 P0 P3 B a1 C B P1 C B C = AB C @ give and the above matrix needs [email protected] A be inverted to P2 A the matrix A: a3 P 0 11 03 1 1000 1 0 0 0 The equations in partB0 1be 0 0C a can written B 1 0 0C B C =B 0 C A = @0 A1 0 @ 3 0 2 1 3 1 1A 1111 1000 a P 0 1 2 3 C B 0 C 2 B 10 C 2 1 B0 1 0 0 a1 C B P1 C B CB @1 1 1 1A @ a2 A = @ P2 A Marks: 0123 a3 P3 4 6 00 equations connecting Pi Pi+1 Pi Pi+1 and a3 a2 a1 a0 . b c Solve the above equations to ﬁnd the values in matrix A in the equation: 01 01 Splines" a0 P0 B a1 C BC B C = A B P1 C @ a2 A @ P2 A a3 P3 Given that the spline patch is one of several making up a more complex curve, and the points are deﬁned as ✓◆ ✓◆ ✓◆ ✓◆ 0 1 5 4 Pi 1 = Pi = P i +1 = P i +2 = 0 3 3 2 0 0 calculate the values of the gradients Pi and Pi+1 using the central difference approximation and hence calculate the values of a3 a2 a1 a0 . d Calculate the co-ordinate and the gradient direction at the midpoint of the spline patch. The four parts carry, respectively, 20%, 30%, 30%, and 20% of the marks. calculate the values of the gradients Pi and Pi+1 using the central difference approximation and hence calculate the values of a3 a2 a1 a0 . 1 0 Pi = 2 Splines" ✓ ◆ 5 3 ✓◆ ✓ ◆ 0 5/2 = 0 3/2 1 = 2 ✓ ◆ 4 2 ✓◆ ✓ ◆ 1 3/2 = 3 1/2 0 P i +1 c)! Using the entries in the matrix A from part b, we have ✓◆ 1 a0 = 3 a1 = ✓ 5/2 3/2 ◆ ✓◆ ✓ ◆ ✓◆ 1 5/2 5 a2 = 3 2 +3 3 3/2 3 ✓ ◆ 11/2 = 5/2 ✓◆ ✓ ◆ 1 5/2 a3 = 2 + 3 3/2 ✓◆ 4 = 1 Marks: ✓ 3/2 1/2 ◆ ✓◆ ✓ ◆ 5 5/2 2 + 3 3/2 6 Department of the co-ordinate and the gradient direction at the midpoint of the spline Conﬁdential d Calculate Computing Examinations – 2011 - 2012 Session patch. MODEL ANSWERS and MARKING SCHEME Splines"Putting µ = 2 into First Examiner: Paul Aljabar d)! Paper: C317 - Graphics Second Examiner: Daniel Rueckert Question: 2 Page 7 of 16 P ( µ ) = a3 µ3 + a2 µ2 + a1 µ + a0 gives the coordinate as ✓◆ 1 1 1 1 P = a3 + a2 + a1 + a0 2 8 4 2 and substituting the values found for a3 a2 a1 a0 gives ✓◆ ✓◆ ✓ ◆ ✓...
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