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# a1soln - Computing Science CMPT 464/764 Instructor...

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Computing Science CMPT 464/764 Spring 2010 Instructor: Richard (Hao) Zhang Simon Fraser University Assignment #1 Solution Problem 2 (3 marks): Parametric curve design Determine the change of basis matrix for a quintic (that is, degree-5) parametric curve which is defined by four points and two tangent vectors, as shown below. Use the transpose of [ P 1 R 1 P 2 P 3 P 4 R 2 ] as your vector of control points or observable quantities. Expressing your solution as the inverse of a computed matrix is sufficient. [Solution] Let T = [1 t t 2 t 3 t 4 t 5 ]. Then x ( t ) = TA = TB –1 G , where = = 2 4 3 2 1 1 and 5 4 3 2 1 0 1 1 1 1 1 1 243 32 81 16 27 8 9 4 3 2 1 243 1 81 1 27 1 9 1 3 1 1 0 0 0 0 1 0 0 0 0 0 0 1 R P P P R P G B Problem 3 (3 marks): Continuity of cubic B-splines Think about how the change-of-basis matrix for cubic B-splines can be derived. Note that you need not submit a solution for this. You need to complete the following however. Recall that given the vector of four control points P = [ P 0 P 1 P 2 P 3 ] T and the monomial basis T = [1 t t 2 t 3 ], the cubic B-spline curve piece defined by P is given by p ( t ) = TM B- spline P , where M B-spline is the cubic B-spline change-of-basis matrix, Prove that piecewise cubic B-spline curves are C 2 . [Solution] Let the first piece of curve, defined by control points P 0 , P 1 , P 2 , P 3 , be denoted by b 1 ( t ). Let the second piece of curve, defined by control points P 1

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a1soln - Computing Science CMPT 464/764 Instructor...

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