We are given the following boundary conditions for a cubic spline section:

P(0) = pk

P(1) = pk+1

P00(0) = pk-1 + pk+1 - 2pk

P00(1) = pk + pk+2 - 2pk+1

In this case Mgeom = [ pk-1 pk pk+1 pk+2 ]T and the boundary conditions can be written

P(0) 0 1 0 0 Pk-1

P(1) 0 0 1 0 PK

P''(0) = 1 -2 1 0 . PK+1

P''(1) 0 1 -2 1 Pk+2

(a) (15 points) Given Mgeom as specified above, derive the 4 by 4 matrix Mspline.

(b) (10 points) Given your solution for Mspline write out the blending functions for this curve.

(c) (10 points) Give the equation P0(u) for the tangent to this curve in terms of Mspline and Mgeom.

(d) (10 points) Do adjacent segments satisfy C1 continuity? Give a mathematical justification.

(e) (10 points) Do adjacent segments satisfy C2 continuity? Give a mathematical justification.

P(0) = pk

P(1) = pk+1

P00(0) = pk-1 + pk+1 - 2pk

P00(1) = pk + pk+2 - 2pk+1

In this case Mgeom = [ pk-1 pk pk+1 pk+2 ]T and the boundary conditions can be written

P(0) 0 1 0 0 Pk-1

P(1) 0 0 1 0 PK

P''(0) = 1 -2 1 0 . PK+1

P''(1) 0 1 -2 1 Pk+2

(a) (15 points) Given Mgeom as specified above, derive the 4 by 4 matrix Mspline.

(b) (10 points) Given your solution for Mspline write out the blending functions for this curve.

(c) (10 points) Give the equation P0(u) for the tangent to this curve in terms of Mspline and Mgeom.

(d) (10 points) Do adjacent segments satisfy C1 continuity? Give a mathematical justification.

(e) (10 points) Do adjacent segments satisfy C2 continuity? Give a mathematical justification.

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