Exercise set 7: Surfaces

1. Compute the transition surface given by border curves (Subscript[c, 0])(v) = {-2 + (v^2), -v, 2 v}, v ∈ [1, 3] and (Subscript[c, 1])(v) = {2 v, 1 + (v^2), (v/3) + (v^2)}, v ∈ [-(3/2), (3/2)]. Plot both the surface and the curves (Subscript[a, 0]), (Subscript[a, 1]) into one image.

2. Consider the following border curves:
(Subscript[a, 0])(v) = {1, (v^2) - v, - v}, v ∈ [0, 1]
(Subscript[a, 1])(v) = {(3/2) v, 2 - v, 1 - (v^2)}, v ∈ [0, 1]
(Subscript[b, 0])(v) = {1 - 2 u + (u^2), 2 u, (u^2)}, u ∈ [0, 1]
(Subscript[b, 1])(v) = {1 + (3/2) u - (u^2), u, -1 + (u^2)}, u ∈ [0, 1]
2.1 Compute the points (Subscript[P, 00]) = (Subscript[a, 0])(0) = (Subscript[b, 0])(0), (Subscript[P, 01]) = (Subscript[a, 0])(1) = (Subscript[b, 1])(0), (Subscript[P, 10]) = (Subscript[a, 1])(0) = (Subscript[b, 0])(1), (Subscript[P, 11]) = (Subscript[a, 1])(1) = (Subscript[b, 1])(1) and verify, that the corresponding curves truly meet at these points.
2.2 Construct a Coons patch using these border curves and plot it into one image with the curves.

3. Consider the following mesh:
{{{0.15042, -0.229705, 0.90914}, {0.076767, 0.764106, 0.114311}, {0.06856, 1.99114, -0.492015}, {-0.196724, 2.86668, 0.09444}, {-0.102773, 4.02536, -0.123333}},
 {{0.85222, 0.191512, -0.45334}, {0.975464, 0.723016, 0.323851}, {1.1029, 1.96797, 0.759833}, {0.890567, 2.93021, -0.947293}, {0.912385, 3.94256, -0.998273}},
 {{1.98681, 0.0489156, 0.509939}, {2.09112, 0.99257, 0.0026571}, {1.88779, 2.05824, -0.602484}, {2.03456, 3.26816, -0.856659}, {2.01448, 4.12713, 0.710227}},
 {{2.95275, 0.296082, -0.0729233}, {3.11238, 1.12073, 0.269672}, {2.86365, 2.13923, 0.442328}, {2.99654, 3.17556, 0.750783}, {3.01425, 3.99257, 0.325363}}}

3.1 Plot a Bézier surface (Subscript[f, 1])(s,t) given by the control mesh. Plot the mesh into the same image. What is the degree of the surface?
3.2 Find a point (Subscript[f, 1])((1/2), (3/2)) on the surface using the DeCasteljau algorithm.
3.3 Plot a B-Spline surface (Subscript[f, 2])(s,t) of degree (2,3) given by the control mesh and the following knot vectors: {0,0,0,1/2,1,1,1}, {0,0,0,0,1/2,1,1,1,1}.
3.4 Plot a NURBS surface (Subscript[f, 3])(s,t) of degree (2,2) given by the control mesh, knot vectors {0,0,0,1/2,1,1,1} and {0,0,0,1/4,3/4,1,1,1}, and the following weights:
{{1, 0.7, 0.3, 1.1, 0.9},{2, 1.05, 0.2, 0.7, 0.1},{1.1, 1.2, 1.8, 0.8, 2.2},{3, 1, 0.5, 1.1, 0.4}}.