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Tensor-Product Surfaces. Dr. Scott Schaefer. Smooth Surfaces. Lagrange Surfaces Interpolating sets of curves Bezier Surfaces B-spline Surfaces. Lagrange Surfaces. Lagrange Surfaces. Lagrange Surfaces. Lagrange Surfaces. Lagrange Surfaces. Lagrange Surfaces. Lagrange Surfaces.
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Tensor-Product Surfaces Dr. Scott Schaefer
Smooth Surfaces • Lagrange Surfaces • Interpolating sets of curves • Bezier Surfaces • B-spline Surfaces
Lagrange Surfaces – Properties • Surface interpolates all control points • The boundaries of the surface are Lagrange curves defined by the control points on the boundary
Interpolating Sets of Curves • Given a set of parametric curves p0(t), p1(t), …, pn(t) , build a surface that interpolates them
Interpolating Sets of Curves • Given a set of parametric curves p0(t), p1(t), …, pn(t) , build a surface that interpolates them • Evaluate each curve at parameter value t, then use these points as the control points for a Lagrange curve of degree n • Evaluate this new curve at parameter value s
Bezier Surfaces – Properties • Surface lies in convex hull of control points • Surface interpolates the four corner control points • Boundary curves are Bezier curves defined only by control points on boundary
Properties • Curve properties/algorithms apply to surfaces too • Convex hull
Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation
Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation • Evaluation algorithms
Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation • Evaluation algorithms • …. • Analog of variation diminishing does not apply!!!
Derivatives of Bezier Surfaces • Exact evaluate in the s-direction and use those control points to compute derivative in t-direction • Exact evaluate in the t-direction and use those control points to compute derivative in s-direction • Use a pyramid algorithm to compute derivatives