Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces.

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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

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Overview June 9- B-Spline Curves June 16- NURBS Curves

Presentation transcript:

Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces

Curves Surfaces

Tensor Product B-Spline

Adding Knots

-Piecewise (16 parametric regions); -Bicubic; -C 2 ; -Local Support; -Local supports cover the plane of the parameters in a regular fashion; -Sum up to one; Tensor Product B-Spline

Uniform cubic B-spline Curves Uniform bicubic B-spline Surfaces Expression Vertices Basis Functions Parameter Space

Local expression

Surfaces and Curves Continuity: C 2

Surface Patch

Four Patches

- C 2 - Counting the surface patches… - Convex Hull - Rotation - Scaling - Translation It requires 16 Control Vertices to define a patch. Control Vertices Patches Properties: Uniform bicubic B-spline Surfaces

Boundary Conditions

Interpolation ?

“Closed” Surfaces

Generalization- Tensor Product Surfaces -Choice of basic functions; -Given the vertices, we may compute the approximation surface; -Given a set of points in the surface, we can compute the vertices of the interpolating surface.

Tensor Product Interpolants Given Wanted

System 2 steps: Solve (Schoenberg-Whitney) (u-direction) (v-direction)

Triangular Patch Surfaces Barycentric Coordinates (r,s,t) Control Vertices Bernstein Polynomials Local Expression of a triangular Bezier Patch

Patch Domain Parameter Space Cubic Triangular Patch

Summary Uniform bicubic B-Spline Functions Generalization- Tensor Product Surfaces Tensor Product Interpolants Triangular Patch Surfaces