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Overview

Overview. June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces. Curves. Surfaces. Tensor Product B-Spline. Adding Knots. Tensor Product B-Spline. Piecewise (16 parametric regions); Bicubic; C 2 ; Local Support;

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Overview

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Presentation Transcript

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

  2. Curves Surfaces

  3. Tensor Product B-Spline

  4. Adding Knots

  5. Tensor Product B-Spline • Piecewise (16 parametric regions); • Bicubic; • C2; • Local Support; • Local supports cover the plane of the parameters in a regular fashion; • Sum up to one;

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

  7. Local expression

  8. Surfaces and Curves Continuity: C2

  9. Surface Patch

  10. Four Patches

  11. Properties: Uniform bicubic B-spline Surfaces • C2 • Counting the surface patches… • Convex Hull • Rotation • Scaling • Translation It requires 16 Control Vertices to define a patch. Control Vertices Patches

  12. Boundary Conditions

  13. Interpolation ?

  14. “Closed” Surfaces

  15. 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.

  16. Tensor Product Interpolants Given Wanted

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

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

  19. Cubic Triangular Patch Patch Domain Parameter Space

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

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