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. -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

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: C 2

9. Surface Patch

10. Four Patches

11. - 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

12. Boundary Conditions

15. Interpolation ?

16. “Closed” Surfaces

18. 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.

19. Tensor Product Interpolants Given Wanted

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

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

22. Patch Domain Parameter Space Cubic Triangular Patch

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