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

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

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

22. Cubic Triangular Patch
Patch Domain
Parameter Space

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