1. Coons Patches and Gregory Patches
Dr. Scott Schaefer

2. Patches With Arbitrary Boundaries
Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners, construct a smooth surface interpolating these curves

3. Patches With Arbitrary Boundaries
Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners, construct a smooth surface interpolating these curves

4. Coons Patches
Build a ruled surface between pairs of curves

5. Coons Patches
Build a ruled surface between pairs of curves

6. Coons Patches
Build a ruled surface between pairs of curves

7. Coons Patches
Build a ruled surface between pairs of curves

8. Coons Patches
“Correct” surface to make boundaries match

9. Coons Patches
“Correct” surface to make boundaries match

10. Properties of Coons Patches
Interpolate arbitrary boundaries
Smoothness of surface equivalent to minimum smoothness of boundary curves
Don’t provide higher continuity across boundaries

11. Hermite Coons Patches
Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners and cross-boundary derivatives along these edges
, construct a smooth surface interpolating these curves and derivatives

12. Hermite Coons Patches
Use Hermite interpolation!!!

13. Hermite Coons Patches
Use Hermite interpolation!!!

14. Hermite Coons Patches
Use Hermite interpolation!!!

15. Hermite Coons Patches Use Hermite interpolation!!!
Requires mixed partials

16. Problems With Bezier Patches

17. Problems With Bezier Patches

18. Problems With Bezier Patches

19. Problems With Bezier Patches
Derivatives along edges not independent!!!

20. Solution

21. Solution

22. Gregory Patches

23. Gregory Patch Evaluation

24. Gregory Patch Evaluation
Derivative along edge decoupled from adjacent edge at interior points

25. Gregory Patch Properties
Rational patches
Independent control of derivatives along edges except at end-points
Don’t have to specify mixed partial derivatives
Interior derivatives more complicated due to rational structure
Special care must be taken at corners (poles in rational functions)

26. Constructing Smooth Surfaces With Gregory Patches
Assume a network of cubic curves forming quad shapes with curves meeting with C1 continuity
Construct a C1 surface that interpolates these curves

27. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!

28. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
Fixed control points!!

29. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!

30. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!

31. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
Derivatives must be linearly dependent!!!

32. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
By construction, property holds at end-points!!!

33. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
Assume weights change linearly

34. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
Assume weights change linearly
A quartic function. Not possible!!!

35. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!
Require v(t) to be quadratic

36. Constructing Smooth Surfaces With Gregory Patches
Need to specify interior points for cross-boundary derivatives
Gregory patches allow us to consider each edge independently!!!

37. Constructing Smooth Surfaces With Gregory Patches
Problem: construction is not symmetric
is quadratic
is cubic

38. Constructing Smooth Surfaces With Gregory Patches
Solution: assume v(t) is linear and use
to find
Same operation to find

39. Constructing Smooth Surfaces With Gregory Patches
Advantages
Simple construction with finite set of (rational) polynomials
Disadvantages
Not very flexible since cross-boundary derivatives are not full cubics
If cubic curves not available, can estimate tangent planes and build hermite curves