1. 1 Dr. Scott Schaefer Coons Patches and Gregory Patches

2. 2/39 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. 3/39 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. 4/39 Coons Patches Build a ruled surface between pairs of curves

5. 5/39 Coons Patches Build a ruled surface between pairs of curves

6. 6/39 Coons Patches Build a ruled surface between pairs of curves

7. 7/39 Coons Patches Build a ruled surface between pairs of curves

8. 8/39 Coons Patches “Correct” surface to make boundaries match

9. 9/39 Coons Patches “Correct” surface to make boundaries match

10. 10/39 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. 11/39 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. 12/39 Hermite Coons Patches Use Hermite interpolation!!!

13. 13/39 Hermite Coons Patches Use Hermite interpolation!!!

14. 14/39 Hermite Coons Patches Use Hermite interpolation!!!

15. 15/39 Hermite Coons Patches Use Hermite interpolation!!! Requires mixed partials

16. 16/39 Problems With Bezier Patches

17. 17/39 Problems With Bezier Patches

18. 18/39 Problems With Bezier Patches

19. 19/39 Problems With Bezier Patches Derivatives along edges not independent!!!

20. Solution 20/39

21. Solution 21/39

22. 22/39 Gregory Patches

23. 23/39 Gregory Patch Evaluation

24. 24/39 Gregory Patch Evaluation Derivative along edge decoupled from adjacent edge at interior points

25. 25/39 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. 26/39 Constructing Smooth Surfaces With Gregory Patches Assume a network of cubic curves forming quad shapes with curves meeting with C 1 continuity Construct a C 1 surface that interpolates these curves

27. 27/39 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. 28/39 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. 29/39 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. 30/39 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. 31/39 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. 32/39 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. 33/39 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. 34/39 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. 35/39 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. 36/39 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. 37/39 Constructing Smooth Surfaces With Gregory Patches Problem: construction is not symmetric  is quadratic  is cubic

38. 38/39 Constructing Smooth Surfaces With Gregory Patches Solution: assume v(t) is linear and use to find Same operation to find

39. 39/39 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