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Coons Patches and Gregory Patches

Coons Patches and Gregory Patches. Dr. Scott Schaefer. 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. Patches With Arbitrary Boundaries.

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Coons Patches and Gregory Patches

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  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

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