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Bezier Triangles and Multi-Sided Patches

Bezier Triangles and Multi-Sided Patches. Dr. Scott Schaefer. Triangular Patches. How do we build triangular patches instead of quads?. Triangular Patches. How do we build triangular patches instead of quads?. Triangular Patches. How do we build triangular patches instead of quads?.

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Bezier Triangles and Multi-Sided Patches

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  1. Bezier Triangles andMulti-Sided Patches Dr. Scott Schaefer

  2. Triangular Patches • How do we build triangular patches instead of quads?

  3. Triangular Patches • How do we build triangular patches instead of quads?

  4. Triangular Patches • How do we build triangular patches instead of quads?

  5. Triangular Patches • How do we build triangular patches instead of quads? Parameterization very distorted Continuity difficult to maintain between patches Not symmetric

  6. Bezier Triangles • Control points pijk defined in triangular array

  7. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  8. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  9. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  10. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  11. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  12. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  13. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  14. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  15. deCasteljau Algorithm for Bezier Triangles • Evaluate at (s,t,u) where s+t+u=1

  16. Properties of Bezier Triangles • Convex hull

  17. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves

  18. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves

  19. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves

  20. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves

  21. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves

  22. Properties of Bezier Triangles • Convex hull • Boundaries are Bezier curves • Explicit polynomial form

  23. Subdividing Bezier Triangles

  24. Subdividing Bezier Triangles

  25. Subdividing Bezier Triangles

  26. Subdividing Bezier Triangles

  27. Subdividing Bezier Triangles

  28. Subdividing Bezier Triangles

  29. Subdividing Bezier Triangles

  30. Subdividing Bezier Triangles • Split along longest edge

  31. Subdividing Bezier Triangles • Split along longest edge

  32. Derivatives of Bezier Triangles

  33. Derivatives of Bezier Triangles

  34. Derivatives of Bezier Triangles

  35. Derivatives of Bezier Triangles Really only 2 directions for derivatives!!!

  36. Continuity Between Bezier Triangles • How do we determine continuity conditions between Bezier triangles?

  37. Continuity Between Bezier Triangles • How do we determine continuity conditions between Bezier triangles?

  38. Continuity Between Bezier Triangles • How do we determine continuity conditions between Bezier triangles? Control points on boundary align for C0

  39. Continuity Between Bezier Triangles • How do we determine continuity conditions between Bezier triangles? What about C1?

  40. Continuity Between Bezier Triangles • Use subdivision in parametric space!!!

  41. Continuity Between Bezier Triangles • Use subdivision in parametric space!!! First k rows of triangles from subdivision yield Ck continuity conditions

  42. Continuity Between Bezier Triangles • C1 continuity

  43. Continuity Between Bezier Triangles • C1 continuity

  44. Continuity Between Bezier Triangles • C1 continuity

  45. Multi-Sided Patches • Multi-sided holes in surfaces can be difficult to fill • Construct a generalized Bezier patch for multi-sided holes

  46. Control Points for Multi-Sided Patches • Five sided control points

  47. Control Points for Multi-Sided Patches • Five sided control points

  48. Control Points for Multi-Sided Patches • Five sided control points Index has number of entries equal to vertices in base shape

  49. Control Points for Multi-Sided Patches • Five sided control points Index has number of entries equal to vertices in base shape Entries positive and sum to d

  50. Control Points for Multi-Sided Patches • Five sided control points

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