1. 1 Dr. Scott Schaefer Generalized Barycentric Coordinates

2. 2/83 Barycentric Coordinates Given find weights such that are barycentric coordinates

3. 3/83 Barycentric Coordinates Given find weights such that are barycentric coordinates Homogenous coordinates

4. 4/83 Barycentric Coordinates Given find weights such that are barycentric coordinates

5. 5/83 Barycentric Coordinates Given find weights such that are barycentric coordinates

6. 6/83 Barycentric Coordinates Given find weights such that are barycentric coordinates

7. 7/83 Barycentric Coordinates Given find weights such that are barycentric coordinates

8. 8/83 Boundary Value Interpolation Given, compute such that Given values at, construct a function Interpolates values at vertices Linear on boundary Smooth on interior

9. 9/83 Boundary Value Interpolation Given, compute such that Given values at, construct a function Interpolates values at vertices Linear on boundary Smooth on interior

10. 10/83 Multi-Sided Patches

11. 11/83 Multi-Sided Patches

12. 12/83 Multi-Sided Patches

13. 13/83 Multi-Sided Patches

14. 14/83 Multi-Sided Patches

15. 15/83 Multi-Sided Patches

16. 16/83 Multi-Sided Patches

17. 17/83 Multi-Sided Patches

18. 18/83 Multi-Sided Patches

19. 19/83 Wachspress Coordinates

20. 20/83 Wachspress Coordinates

21. 21/83 Wachspress Coordinates

22. 22/83 Wachspress Coordinates

23. 23/83 Wachspress Coordinates

24. 24/83 Wachspress Coordinates

25. Smooth Wachspress Coordinates Given find weights such that

26. Smooth Wachspress Coordinates Given find weights such that

27. Smooth Wachspress Coordinates Given find weights such that

28. 28/83 Wachspress Coordinates – Summary Coordinate functions are rational and of low degree Coordinates are only well-defined for convex polygons w i are positive inside of convex polygons 3D and higher dimensional extensions (for convex shapes) do exist

29. 29/83 Mean Value Coordinates

30. 30/83 Mean Value Coordinates

31. 31/83 Mean Value Coordinates

32. 32/83 Mean Value Coordinates

33. 33/83 Mean Value Coordinates

34. 34/83 Mean Value Coordinates

35. 35/83 Mean Value Coordinates

36. 36/83 Mean Value Coordinates

37. 37/83 Mean Value Coordinates Apply Stokes’ Theorem

38. 38/83 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

39. 39/83 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

40. 40/83 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

41. 41/83 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

42. 42/83 3D Mean Value Coordinates

43. 43/83 Exactly same as 2D but must compute mean vector for a given spherical triangle 3D Mean Value Coordinates

44. 44/83 3D Mean Value Coordinates Exactly same as 2D but must compute mean vector for a given spherical triangle Build wedge with face normals

45. 45/83 Exactly same as 2D but must compute mean vector for a given spherical triangle Build wedge with face normals Apply Stokes’ Theorem, 3D Mean Value Coordinates

46. 46/83 Deformations using Barycentric Coordinates

47. 47/83 Deformations using Barycentric Coordinates

48. 48/83 Deformations using Barycentric Coordinates

49. 49/83 Deformations using Barycentric Coordinates

50. 50/83 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 216 triangles30,000 triangles0.7 seconds0.02 seconds

51. 51/83 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 216 triangles30,000 triangles0.7 seconds0.02 seconds Real-time!

52. 52/83 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 98 triangles96,966 triangles1.1 seconds0.05 seconds

53. 53/83 Mean Value Coordinates – Summary Coordinate functions are NOT rational Coordinates are only well-defined for any closed, non-self-intersecting polygon/surface w i are positive inside of convex polygons, but not in general

54. Constructing a Laplacian Operator 54/83

55. Constructing a Laplacian Operator 55/83 Laplacian

56. Constructing a Laplacian Operator 56/83 Euler-Lagrange Theorem

57. Constructing a Laplacian Operator 57/83

58. Constructing a Laplacian Operator 58/83

59. Constructing a Laplacian Operator 59/83

60. Constructing a Laplacian Operator 60/83

61. Constructing a Laplacian Operator 61/83

62. Constructing a Laplacian Operator 62/83

63. Constructing a Laplacian Operator 63/83

64. Constructing a Laplacian Operator 64/83

65. Constructing a Laplacian Operator 65/83

66. Constructing a Laplacian Operator 66/83

67. Constructing a Laplacian Operator 67/83

68. Constructing a Laplacian Operator 68/83

69. Constructing a Laplacian Operator 69/83

70. Constructing a Laplacian Operator 70/83

71. Constructing a Laplacian Operator 71/83

72. Constructing a Laplacian Operator 72/83

73. Constructing a Laplacian Operator 73/83

74. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 74/83

75. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 75/83

76. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 76/83

77. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 77/83

78. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 78/83 i th row contains laplacian for i th vertex

79. Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 79/83

80. Harmonic Coordinates 80/83

81. 81/83 Harmonic Coordinates – Summary Positive, smooth coordinates for all polygons Fall off with respect to geodesic distance, not Euclidean distance Only approximate solutions exist and require matrix solve whose size is proportional to accuracy

82. 82/83 Harmonic Coordinates – Summary Positive, smooth coordinates for all polygons Fall off with respect to geodesic distance, not Euclidean distance Only approximate solutions exist and require matrix solve whose size is proportional to accuracy

83. 83/83 Barycentric Coordinates – Summary Infinite number of barycentric coordinates Constructions exists for smooth shapes too Challenge is finding coordinates that are:  well-defined for arbitrary shapes  positive on the interior of the shape  easy to compute  smooth

84. Polar Duals of Convex Polygons Given a convex polyhedron P containing the origin, the polar dual is 84/83

85. Properties of Polar Duals is dual to a face with plane equation Each face with normal and vertex is dual to the vertex 85/83

86. Properties of Polar Duals is dual to a face with plane equation Each face with normal and vertex is dual to the vertex 86/83

87. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 87/83

88. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 88/83

89. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 89/83

90. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 90/83

91. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 91/83

92. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 92/83

93. Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 93/83

94. Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 94/83

95. Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 95/83

96. Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 96/83

97. Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 97/83 Identical to Wachspress Coordinates!

98. Extensions into Higher Dimensions Compute polar dual Volume of pyramid from dual face to origin is barycentric coordinate 98/83