1 Dr. Scott Schaefer Generalized Barycentric Coordinates

2/95 Barycentric Coordinates Given find weights such that are barycentric coordinates

3/95 Barycentric Coordinates Given find weights such that are barycentric coordinates Homogenous coordinates

4/95 Barycentric Coordinates Given find weights such that are barycentric coordinates

5/95 Barycentric Coordinates Given find weights such that are barycentric coordinates

6/95 Barycentric Coordinates Given find weights such that are barycentric coordinates

7/95 Barycentric Coordinates Given find weights such that are barycentric coordinates

8/95 Boundary Value Interpolation Given, compute such that Given values at, construct a function Interpolates values at vertices Linear on boundary Smooth on interior

9/95 Boundary Value Interpolation Given, compute such that Given values at, construct a function Interpolates values at vertices Linear on boundary Smooth on interior

10/95 Multi-Sided Patches

11/95 Multi-Sided Patches

12/95 Multi-Sided Patches

13/95 Multi-Sided Patches

14/95 Multi-Sided Patches

15/95 Multi-Sided Patches

16/95 Multi-Sided Patches

17/95 Multi-Sided Patches

18/95 Multi-Sided Patches

19/95 Wachspress Coordinates

20/95 Wachspress Coordinates

21/95 Wachspress Coordinates

22/95 Wachspress Coordinates

23/95 Wachspress Coordinates

24/95 Wachspress Coordinates

Polar Duals of Convex Polygons Given a convex polyhedron P containing the origin, the polar dual is 25/95

Properties of Polar Duals is dual to a face with plane equation Each face with normal and vertex is dual to the vertex 26/95

Properties of Polar Duals is dual to a face with plane equation Each face with normal and vertex is dual to the vertex 27/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 28/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 29/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 30/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 31/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 32/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 33/95

Coordinates From Polar Duals Given a point v, translate v to origin Construct polar dual 34/95

Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 35/95

Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 36/95

Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 37/95

Given a point v, translate v to origin Construct polar dual Coordinates From Polar Duals 38/95 Identical to Wachspress Coordinates!

Extensions into Higher Dimensions Compute polar dual Volume of pyramid from dual face to origin is barycentric coordinate 39/95

40/95 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

41/95 Mean Value Coordinates

42/95 Mean Value Coordinates

43/95 Mean Value Coordinates

44/95 Mean Value Coordinates

45/95 Mean Value Coordinates

46/95 Mean Value Coordinates

47/95 Mean Value Coordinates

48/95 Mean Value Coordinates

49/95 Mean Value Coordinates Apply Stokes’ Theorem

50/95 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

51/95 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

52/95 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

53/95 Comparison convex polygons (Wachspress Coordinates) closed polygons (Mean Value Coordinates)

54/95 3D Mean Value Coordinates

55/95 Exactly same as 2D but must compute mean vector for a given spherical triangle 3D Mean Value Coordinates

56/95 3D Mean Value Coordinates Exactly same as 2D but must compute mean vector for a given spherical triangle Build wedge with face normals

57/95 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

58/95 Deformations using Barycentric Coordinates

59/95 Deformations using Barycentric Coordinates

60/95 Deformations using Barycentric Coordinates

61/95 Deformations using Barycentric Coordinates

62/95 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 216 triangles30,000 triangles1.9 seconds0.03 seconds

63/95 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 216 triangles30,000 triangles1.9 seconds0.03 seconds Real-time!

64/95 Deformation Examples Control MeshSurfaceComputing WeightsDeformation 98 triangles96,966 triangles3.3 seconds0.09 seconds

65/95 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

Constructing a Laplacian Operator 66/95

Constructing a Laplacian Operator 67/95 Laplacian

Constructing a Laplacian Operator 68/95 Euler-Lagrange Theorem

Constructing a Laplacian Operator 69/95

Constructing a Laplacian Operator 70/95

Constructing a Laplacian Operator 71/95

Constructing a Laplacian Operator 72/95

Constructing a Laplacian Operator 73/95

Constructing a Laplacian Operator 74/95

Constructing a Laplacian Operator 75/95

Constructing a Laplacian Operator 76/95

Constructing a Laplacian Operator 77/95

Constructing a Laplacian Operator 78/95

Constructing a Laplacian Operator 79/95

Constructing a Laplacian Operator 80/95

Constructing a Laplacian Operator 81/95

Constructing a Laplacian Operator 82/95

Constructing a Laplacian Operator 83/95

Constructing a Laplacian Operator 84/95

Constructing a Laplacian Operator 85/95

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 86/95

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 87/95

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 88/95

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 89/95

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 90/95 i th row contains laplacian for i th vertex

Harmonic Coordinates Solution to Laplace’s equation with boundary constraints 91/95

Harmonic Coordinates 92/95

93/95 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

94/95 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

95/95 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