Generalized Barycentric Coordinates Dr Scott Schaefer 1 Barycentric

Generalized Barycentric Coordinates Dr. Scott Schaefer 1

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

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

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

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

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

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

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

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

Multi-Sided Patches 10/83

Multi-Sided Patches 11/83

Multi-Sided Patches 12/83

Multi-Sided Patches 13/83

Multi-Sided Patches 14/83

Multi-Sided Patches 15/83

Multi-Sided Patches 16/83

Multi-Sided Patches 17/83

Multi-Sided Patches 18/83

Wachspress Coordinates 19/83

Wachspress Coordinates 20/83

Wachspress Coordinates 21/83

Wachspress Coordinates 22/83

Wachspress Coordinates 23/83

Wachspress Coordinates 24/83

Smooth Wachspress Coordinates n Given find weights such that

Smooth Wachspress Coordinates n Given find weights such that

Smooth Wachspress Coordinates n Given find weights such that

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

Mean Value Coordinates 29/83

Mean Value Coordinates 30/83

Mean Value Coordinates 31/83

Mean Value Coordinates 32/83

Mean Value Coordinates n 33/83

Mean Value Coordinates n n 34/83

Mean Value Coordinates n n n 35/83

Mean Value Coordinates 36/83

Mean Value Coordinates n Apply Stokes’ Theorem 37/83

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

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

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

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

3 D Mean Value Coordinates 42/83

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

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

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

Deformations using Barycentric Coordinates 46/83

Deformations using Barycentric Coordinates 47/83

Deformations using Barycentric Coordinates 48/83

Deformations using Barycentric Coordinates 49/83

Deformation Examples Control Mesh Surface Computing Weights Deformation 216 triangles 30, 000 triangles 0. 7 seconds 0. 02 seconds 50/83

Deformation Examples Control Mesh Surface Computing Weights Deformation 216 triangles 30, 000 triangles 0. 7 seconds 0. 02 seconds Real-time! 51/83

Deformation Examples Control Mesh Surface Computing Weights Deformation 98 triangles 96, 966 triangles 1. 1 seconds 0. 05 seconds 52/83

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

Constructing a Laplacian Operator 54/83

Constructing a Laplacian Operator Laplacian 55/83

Constructing a Laplacian Operator Euler-Lagrange Theorem 56/83

Constructing a Laplacian Operator 57/83

Constructing a Laplacian Operator 58/83

Constructing a Laplacian Operator 59/83

Constructing a Laplacian Operator 60/83

Constructing a Laplacian Operator 61/83

Constructing a Laplacian Operator 62/83

Constructing a Laplacian Operator 63/83

Constructing a Laplacian Operator 64/83

Constructing a Laplacian Operator 65/83

Constructing a Laplacian Operator 66/83

Constructing a Laplacian Operator 67/83

Constructing a Laplacian Operator 68/83

Constructing a Laplacian Operator 69/83

Constructing a Laplacian Operator 70/83

Constructing a Laplacian Operator 71/83

Constructing a Laplacian Operator 72/83

Constructing a Laplacian Operator 73/83

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

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

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

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

Harmonic Coordinates n Solution to Laplace’s equation with boundary constraints ith row contains laplacian for ith vertex 78/83

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

Harmonic Coordinates 80/83

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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