Dr. Scott Schaefer

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Generalized Barycentric Coordinates

• Dr. Scott Schaefer

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

Homogenous coordinates
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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Barycentric Coordinates

• Given find weights such that
• are barycentric coordinates

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Boundary Value Interpolation

• Given , compute such that
• Given values at , construct a function
• Interpolates values at vertices
• Linear on boundary
• Smooth on interior

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Boundary Value Interpolation

• Given , compute such that
• Given values at , construct a function
• Interpolates values at vertices
• Linear on boundary
• Smooth on interior

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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Multi-Sided Patches
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Wachspress Coordinates
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Wachspress Coordinates
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Wachspress Coordinates
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Wachspress Coordinates
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Wachspress Coordinates
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Wachspress Coordinates
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Smooth Wachspress Coordinates

• Given find weights such that

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Smooth Wachspress Coordinates

• Given find weights such that

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Smooth Wachspress Coordinates

• Given find weights such that

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Wachspress Coordinates Summary

• Coordinate functions are rational and of low
  degree
• Coordinates are only well-defined for convex
  polygons
• wi are positive inside of convex polygons
• 3D and higher dimensional extensions (for convex
  shapes) do exist

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Mean Value Coordinates
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Mean Value Coordinates
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Mean Value Coordinates
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Mean Value Coordinates
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Mean Value Coordinates

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Mean Value Coordinates

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Mean Value Coordinates

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Mean Value Coordinates
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Mean Value Coordinates

• Apply Stokes Theorem

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Comparison
convex polygons (Wachspress Coordinates)
closed polygons (Mean Value Coordinates)
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Comparison
convex polygons (Wachspress Coordinates)
closed polygons (Mean Value Coordinates)
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Comparison
convex polygons (Wachspress Coordinates)
closed polygons (Mean Value Coordinates)
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Comparison
convex polygons (Wachspress Coordinates)
closed polygons (Mean Value Coordinates)
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3D Mean Value Coordinates
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3D Mean Value Coordinates

• Exactly same as 2D but must compute mean vector
  for a given spherical triangle

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3D Mean Value Coordinates

• Exactly same as 2D but must compute mean vector
  for a given spherical triangle
• Build wedge with face normals

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3D Mean Value Coordinates

• Exactly same as 2D but must compute mean vector
  for a given spherical triangle
• Build wedge with face normals
• Apply Stokes Theorem,

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Deformations using Barycentric Coordinates
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Deformations using Barycentric Coordinates
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Deformations using Barycentric Coordinates
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Deformations using Barycentric Coordinates
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Deformation Examples
Control Mesh Surface Computing Weights Deformation
216 triangles 30,000 triangles 0.7 seconds 0.02 seconds
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Deformation Examples
Control Mesh Surface Computing Weights Deformation
216 triangles 30,000 triangles 0.7 seconds 0.02 seconds
Real-time!
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Deformation Examples
Control Mesh Surface Computing Weights Deformation
98 triangles 96,966 triangles 1.1 seconds 0.05 seconds
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Mean Value Coordinates Summary

• Coordinate functions are NOT rational
• Coordinates are only well-defined for any closed,
  non-self-intersecting polygon/surface
• wi are positive inside of convex polygons, but
  not in general

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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
Laplacian
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Constructing a Laplacian Operator
Euler-Lagrange Theorem
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Constructing a Laplacian Operator
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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

ith row contains laplacian for ith vertex
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Harmonic Coordinates

• Solution to Laplaces equation with boundary
  constraints

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Harmonic Coordinates
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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

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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

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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

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Polar Duals of Convex Polygons

• Given a convex polyhedron P containing the
  origin, the polar dual is

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Properties of Polar Duals

• is dual to a face with plane equation
• Each face with normal and vertex is
  dual to the vertex

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Properties of Polar Duals

• is dual to a face with plane equation
• Each face with normal and vertex is
  dual to the vertex

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

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Coordinates From Polar Duals

• Given a point v, translate v to origin
• Construct polar dual

Identical to Wachspress Coordinates!
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Extensions into Higher Dimensions

• Compute polar dual
• Volume of pyramid from dual face to origin is
  barycentric coordinate