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Exploring Barycentric Coordinates in Geometry & Interpolation

Learn about Barycentric Coordinates, Homogeneous Coordinates, Boundary Value Interpolation, Wachspress Coordinates, Mean Value Coordinates, and Deformations using Barycentric Coordinates in geometry. Understand their applications, properties, and comparisons in different dimensions.

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Exploring Barycentric Coordinates in Geometry & Interpolation

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  1. Generalized Barycentric Coordinates Dr. Scott Schaefer

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

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

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

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

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

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

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

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

  10. Multi-Sided Patches

  11. Multi-Sided Patches

  12. Multi-Sided Patches

  13. Multi-Sided Patches

  14. Multi-Sided Patches

  15. Multi-Sided Patches

  16. Multi-Sided Patches

  17. Multi-Sided Patches

  18. Multi-Sided Patches

  19. Wachspress Coordinates

  20. Wachspress Coordinates

  21. Wachspress Coordinates

  22. Wachspress Coordinates

  23. Wachspress Coordinates

  24. 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. 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

  29. Mean Value Coordinates

  30. Mean Value Coordinates

  31. Mean Value Coordinates

  32. Mean Value Coordinates

  33. Mean Value Coordinates

  34. Mean Value Coordinates

  35. Mean Value Coordinates

  36. Mean Value Coordinates

  37. Mean Value Coordinates • Apply Stokes’ Theorem

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

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

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

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

  42. 3D Mean Value Coordinates

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

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

  45. 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,

  46. Deformations using Barycentric Coordinates

  47. Deformations using Barycentric Coordinates

  48. Deformations using Barycentric Coordinates

  49. Deformations using Barycentric Coordinates

  50. Deformation Examples

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