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Moving Least Squares Coordinates for Barycentric Interpolation in Polygon Domains

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This work by Josiah Manson and Scott Schaefer from Texas A&M University introduces a novel family of barycentric coordinates tailored for polygon domains, focusing on the moving least squares method. It addresses boundary interpolation, emphasizing smoothness, positivity, and extended coordinate properties applicable to various computational geometry scenarios. The methodologies discussed enable polynomial boundary values and derivatives interpolation, and offer essential insights for applications in finite element methods, surface parameterization, and free-form deformations.

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Moving Least Squares Coordinates for Barycentric Interpolation in Polygon Domains

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  1. Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University

  2. Barycentric Coordinates • Polygon Domain

  3. Barycentric Coordinates • Polygon Domain

  4. Barycentric Coordinates • Polygon Domain

  5. Barycentric Coordinates • Polygon Domain

  6. Barycentric Coordinates • Polygon Domain • Boundary Interpolation

  7. Barycentric Coordinates • Polygon Domain • Boundary Interpolation

  8. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  9. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  10. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  11. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  12. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  13. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions

  14. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision

  15. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision

  16. Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision

  17. Other Properties • Desirable Features • Smoothness • Closed-form solution • Positivity • Extended Coordinates • Polynomial Boundary Values • Polynomial Precision • Interpolation of Derivatives • Curved Boundaries

  18. Applications • Finite Element Methods [Wachspress 1975]

  19. Applications • Boundary Value Problems [Ju et al. 2005]

  20. Applications • Free-Form Deformations[Sederberg et al. 1986], [MacCracken et al. 1996], [Ju et al. 2005], [Joshi et al. 2007]

  21. Applications • Surface Parameterization [Hormann et al. 2000], [Desbrun et al. 2002]

  22. Comparison of Methods Open Boundaries Concave Shapes Poly. Boundary Poly. Precision Closed-Form Derivatives Smooth Positive Wachspress Mean Val. Pos. Mean Val. Max Entropy Harmonic Hermite MVC Moving Least Sqr.

  23. Moving Least Squares Coordinates • A new family of barycentric coordinates • Solves a least squares problem • Solution depends on point of evaluation

  24. Fit a Polynomial to Points

  25. Fit a Polynomial to Points

  26. Fit a Polynomial to Points

  27. Interpolating Points

  28. Interpolating Points

  29. Interpolating Points

  30. Interpolating Line Segments

  31. Interpolating Line Segments

  32. Interpolating Line Segments

  33. Interpolating Line Segments

  34. Line Basis Functions

  35. Line Basis Functions

  36. Line Basis Functions

  37. Line Basis Functions

  38. Line Basis Functions

  39. Line Basis Functions

  40. Polygon Basis Functions

  41. Polygon Basis Functions

  42. Polygon Basis Functions

  43. Polygon Basis Functions

  44. Polygon Basis Functions

  45. Polynomial Boundary Values

  46. Polynomial Boundary Values

  47. Polynomial Boundary Values

  48. Polynomial Precision

  49. Polynomial Precision Linear Quadratic

  50. Interpolation of Derivatives

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