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Representing analytic functions using Poisson expansions and s-power series

Representing analytic functions using Poisson expansions and s-power series. Reporter: Lincong Fang Mar 29,2006. Outline. Introduction Poisson expansions S-power series. Introduction. Parametric curves and surfaces in CAGD Polynomial Rational functions Analytic functions

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Representing analytic functions using Poisson expansions and s-power series

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  1. Representing analytic functions using Poisson expansions and s-power series Reporter: Lincong Fang Mar 29,2006

  2. Outline • Introduction • Poisson expansions • S-power series

  3. Introduction • Parametric curves and surfaces in CAGD • Polynomial • Rational functions • Analytic functions • Not encompassed by the standard rational model • i.e. nonalgebraic curves or surfaces

  4. Introduction • Reason for represent analytic functions • Modeling tools • Some geometry processing operations • Offset curves and surfaces • Arc lengths, surface areas • Etc.

  5. Method I : B-basis • Peña,1999 • Advantages • Control polygon • All positive properties of Bézier scheme • de Casteljau-type algorithm • Disadvantages • No uniform basis

  6. Method II : Taylor series • Advantage • Good at approximation over a symmetric interval around 0. • Disadvantage • Lead to gaps when connect several expansions.

  7. Method III : Poisson expansions • Trimming analytic functions using right sided Poisson subdivision. CAD33(11)812-824.(2001) • Author: • Géraldine Morin • Ron Goldman • Rice University, Houston.

  8. Method IV: s-power series • S-power series: an alternative to Poisson expansions for representing analytic functions. CAGD 22,103-119.(2005) • Author: • J. Sánchez-Reyes • J.M. Chacón • University of Castilla-La Mancha(Spain)

  9. Poisson basis

  10. Poisson basis

  11. Poisson curves

  12. Poisson expansions

  13. as control points • Mimics the shape of f(t) • Tangent to f(t) at • If the series converges on [0, ∞), the convex hull and V.D. properties hold • Linear precision:

  14. Convergence of Poisson series

  15. Bézier subdivision

  16. Bézier subdivision

  17. Left-sided Poisson subdivision

  18. Left-sided Poisson subdivision

  19. Trimming: right-sided Poisson subdivision

  20. Trimming: right-sided Poisson subdivision

  21. Trimming: right-sided Poisson subdivision

  22. Convergence issues

  23. Convergence issues

  24. Extend to surfaces

  25. Intersection

  26. Conclusion • Advantages • Better convergence than Taylor series • The coefficients of the series as Control points • Analytic blossom

  27. Conclusion • Disadvantages (Truncated control polygon) • No V.D and convex hull properties • Piecewise linear approximation • Not interpolate at right endpoint

  28. Poisson subdivision

  29. S-power series

  30. The order-k Hermite interpolation:

  31. Convergence of s-power series • 0<r<1/2: two Ovals of Cassini • r=1/2: Lemniscate L of Bernoulli • R>1/2: a closed contour

  32. Convergence of s-power series

  33. Convergence of s-power series

  34. Compare with Poisson series

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