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Conformal parameterization for multiply connected domains: combining finite elements and complex analysis

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Abstract

Conformal parameterization plays an important role in isogeometric analysis. Genus zero surfaces with multiple boundary components (multiply connected domains) can be conformally mapped onto planar domains with circular holes (circle domains). This work introduces a novel method to compute such conformal mappings combining finite element and complex analysis methods. First, the surface is mapped to planar annulus with concentric circular slits using holomorphic differentials, which is carried out using a finite element method based on Hodge decomposition; second the slit domain is conformally mapped to a circle domain by a Laurent series method. Compared with existing algorithms, the proposed method is more efficient and robust. Numerical experiments demonstrate the efficiency and efficacy of the method.

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Notes

  1. To see this, transform C j to the real axis by a Möbius map, where reflection is simply complex conjugation, and then invert the previously applied Möbius map.

References

  1. Allgower EL, Georg K (1990) Numerical continuation methods: an introduction. Springer, New York

    Book  MATH  Google Scholar 

  2. Bobenko AI, Springborn BA (2004) Variational principles for circle patterns and Koebe’s theorem. Trans Am Math Soc 356:659–689

    Article  MathSciNet  MATH  Google Scholar 

  3. Bobenko AI, Springborn BA, Pinkall U (2010) Discrete conformal equivalence and ideal hyperbolic polyhedra arXiv:1005.2698

  4. Bowers PL, Hurdal MK (2003) Planar conformal mapping of piecewise flat surfaces. In: Visualization and Mathematics III. Springer, Berlin, pp 3–34

  5. Chow B, Luo F (2003) Combinatorial Ricci flows on surfaces. J Differ Geom 63(1):97–129

    MathSciNet  MATH  Google Scholar 

  6. Colin de verdiére Y (1991) Un principe variationnel pour les empilements de cercles. Invent Math 104(3):655–669

    Article  MathSciNet  MATH  Google Scholar 

  7. Collins C, Stephenson K (2003) A circle packing algorithm. Comput Geom Theory Appl 25:233–256

    Article  MathSciNet  MATH  Google Scholar 

  8. DeLillo TK (1994) The accuracy of numerical conformal mapping methods: a survey of examples and results. SIAM J Numer Anal 31(3):788–12

    Article  MathSciNet  MATH  Google Scholar 

  9. DeLillo TK (2008) Radial and circular slit maps of unbounded multiply connected circle domains. Proc R Soc A 464:1719–1737

    Google Scholar 

  10. DeLillo TK, Elcrat AR, Kropf EH Calculation of resistances for multiply connected domains using Schwarz–Christoffel transformations. Comput Methods Funct Theory 11(2):725–745

  11. DeLillo TK, Elcrat AR, Kropf EH, Pfaltzgraff JA (2013) Efficient calculation of Schwarz–Christoffel transformations for multiply connected domains using Laurent series. Comput Methods Funct Theory. doi:10.1007/s40315-013-0023-1

  12. DeLillo TK, Elcrat AR, Pfaltzgraff JA (2004) Schwarz–Christoffel mapping of multiply connected domains. Journal d’Analyse Mathématique 94(1):17–47

    Article  MathSciNet  MATH  Google Scholar 

  13. DeLillo TK, Kropf EH (2010) Slit maps and Schwarz–Christoffel maps for multiply connected domains. Electron Trans Numer Anal 36:195–223

    MathSciNet  Google Scholar 

  14. Farkas HM, Kra I (1991) Riemann Surfaces. In: Graduate texts in mathematics, vol 71. Springer, Berlin

  15. Finn MD, Cox SM, Byrne HM (2003) Topological chaos in inviscid and viscous mixers. J Fluid Mech 493:345–361

    Article  MathSciNet  MATH  Google Scholar 

  16. Floater MS, Hormann K (2005) Surface parameterization: a tutorial and survey. In: Advances in multiresolution for geometric modelling. Springer, Berlin, pp 157–186

  17. Gortler SJ, Gotsman C, Thurston D (2005) Discrete one-forms on meshes and applications to 3D mesh parameterization. Comput Aided Geom Design 23(2):83–112

    Article  MathSciNet  Google Scholar 

  18. Gu X, Yau S-T (2003) Global conformal parameterization. In: Symposium on Geometry Processing, pp 127–137

  19. Gu X, Yau S-T (2007) Computational Conformal Geometry. In: Advanced lectures in mathematics, vol 3. International Press and Higher Education Press, Boston and Beijing

  20. Guennbaud G, Jacob B et al (2010) Eigen v3, http://eigen.tuxfamily.org

  21. Hamilton RS (1988) The Ricci flow on surfaces. In: Mathematics and general relativity (Santa Cruz, CA, 1986), vol 71. Contemp. Math. Am. Math. Soc. Providence,

  22. Hamilton RS (1982) Three manifolds with positive Ricci curvature. J Differ Geom 17:255–306

    MATH  Google Scholar 

  23. Henrici P (1993) Applied and computational complex analysis, discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions, vol 3. Wiley, New York

  24. Hirani AN (2003) Discrete exterior calculus. PhD thesis, California Institute of Technology

  25. Huang P, Zhang C, Pen F (2003) High-speed 3-D shape measurement based on digital fringe projection. Opt Eng 42(1):163–168

    Article  Google Scholar 

  26. Ivanov VI, Trubetskov MK (1995) Handbook of conformal mapping with computer-aided visualization. CRC Press, Boca Raton

    MATH  Google Scholar 

  27. Jin M, Kim J, Luo F, Gu X (2008) Discrete surface Ricci flow. IEEE Trans Vis Comput Gr (TVCG) 14(5):1030–1043

    Article  Google Scholar 

  28. Kharevych L, Springborn B, Schröder P (2006) Discrete conformal mappings via circle patterns. ACM Trans Gr 25(2):412–438

    Article  Google Scholar 

  29. Koebe P (1916) Abhandlungen zur theorie der konformen abbildung. IV. Abbildung mehrfach zusammenhängender schlicter bereiche auf schlitzbereiche. Acta Math 41:305–344

    Article  MathSciNet  Google Scholar 

  30. Koebe P (1936) Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88:141–164

    Google Scholar 

  31. Kraevoy V, Sheffer A (2004) Cross-parameterization and compatible remeshing of 3D models. ACM Trans Gr 23(3):861–869

    Article  Google Scholar 

  32. Kropf EH (2012) Numerical computation of Schwarz–Christoffel transformations and slit maps for multiply connected domains. PhD Dissertation, Wichita State University

  33. Luo F (2004) Combinatorial Yamabe flow on surfaces. Commun Contemp Math 6(5):765–780

    Article  MathSciNet  MATH  Google Scholar 

  34. Lui LM, Zeng W, Yau ST, Gu XF (2013) Shape analysis of planar multiply-connected objects using conformal welding. IEEE Trans Pattern Mach Intell

  35. Mercat C (2004) Discrete Riemann surfaces and the Ising model. Commun Math Phys 218(1):177–216

    Article  MathSciNet  Google Scholar 

  36. Nasser M (2011) Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebes canonical slit domains. J Math Anal Appl 382(1):47–56

    Article  MathSciNet  MATH  Google Scholar 

  37. Nasser M (2012) Numerical conformal mapping of multiply connected regions onto the fifth category of Koebes canonical slit regions. J Math Anal Appl 398(2):729–743

    Article  MathSciNet  Google Scholar 

  38. Nehari Z (1975) Conformal Mapping. 1952. Reprint, Dover

  39. Pinkall U, Polthier K (1993) Computing discrete minimal surfaces and their conjugates. Exp Math 2(1):15–36

    Article  MathSciNet  MATH  Google Scholar 

  40. Rodin B, Sullivan D (1987) The convergence of circle packings to the Riemann mapping. J Differ Geom 26(2):349–360

    MathSciNet  MATH  Google Scholar 

  41. Sharon E, Mumford DB (2006) 2D-shape analysis using conformal mapping. Int J Comput Vis 70(1):55–75

    Article  Google Scholar 

  42. Tewari G, Gotsman C, Gortler SJ (2006) Meshing genus-1 point clouds using discrete one-forms. Comput Gr 30(6):917–926

    Article  Google Scholar 

  43. Thurston WP (1980) Geometry and topology of three-manifolds. In: Lecture notes at Princeton university

  44. Thurston WP (1985) The finite Riemann mapping theorem

  45. Tong Y, Alliez P, Cohen-Steiner D, Desbrun M (2006) Designing quadrangulations with discrete harmonic forms, In: Symposium on geometry processing, pp 201–210

  46. Trefethen LN (1980) Numerical computation of the Schwarz–Christoffel transformation. SIAM J Sci Stat Comput 1(1):82–102

    Article  MathSciNet  MATH  Google Scholar 

  47. Trefethen LN (1984) Analysis and design of polygonal resistors by conformal mapping. Zeitschrift für angewandte Mathematik und Physik 35:692–703

    Article  MathSciNet  Google Scholar 

  48. Trefethen LN (2005) Ten digit algorithms, Report No. 05/13, Oxford Univ. Comput. Lab.

  49. Wegmann R (2005) Methods for numerical conformal mapping. In: Handbook of complex analysis: geometric function theory, vol. 2. Elsevier, Amsterdam, pp 351–377

  50. Zeng W, Jin M, Luo F, Gu X (2009) Computing canonical homotopy class representative using hyperbolic structure. In: IEEE International Conference on Shape Modeling and Applications (SMI)

  51. Zeng W, Luo F, Yau S-T, Gu X (2009) Surface quasi-conformal mapping by solving Beltrami equations. In: IMA conference on the mathematics of surfaces, pp 391-408

  52. Zeng W, Yin X, Zhang M, Luo F, Gu X (2009) Generalized Koebe’s method for conformal mapping multiply connected domains. In: SIAM/ACM joint conference on geometric and physical modeling (SPM), pp 89–100

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Authors and Affiliations

  1. Mathematics Department, Harvard University, Cambridge, USA

    Everett Kropf, Xiaotian Yin & Shing-Tung Yau

  2. Computer Science, Stony Brook University, Stony Brook, USA

    Xianfeng David Gu

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  1. Everett Kropf
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  2. Xiaotian Yin
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  3. Shing-Tung Yau
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  4. Xianfeng David Gu
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Correspondence to Everett Kropf.

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Kropf, E., Yin, X., Yau, ST. et al. Conformal parameterization for multiply connected domains: combining finite elements and complex analysis. Engineering with Computers 30, 441–455 (2014). https://doi.org/10.1007/s00366-013-0348-4

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