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Conformal Mapping in Linear Time

Abstract

Given any ε>0 and any planar region Ω bounded by a simple n-gon P we construct a (1+ε)-quasiconformal map between Ω and the unit disk in time C(ε)n. One can take \(C(\epsilon)=C+C\log \frac{1}{\epsilon}\log \log \frac{1}{\epsilon}\).

References

  1. Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the Voronoĭ diagram of a convex polygon. Discrete Comput. Geom. 4(6), 591–604 (1989)

    MathSciNet  Google Scholar 

  2. Ahlfors, L.V.: Lectures on Quasiconformal Mappings. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1987). With the assistance of Clifford J. Earle, Jr., reprint of the 1966 original

    Google Scholar 

  3. Aho, A.V., Steiglitz, K., Ullman, J.D.: Evaluating polynomials at fixed sets of points. SIAM J. Comput. 4(4), 533–539 (1975)

    MathSciNet  Google Scholar 

  4. Aurenhammer, F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)

    Google Scholar 

  5. Aurenhammer, F., Klein, R.: Voronoi Diagrams. In: Handbook of Computational Geometry, pp. 201–290. North-Holland, Amsterdam (2000)

    Google Scholar 

  6. Banchoff, T.F., Giblin, P.J.: Global theorems for symmetry sets of smooth curves and polygons in the plane. Proc. R. Soc. Edinb. A 106(3–4), 221–231 (1987)

    MathSciNet  Google Scholar 

  7. Banjai, L., Trefethen, L.N.: A multipole method for Schwarz–Christoffel mapping of polygons with thousands of sides. SIAM J. Sci. Comput. 25(3), 1042–1065 (2003) (electronic)

    MathSciNet  Google Scholar 

  8. Bern, M., Eppstein, D.: Mesh generation and optimal triangulation. In: Computing in Euclidean Geometry. Lecture Notes Ser. Comput, vol. 1, pp. 23–90. World Scientific, River Edge (1992)

    Google Scholar 

  9. Beurling, A., Ahlfors, L.: The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956)

    MathSciNet  Google Scholar 

  10. Binder, I., Braverman, M., Yampolsky, M.: On the computational complexity of the Riemann mapping. Ark. Mat. 45(2), 221–239 (2007)

    MathSciNet  Google Scholar 

  11. Bishop, C.J.: Divergence groups have the Bowen property. Ann. Math. (2) 154(1), 205–217 (2001)

    MathSciNet  Google Scholar 

  12. Bishop, C.J.: BiLipschitz approximations of quasiconformal maps. Ann. Acad. Sci. Fenn. Math. 27(1), 97–108 (2002)

    MathSciNet  Google Scholar 

  13. Bishop, C.J.: Quasiconformal Lipschitz maps, Sullivan’s convex hull theorem and Brennan’s conjecture. Ark. Mat. 40(1), 1–26 (2002)

    MathSciNet  Google Scholar 

  14. Bishop, C.J.: An explicit constant for Sullivan’s convex hull theorem. In: In the Tradition of Ahlfors and Bers, III. Contemp. Math., vol. 355, pp. 41–69. Am. Math. Soc., Providence (2004)

    Google Scholar 

  15. Bishop, C.J., Hakobyan, H.: A central set of dimension 2. Proc. Am. Math. Soc. 136(7), 2453–2461 (2008)

    MathSciNet  Google Scholar 

  16. Bishop, C.J.: Bounds for the CRDT conformal mapping algorithm. Comput. Methods Funct. Theory 10(1), 325–366 (2010)

    MathSciNet  Google Scholar 

  17. Bishop, C.J.: A fast QC-mapping theorem for polygons. Preprint (2009)

  18. Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279. Springer, Berlin (1985)

    Google Scholar 

  19. Blum, H.: A transformation for extracting new descriptors of shape. In: Dunn, W.W. (ed.) Proc. Sympos. Models for the Perception of Speech and Visual Form, pp. 362–380. MIT Press, Cambridge (1967)

    Google Scholar 

  20. Blum, H.: Biological shape and visual science. J. Theoret. Biol. 38, 205–287 (1973)

    Google Scholar 

  21. Blum, H., Nagel, R.N.: Shape descriptors using weighted symmetric axis functions. Pattern Recogn. 10(3), 167–180 (1978)

    Google Scholar 

  22. Bridgeman, M.: Average bending of convex pleated planes in hyperbolic three-space. Invent. Math. 132(2), 381–391 (1998)

    MathSciNet  Google Scholar 

  23. Bridgeman, M.: Average curvature of convex curves in H 2. Proc. Am. Math. Soc. 126(1), 221–224 (1998)

    MathSciNet  Google Scholar 

  24. Bridgeman, M., Canary, R.D.: From the boundary of the convex core to the conformal boundary. Geom. Dedicata 96, 211–240 (2003)

    MathSciNet  Google Scholar 

  25. Bruce, J.W., Giblin, P.J., Gibson, C.G.: Symmetry sets. Proc. R. Soc. Edinb. A 101(1–2), 163–186 (1985)

    MathSciNet  Google Scholar 

  26. Canary, R.D.: The conformal boundary and the boundary of the convex core. Duke Math. J. 106(1), 193–207 (2001)

    MathSciNet  Google Scholar 

  27. Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. (2) 76, 547–559 (1962)

    MathSciNet  Google Scholar 

  28. Chazai, F., Lieutier, A.: Stability and homotopy of a subset of the medial axis. In: Proc. 9th ACM Sympos. Solid Modeling Appl. (2004)

  29. Chazal, F., Soufflet, R.: Stability and finiteness properties of medial axis and skeleton. J. Dynam. Control Syst. 10(2), 149–170 (2004)

    MathSciNet  Google Scholar 

  30. Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(5), 485–524 (1991)

    MathSciNet  Google Scholar 

  31. Cheng, H.: A constructive Riemann mapping theorem. Pac. J. Math. 44, 435–454 (1973)

    Google Scholar 

  32. Chiang, C.-S., Hoffmann, C.M.: The medial axis transform for 2d regions. ACM Transactions on graphics (1982)

  33. Chin, F., Snoeyink, J., Wang, C.A.: Finding the medial axis of a simple polygon in linear time. Discrete Comput. Geom. 21(3), 405–420 (1999)

    MathSciNet  Google Scholar 

  34. Choi, H.I., Choi, S.W., Moon, H.P.: Mathematical theory of medial axis transform. Pac. J. Math. 181(1), 57–88 (1997)

    MathSciNet  Google Scholar 

  35. Choi, S.W., Seidel, H.-P.: Hyperbolic Hausdorff distance for medial axis transformation. Graph. Models 63, 369–384 (2001)

    Google Scholar 

  36. Choi, S.W., Seidel, H.-P.: Linear one-sided stability of MAT for weakly injective domain. J. Math. Imaging Vis. 17(3), 237–247 (2002)

    MathSciNet  Google Scholar 

  37. Choi, S.W., Lee, S.-W.: Stability analysis of medial axis transform. In Proc. 15th ICPR Barcelona, Spain, vol. 3, pp. 139–142 (2000)

  38. Christoffle, E.B.: Sul problema della tempurature stazonaire e la rappresetazione di una data superficie. Ann. Mat. Pura Appl. Ser. II, pp. 89–103 (1867)

  39. Cipra, B.: The best of the 20th century: Editors name top 10 algorithms. SIAM News 33(4), 1 (2000)

    Google Scholar 

  40. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    MathSciNet  Google Scholar 

  41. Culver, T., Keyser, J., Manocha, D.: Accurate computation of the medial axis of a polyhedron. In: Proceedings of the Fifth ACM Symposium on Solid Modeling and Applications, June 8–11, 1999, Ann Arbor, MI, USA, pp. 179–190 (1999)

  42. Daripa, P.: A fast algorithm to solve nonhomogeneous Cauchy–Riemann equations in the complex plane. SIAM J. Sci. Stat. Comput. 13(6), 1418–1432 (1992)

    MathSciNet  Google Scholar 

  43. Daripa, P.: A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings. J. Comput. Phys. 106(2), 355–365 (1993)

    MathSciNet  Google Scholar 

  44. Daripa, P., Mashat, D.: An efficient and novel numerical method for quasiconformal mappings of doubly connected domains. Numer. Algorithms 18(2), 159–175 (1998)

    MathSciNet  Google Scholar 

  45. Davis, R.T.: Numerical methods for coordinate generation based on Schwarz–Christoffel transformations. In: 4th AIAA Comput. Fluid Dynamics Conf., Williamsburg, VA, pp. 1–15 (1979)

  46. DeLillo, T.K.: The accuracy of numerical conformal mapping methods: a survey of examples and results. SIAM J. Numer. Anal. 31(3), 788–812 (1994)

    MathSciNet  Google Scholar 

  47. Dirichlet, G.L.: Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math. 40, 209–227 (1850)

    Google Scholar 

  48. Driscoll, T.A., Trefethen, L.N.: Schwarz–Christoffel Mapping. Cambridge Monographs on Applied and Computational Mathematics, vol. 8. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  49. Driscoll, T.A., Vavasis, S.A.: Numerical conformal mapping using cross-ratios and Delaunay triangulation. SIAM J. Sci. Comput. 19(6), 1783–1803 (1998) (electronic),

    MathSciNet  Google Scholar 

  50. Duan, H.B., Rees, E.: The existence of bitangent spheres. Proc. R. Soc. Edinb. A 111(1–2), 85–87 (1989)

    MathSciNet  Google Scholar 

  51. Dutt, A., Gu, M., Rokhlin, V.: Fast algorithms for polynomial interpolation, integration, and differentiation. SIAM J. Numer. Anal. 33(5), 1689–1711 (1996)

    MathSciNet  Google Scholar 

  52. Epstein, D.B.A., Marden, A.: Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In: Analytical and Geometric Aspects of Hyperbolic Space, Coventry/Durham, 1984. London Math. Soc. Lecture Note Ser., vol. 111, pp. 113–253. Cambridge Univ. Press, Cambridge (1987)

    Google Scholar 

  53. Epstein, D.B.A., Marden, A., Markovic, V.: Quasiconformal homeomorphisms and the convex hull boundary. Ann. Math. (2) 159(1), 305–336 (2004)

    MathSciNet  Google Scholar 

  54. Epstein, D.B.A., Marden, A., Markovic, V.: Complex earthquakes and deformations of the unit disk. J. Differ. Geom. 73(1), 119–166 (2006)

    MathSciNet  Google Scholar 

  55. Epstein, D.B.A., Marden, A., Markovic, V.: Convex regions in the plane and their domes. Proc. Lond. Math. Soc. (3) 92(3), 624–654 (2006)

    MathSciNet  Google Scholar 

  56. Epstein, D.B.A., Markovic, V.: The logarithmic spiral: a counterexample to the K=2 conjecture. Ann. Math. (2) 161(2), 925–957 (2005)

    MathSciNet  Google Scholar 

  57. Erdős, P.: On the Hausdorff dimension of some sets in Euclidean space. Bull. Am. Math. Soc. 52, 107–109 (1946)

    Google Scholar 

  58. Evans, G., Middleditch, A., Miles, N.: Stable computation of the 2D medial axis transform. Int. J. Comput. Geom. Appl. 8(5–6), 577–598 (1998)

    MathSciNet  Google Scholar 

  59. Evans, W.D., Harris, D.J.: Sobolev embeddings for generalized ridged domains. Proc. Lond. Math. Soc. (3) 54(1), 141–175 (1987)

    MathSciNet  Google Scholar 

  60. Fortune, S.: Voronoĭ diagrams and Delaunay triangulations. In: Computing in Euclidean Geometry. Lecture Notes Ser. Comput, vol. 1, pp. 193–233. World Scientific, River Edge (1992)

    Google Scholar 

  61. Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Handbook of Discrete and Computational Geometry. CRC Press Ser. Discrete Math. Appl., pp. 377–388. CRC, Boca Raton (1997)

    Google Scholar 

  62. Fremlin, D.H.: Skeletons and central sets. Proc. Lond. Math. Soc. (3) 74(3), 701–720 (1997)

    MathSciNet  Google Scholar 

  63. Gaier, D.: Konstruktive Methoden der konformen Abbildung. Springer Tracts in Natural Philosophy, vol. 3. Springer, Berlin (1964)

    Google Scholar 

  64. Garnett, J.B.: Bounded Analytic Functions. Pure and Applied Mathematics, vol. 96. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1981)

    Google Scholar 

  65. Gaudeau, C., Boiron, M., Thouvenot, J.: Squelettisation et anamorphose dans l’étude de la dynamique des déformations des structures: application à l’analyse de la motricité gastrique. In: Recognition of Shapes and Artificial Intelligence (Second AFCET-IRIA Cong., Toulouse, 1979), vol. III, pp. 57–63. IRIA, Rocquencourt (1979) (in French)

    Google Scholar 

  66. Gehring, F.W.: The definitions and exceptional sets for quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I 281, 28 (1960)

    MathSciNet  Google Scholar 

  67. Gehring, F.W.: Symmetrization of rings in space. Trans. Am. Math. Soc. 101, 499–519 (1961)

    MathSciNet  Google Scholar 

  68. Giblin, P.: Symmetry sets and medial axes in two and three dimensions. In: The Mathematics of Surfaces, IX, Cambridge, 2000, pp. 306–321. Springer, London (2000)

    Google Scholar 

  69. Giblin, P.J., O’Shea, D.B.: The bitangent sphere problem. Am. Math. Mon. 97(1), 5–23 (1990)

    MathSciNet  Google Scholar 

  70. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    MathSciNet  Google Scholar 

  71. Gursoy, H.N., Patrikalakis, N.M.: Automated interrogation and adaptive subdivision of shape using medial axis transform. Adv. Eng. Softw. Workstations 13(5/6), 287–302 (1991)

    Google Scholar 

  72. He, Z.-X., Schramm, O.: On the convergence of circle packings to the Riemann map. Invent. Math. 125(2), 285–305 (1996)

    MathSciNet  Google Scholar 

  73. He, Z.-X., Schramm, O.: The C -convergence of hexagonal disk packings to the Riemann map. Acta Math. 180(2), 219–245 (1998)

    MathSciNet  Google Scholar 

  74. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)

    MathSciNet  Google Scholar 

  75. Henrici, P.: Applied and Computational Complex Analysis. Pure and Applied Mathematics, vol. 3. Wiley, New York (1986). Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions, A Wiley-Interscience Publication

    Google Scholar 

  76. Hertling, P.: An effective Riemann mapping theorem. Theoret. Comput. Sci. 219(1–2), 225–265 (1999). Computability and complexity in analysis (Castle Dagstuhl, 1997)

    MathSciNet  Google Scholar 

  77. Hoffmann, C.M.: Computer vision, descriptive geometry and classical mechanics. In: Computer Graphics and Mathematics. Eurographics Series, pp. 229–244. Springer, Berlin (1992)

    Google Scholar 

  78. Hoffmann, C.M., Dutta, D.: On the skeleton of simple CSG objects. Trans. ASME 115, 87–94 (1993)

    Google Scholar 

  79. Holopainen, I.: Rough isometries and p-harmonic functions with finite Dirichlet integral. Rev. Mat. Iberoam. 10(1), 143–176 (1994)

    MathSciNet  Google Scholar 

  80. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, 2nd edn. Springer, Berlin (1990). Distribution theory and Fourier analysis

    Google Scholar 

  81. Howell, L.H.: Numerical conformal mapping of circular arc polygons. J. Comput. Appl. Math. 46(1–2), 7–28 (1993)

    MathSciNet  Google Scholar 

  82. Hu, C.: Algorithm 785: a software package for computing Schwarz–Christoffel conformal transformation for doubly connected polygonal regions. ACM Trans. Math. Softw. 24(3), 317–333 (1998)

    Google Scholar 

  83. Ivanov, V.I., Trubetskov, M.K.: Handbook of Conformal Mapping with Computer-Aided Visualization. CRC Press, Boca Raton (1995). With 1 IBM-PC floppy disk (5.25 inch; HD)

    Google Scholar 

  84. Jinkerson, R.A., Abrams, S.L., Bardis, L., Chryssostomidis, C., Clement, A., Patrikalakis, N.M., Wolter, F.E.: Inspection and feature extraction of marine propellers. J. Ship Product. 9(2), 88–106 (1993)

    Google Scholar 

  85. Karatsuba, A., Ofman, Yu.: Multiplication of many-digital numbers by automatic computers. Dokl. Akad. Nauk SSSR 145, 293–294 (1962)

    Google Scholar 

  86. Klein, R., Lingas, A.: A linear-time randomized algorithm for the bounded Voronoi diagram of a simple polygon. Int. J. Comput. Geom. Appl. 6(3), 263–278 (1996). ACM Symposium on Computational Geometry (San Diego, CA, 1993)

    MathSciNet  Google Scholar 

  87. Kythe, P.K.: Computational Conformal Mapping. Birkhäuser, Boston (1998)

    Google Scholar 

  88. Lee, D.-T.: The medial axis transform of a planar shape. IEEE Trans. Pattern Anal. Mach. Intell. 4(4), 363–369 (1982)

    Google Scholar 

  89. Lee, D.T., Drysdale, R.L. III: Generalization of Voronoĭ diagrams in the plane. SIAM J. Comput. 10(1), 73–87 (1981)

    MathSciNet  Google Scholar 

  90. Lee, Y.-H., Horng, S.-J.: The equivalence of the chessboard distance transform and the medial axis transform. Int. J. Comput. Math. 65(3–4), 165–177 (1997)

    MathSciNet  Google Scholar 

  91. Maekawa, T., Patrikalakis, N.M.: Computation of singularities and intersections of offsets of planar curves. Comput. Aided Geom. Design 10(5), 407–429 (1993)

    MathSciNet  Google Scholar 

  92. Maekawa, T., Patrikalakis, N.M.: Interrogation of differential geometry properties for design and manufacture. Vis. Comput. 10(4), 216–237 (1994)

    Google Scholar 

  93. Marshall, D.E., Rohde, S.: Convergence of a variant of the zipper algorithm for conformal mapping. SIAM J. Numer. Anal. 45(6), 2577–2609 (2007) (electronic)

    MathSciNet  Google Scholar 

  94. Milman, D.: The central function of the boundary of a domain and its differentiable properties. J. Geom. 14(2), 182–202 (1980)

    MathSciNet  Google Scholar 

  95. Milman, D., Waksman, Z.: On topological properties of the central set of a bounded domain in R m. J. Geom. 15(1), 1–7 (1981)

    MathSciNet  Google Scholar 

  96. Mohar, B.: A polynomial time circle packing algorithm. Discrete Math. 117(1–3), 257–263 (1993)

    MathSciNet  Google Scholar 

  97. Mohar, B.: Circle packings of maps in polynomial time. Eur. J. Combin. 18(7), 785–805 (1997)

    MathSciNet  Google Scholar 

  98. Nehari, Z.: Conformal Mapping. Dover, New York (1975). Reprinting of the 1952 edition

    Google Scholar 

  99. O’Donnell, S.T., Rokhlin, V.: A fast algorithm for the numerical evaluation of conformal mappings. SIAM J. Sci. Stat. Comput. 10(3), 475–487 (1989)

    MathSciNet  Google Scholar 

  100. O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  101. Papamichael, N., Saff, E.B. (eds.): Computational complex analysis. J. Comput. Appl. Math. 46(1–2) (1993)

  102. Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Berlin (2002)

    Google Scholar 

  103. Pottmann, H., Wallner, J.: Computational Line Geometry. Mathematics+Visualization. Springer, Berlin (2001)

    Google Scholar 

  104. Preparata, F.P.: The medial axis of a simple polygon. In: Mathematical Foundations of Computer Science, Proc. Sixth Sympos., Tatranská Lomnica, 1977. Lecture Notes in Comput. Sci., vol. 53, pp. 443–450. Springer, Berlin (1977)

    Google Scholar 

  105. Preparata, F.P., Shamos, M.I.: Computational Geometry. Texts and Monographs in Computer Science. Springer, New York (1985). An introduction

    Google Scholar 

  106. Rajan, V.T.: Optimality of the Delaunay triangulation in R d. Discrete Comput. Geom. 12(2), 189–202 (1994)

    MathSciNet  Google Scholar 

  107. Reif, J.H.: Approximate complex polynomial evaluation in near constant work per point. SIAM J. Comput. 28(6), 2059–2089 (1999) (electronic)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  109. Schwarz, H.A.: Conforme Abbildung der Oberfläche eines Tetraeders auf die Oberfläche einer Kugel. J. Reine Ange. Math., pp. 121–136, 1869. Also in collected works, Gesammelte Mathematische Abhandlungen, pp. 84–101. Springer, Berlin (1890)

  110. Schwarz, H.A.: Gesammelte Mathematische Abhandlungen. Springer, Berlin (1890)

    Google Scholar 

  111. Shamos, M.I., Hoey, D.: Closest-point problems. In: 16th Annual Symposium on Foundations of Computer Science (Berkeley, CA, 1975), pp. 151–162. IEEE Computer Society, Long Beach (1975)

    Google Scholar 

  112. Sherbrooke, E.C., Patrikalakis, N.M., Brisson, E.: Computation of the medial axis transform of 3-d. In: Symposium on Solid Modeling and Applications, pp. 187–200 (1995)

  113. Sherbrooke, E.C., Patrikalakis, N.M., Brisson, E.: An algorithm for the medial axis transform of 3d polyhedral solids. IEEE Trans. Vis. Comput. Graph. 2(1), 44–61 (1996)

    Google Scholar 

  114. Sherbrooke, E.C., Patrikalakis, N.M., Wolter, F.-E.: Differential and topological properties of medial axis transforms. CVGIP: Graph. Model Image Process. 58(6), 574–592 (1996)

    Google Scholar 

  115. Smith, W.D.: Accurate circle configurations and numerical conformal mapping in polynomial time. Unpublished technical memorandum, NEC Research Institute, Princeton, NJ (1991)

  116. Stephenson, K.: Circlepack. Software available from http://www.math.utk.edu/~kens/

  117. Stephenson, K.: The approximation of conformal structures via circle packing. In: Computational Methods and Function Theory 1997 (Nicosia). Ser. Approx. Decompos., vol. 11, pp. 551–582. World Scientific, River Edge (1999)

    Google Scholar 

  118. Stephenson, K.: Circle packing and discrete analytic function theory. In: Handbook of Complex Analysis: Geometric Function Theory, vol. 1, pp. 333–370. North-Holland, Amsterdam (2002)

    Google Scholar 

  119. Stephenson, K.: Circle packing: a mathematical tale. Not. Am. Math. Soc. 50(11), 1376–1388 (2003)

    MathSciNet  Google Scholar 

  120. Strebel, K.: On the existence of extremal Teichmueller mappings. J. Anal. Math. 30, 464–480 (1976)

    MathSciNet  Google Scholar 

  121. Sullivan, D.: Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibrées sur S 1. In: Bourbaki Seminar, vol. 1979/80, pp. 196–214. Springer, Berlin (1981)

    Google Scholar 

  122. Tang, Z.: Fast Transformations Based on Structured Matrices with Applications to the Fast Multipole Method. PhD thesis, University of Maryland, College Park, Maryland (2004)

  123. Thom, R.: Sur le cut-locus d’une variété plongée. J. Differ. Geom. 6, 577–586 (1972). Collection of articles dedicated to S.S. Chern and D.C. Spencer on their sixtieth birthdays

    MathSciNet  Google Scholar 

  124. Thurston, W.P.: The Geometry and Topology of 3-Manifolds. The Geometry Center, University of Minnesota (1979)

  125. Trefethen, L.N. (ed.): Numerical Conformal Mapping. North-Holland, Amsterdam (1986). Reprint of J. Comput. Appl. Math. 14(1–2) (1986)

    Google Scholar 

  126. Trefethen, L.N., Driscoll, T.A.: Schwarz–Christoffel mapping in the computer era. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 533–542 Berlin (1998) (electronic)

  127. van der Hoeven, J.: Fast evaluation of holonomic functions. Theoret. Comput. Sci. 210(1), 199–215 (1999)

    MathSciNet  Google Scholar 

  128. van der Hoeven, J.: Relax, but don’t be too lazy. J. Symb. Comput. 34(6), 479–542 (2002)

    Google Scholar 

  129. von Koppenfels, W., Stallmann, F.: Praxis der konformen Abbildung. Die Grundlehren der mathematischen Wissenschaften, vol. 100. Springer, Berlin (1959)

    Google Scholar 

  130. Voronoi, G.M.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. recherches sur les parallélloèdres primitifs. J. Reine Angew. Math. 134, 198–287 (1908)

    Google Scholar 

  131. Wang, J.: Medial axis and optimal locations for min-max sphere packing. J. Comput. Optim. 4(4), 487–503 (2000)

    Google Scholar 

  132. Wegmann, R.: Methods for numerical conformal mapping. In: Handbook of Complex Analysis: Geometric Function Theory, vol. 2, pp. 351–477. Elsevier, Amsterdam (2005)

    Google Scholar 

  133. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  134. Wolter, E.-F.: Cut locus and the medial axis in global shape interrogation and representation. MIT, Dept. of Ocean Engineering, Design Laboratory Memorandum 92-2 (1993)

  135. Wu, Q.J.: Sphere packing using morphological analysis. In: Discrete Mathematical Problems with Medical Applications, New Brunswick, NJ, 1999. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 55, pp. 45–54. Am. Math. Soc., Providence (2000)

    Google Scholar 

  136. Yao, C., Rokne, J.G.: A straightforward algorithm for computing the medial axis of a simple polygon. Int. J. Comput. Math 39, 51–60 (1991)

    Google Scholar 

  137. Yap, C.-K.: An O(n log n) algorithm for the Voronoĭ diagram of a set of simple curve segments. Discrete Comput. Geom. 2(4), 365–393 (1987)

    MathSciNet  Google Scholar 

  138. Zhou, Q.: Computable real-valued functions on recursive open and closed subsets of Euclidean space. Math. Logic Q. 42(3), 379–409 (1996)

    Google Scholar 

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Correspondence to Christopher J. Bishop.

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The author is partially supported by NSG Grant DMS 10-06309.

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Bishop, C.J. Conformal Mapping in Linear Time. Discrete Comput Geom 44, 330–428 (2010). https://doi.org/10.1007/s00454-010-9269-9

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Keywords

  • Numerical conformal mappings
  • Schwarz–Christoffel formula
  • Hyperbolic 3-manifolds
  • Sullivan’s theorem
  • Convex hulls
  • Quasiconformal mappings
  • Quasisymmetric mappings
  • Medial axis
  • CRDT algorithm