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## Conformal mapping in linear time (2006)

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2989 |
The analysis of linear partial differential operators I,
- Hormander
- 1990
(Show Context)
Citation Context ...ensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include [8], [28], [55], [66], [77], [78], =-=[91]-=-, [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the ... |

1152 | A fast algorithm for particle simulations
- Greengard, Rokhlin
(Show Context)
Citation Context ...nd then simply integrate the series term-by-term. We need to compute this with error at most on a Whitney square Qj. Cɛ 2 diam(Qj) , We shall use the fast multipole algorithm of Rokhlin and Greengard =-=[79]-=-. This method was named one of the top ten algorithms of the 20th century in [44]. The basic idea is that we have n empty regions where want to compute a series expansion, each of which is influenced ... |

1120 |
An Algorithm for the Machine Calculation of Complex Fourier Series
- Cooley, Tukey
- 1965
(Show Context)
Citation Context ... . . . . .. . 1 ω ω2 . . . ωn−1 1 ω2 ω4 . . . ω2(n−1) 1 ω n−1 ω 2(n−1) . . . ω (n−1)(n−2) where ω is an nth root of unity. The fast Fourier transform (FFT) applies Fn to an nvector in time O(n log n) =-=[45]-=-. Fn is unitary and its conjugate transpose, F ∗ , can also be applied in O(n log n) time. The discrete Fourier transform (DFT) takes a n-long sequence of complex numbers {ak} n−1 0 and a n-root of un... |

743 | Voronoi diagrams, a survey of a fundamental geometric data structure,”
- Aurenhammer
- 1991
(Show Context)
Citation Context ...t [52] (indeed, in the theory of Kleinian groupsCONFORMAL MAPPING IN LINEAR TIME 21 the Voronoi cells of an orbit are called Dirichlet fundamental domains). For more about Voronoi diagrams see e.g., =-=[5]-=-, [6], [10], [67], [68], [116], [121]. Figure 16. This shows the Voronoi cells when all the edges and vertices are sites. However, the dashed edges must be removed to give the medial axis. It is a the... |

643 |
A transformation for extracting new descriptors of shape. Models for the Perception of Speech and Visual Form,
- Blum
- 1967
(Show Context)
Citation Context ...edial axis include [8], [28], [55], [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes =-=[21]-=-, [22], [23]. A few papers consider the theory of the medial axis (e.g., [38], [39], [40], [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like... |

323 |
Triangulating a Simple Polygon in Linear Time,
- Chazelle
- 1990
(Show Context)
Citation Context ...ch piece with work O(k) for a piece with k sides and finally (3) merge the Voronoi diagrams of the pieces using at most O(n) work. The first step is accomplished using a celebrated result of Chazelle =-=[34]-=- that one can cut interior of P into trapezoids with vertical sides in linear time (this is equivalent to triangulating the polygon in linear time). Klein and Lingas [101] showed how to use Chazelle’s... |

265 | Quasiconformal maps in metric spaces with controlled geometry.
- Heinonen, Koskela
- 1998
(Show Context)
Citation Context ...sup r→0 maxy:|x−y|=r |f(x) − f(y)| miny:|x−y|=r |f(x) − f(y)| ≤ K. For a proof of the equivalence of the first two, see [2] and for a discussion of the third and a generalization to metric spaces see =-=[83]-=- and its references. In Euclidean space the equivalence of the three definitions is due to Gehring [74], [75], [75]. A composition of a K1-quasiconformal map with a K2-quasiconformal map is (K1K2)quas... |

250 | Voronoi diagrams and Delaunay triangulations, Handbook of discrete and computational geometry
- Fortune
- 1997
(Show Context)
Citation Context ...in the theory of Kleinian groupsCONFORMAL MAPPING IN LINEAR TIME 21 the Voronoi cells of an orbit are called Dirichlet fundamental domains). For more about Voronoi diagrams see e.g., [5], [6], [10], =-=[67]-=-, [68], [116], [121]. Figure 16. This shows the Voronoi cells when all the edges and vertices are sites. However, the dashed edges must be removed to give the medial axis. It is a theorem of Chin, Sno... |

214 | Mesh generation and optimal triangulation.
- Bern, Eppstein
- 1995
(Show Context)
Citation Context ...deed, in the theory of Kleinian groupsCONFORMAL MAPPING IN LINEAR TIME 21 the Voronoi cells of an orbit are called Dirichlet fundamental domains). For more about Voronoi diagrams see e.g., [5], [6], =-=[10]-=-, [67], [68], [116], [121]. Figure 16. This shows the Voronoi cells when all the edges and vertices are sites. However, the dashed edges must be removed to give the medial axis. It is a theorem of Chi... |

186 |
Biological Shape and Visual Science,
- Blum
- 1973
(Show Context)
Citation Context ...axis include [8], [28], [55], [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], =-=[22]-=-, [23]. A few papers consider the theory of the medial axis (e.g., [38], [39], [40], [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like patte... |

184 |
Interpolations by bounded analytic functions and the corona problem.
- Carleson
- 1962
(Show Context)
Citation Context ...olic size. See Figure 29. Carleson squares are named after40 CHRISTOPHER J. BISHOP Lennart Carleson who used them in his solution of the corona problem and they are now ubiquitous in function theory =-=[30]-=-, [72]. Q T(Q) Figure 29. A decomposition of a Carleson square into dyadic Whitney boxes. [72] Dyadic Carleson squares form a tree under intersection of the interiors. Each square has a unique parent ... |

178 | Voronoi diagrams.
- Aurenhammer, Klein
- 2000
(Show Context)
Citation Context ...] (indeed, in the theory of Kleinian groupsCONFORMAL MAPPING IN LINEAR TIME 21 the Voronoi cells of an orbit are called Dirichlet fundamental domains). For more about Voronoi diagrams see e.g., [5], =-=[6]-=-, [10], [67], [68], [116], [121]. Figure 16. This shows the Voronoi cells when all the edges and vertices are sites. However, the dashed edges must be removed to give the medial axis. It is a theorem ... |

169 | The Nielsen realization problem,
- Kerckhoff
- 1983
(Show Context)
Citation Context ...rphisms of the circle. Homeomorphisms of this form are called “earthquakes” (the terminology is due to Thurston) and there is an extensive literature about such maps, e.g., [71], [152], [108], [142], =-=[99]-=-, [100], [24], [126]. The boundary of our domains are moving in what is called a “holomorphic motion”. There is an extensive theory holomorphic motions e.g., [4] and its references. Complex scaling al... |

161 |
Convex hulls in hyperbolic spaces, a theorem of Sullivan, and measured pleated surfaces
- Epstein, Marden
- 1987
(Show Context)
Citation Context ... or the complement of a circular arc) and let S be its dome. Then (S, ρS) is isometric to the hyperbolic unit disk. We will denote the isometry by ι : S → D. Theorem 7 (Sullivan [139], Epstein-Marden =-=[57]-=-). Suppose Ω is a simply connected plane domain (other than than the whole plane or the complement of a circular arc). There is a K-quasiconformal map σ : Ω → S which extends continuously to the ident... |

117 |
Medial axis transformation of a planar shape",
- Lee
- 1982
(Show Context)
Citation Context ...rn recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], [80], [87], [88], [89], [96], =-=[103]-=-, [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding Voronoi diagram divides the pl... |

107 |
The boundary correspondence under quasiconformal mappings.
- Beurling, Ahlfors
- 1956
(Show Context)
Citation Context ...between the crescents, it defines a quasi-conformal map with minimal possible dilatation extending the given boundary values (e.g. Theorem 3.1 of [63] for a simple proof; strict equality actual holds =-=[12]-=-, [138]). We shall call such a map an affine-crescent map.CONFORMAL MAPPING IN LINEAR TIME 123 Suppose we are given an ɛ-Delaunay triangulation in a region Ω and a map f : Ω → Ω ′ which sends the ver... |

98 | Mathematical Theory Of Medial Axis Transform, Pacific J.Math. 181
- Choi, Choi, et al.
- 1997
(Show Context)
Citation Context ...1]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the medial axis (e.g., =-=[38]-=-, [39], [40], [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packin... |

93 | Computing minimum length paths of a given homotopy class.
- Hershberger, Snoeyink
- 1994
(Show Context)
Citation Context ... two points inside a simple polygon is super-quadratic if one takes into account the number of bits that must be maintained, even though the problem takes only linear time in infinite precision [76], =-=[85]-=-. A similar analysis for conformal mapping would be interesting and perhaps useful, but we will be content with the infinite precision assumption here to allow us to introduce the algorithm as simply ... |

83 |
A linear-time algorithm for computing the Voronoi diagram of a convex polygon
- Aggarwal, Guibas, et al.
- 1989
(Show Context)
Citation Context ...e Voronoi diagram of a monotone histogram can be computed in linear time. The argument given in [37] follows the elegant argument of Aggarwal, Guibas, Saxe and Shor for the case of convex domains. In =-=[1]-=- the four authors use duality to reduce the problem to finding the three dimensional convex22 CHRISTOPHER J. BISHOP hull of n points whose vertical projections onto the plane are the vertices of a co... |

80 | Finding the medial axis of a simple polygon in linear time
- Chin, Snoeyink, et al.
(Show Context)
Citation Context ...nded derivative. See the end of Section 4. The construction of ι in linear time depends on the fact that the medial axis of a n-gon can be computed in linear time, a result of Chin, Snoeyink and Wang =-=[37]-=-. The next idea is to decompose polygons into pieces, again following a motivation from hyperbolic geometry. A standard technique in the theory of hyperbolic manifolds is to partition the manifold int... |

72 |
Multiplication of many-digital numbers by automatic computers.
- Karatsuba, Ofman
- 1963
(Show Context)
Citation Context ...number of field operations it takes to multiply two power series of length n. The usual process of convolving the coefficients shows M(n) = O(n 2 ). A divide and conquer method of Karatsuba and Ofman =-=[98]-=- improves this to O(n α ) with α = log 3/ log 2, but the fastest known method uses the Fast Fourier Transform [45], which shows M(n) = O(n log n) (two power series of length n can be multiplied by tak... |

60 |
Konstruktive Methoden der konformen Abbildung, Springer Tracts
- Gaier
- 1964
(Show Context)
Citation Context ...e conformal maps using the fast multipole method to solve an integral equation arising from the Kerzman-Stein formula. For surveys of different numerical conformal mapping techniques see, e.g., [51], =-=[70]-=-, [84], [95], [117], [144], [148], [151]. A circle packing of a domain is a collection of disjoint (except for tangencies) disks in the domain. The Andreev-Thurston theorem say that given such a packi... |

49 |
R-trees in topology, geometry and group theory,
- Bestvina
- 1999
(Show Context)
Citation Context ...which is isomorphic to an interval in R). For the bending lamination of a finitely bent domain, this dual is a finite tree in the sense of graph theory and can be identified with the medial axis. See =-=[11]-=-, [113]. A finite lamination Γ in the disk lies in the hyperbolic convex hull of its endpoints. If it triangulates the convex hull we say it is complete. We shall assume that our bending laminations a... |

49 |
Constructive analysis, volume 279 of Grundlehren der Mathematischen Wissenschaften
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...andom walk on an ɛ-grid stopped by the oracle). This is related to other notions of the computability of conformal maps, such as constructibility in the sense of Brouwder and Errett Bishop, e.g., see =-=[20]-=-, [35], [86], [158].s100 CHRISTOPHER J. BISHOP Möbius transformations are the only 1-1, onto holomorphic maps of the Riemann sphere to itself. In the complex plane we write these maps as z → (az + b)/... |

46 | Accurate computation of the medial axis of a polyhedron,
- Culver, Keyser, et al.
- 1999
(Show Context)
Citation Context ...ing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], =-=[46]-=-, [65], [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites),... |

44 |
Quasiconformal homeomorphisms and the convex hull boundary
- Epstein, Marden, et al.
(Show Context)
Citation Context ...action map is 2-quasiconformal. The nearest retraction map R is C-Lipschitz from ρΩ to ρS for some C < ∞ (e.g., see [16]). It is easy to prove this for some C; the sharp estimate of C = 2 is given in =-=[60]-=- and earlier results are given in [27], [29], [57]. Now suppose Ω is a finitely bent domain. Then the dome S of Ω is a finite union of geodesic faces. On the interior of each face the retraction map h... |

39 |
Fast algorithms for polynomial interpolation, integration,
- Dutt, Gu, et al.
- 1996
(Show Context)
Citation Context ... n points takes O(n log 2 n), so our approximate method is faster for the cases we consider. Multipole methods can also be used to give faster approximate evaluation and interpolation algorithms. See =-=[56]-=-, [123]. These are O(n log 1) where n is the degree and ɛ is the ɛ desired accuracy. In our case, however, n ∼ log 1, so this is not faster than exact ɛ calculation.116 CHRISTOPHER J. BISHOP Appendix... |

33 | Numerical conformal mapping using cross-ratios and Delaunay triangulation
- DRISCOLL, VAVASIS
- 1998
(Show Context)
Citation Context ...l, Nick Trefethen, Jack Snoeyink and Steve Vavasis. Many thanks to them and the others who helped me reach the results described here. Two papers in particular helped motivate me to write this paper; =-=[54]-=- and [92]. The first is by Tobin Driscoll and Steve Vavasis and describes their CRDT algorithm for conformal mappings; this is very closely related to medial axes, domes and ι although they use the la... |

33 |
Sobolev embeddings for generalized ridged domains
- Evans, Harris
- 1987
(Show Context)
Citation Context ...has σ-finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include [8], [28], [55], =-=[66]-=-, [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider ... |

31 |
Evaluating polynomials at fixed sets of points.
- Aho, Steiglitz, et al.
- 1975
(Show Context)
Citation Context ...this means that a truncation ht of h satisfies |ht(z) − g ◦ f −1 (z)| = O(ɛ 2(1−β) ), on D(0, r) where r = R β . □ Exact evaluation of a degree n polynomial at m points takes O((n + m) log 2 (n + m)) =-=[3]-=- and recovering a degree n − 1 polynomial from its values at n points takes O(n log 2 n), so our approximate method is faster for the cases we consider. Multipole methods can also be used to give fast... |

31 |
The “λ-medial axis”.
- Chazal, Lieutier
- 2005
(Show Context)
Citation Context ...th algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: =-=[31]-=-, [32], [33], [36], [46], [65], [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjo... |

26 |
Numerical Methods for Coordinate Generation Based on Schwarz-Christoffel Transformation
- Davis
- 1979
(Show Context)
Citation Context ... are various iterative methods for finding the points z starting from an initial guess (often taken to be n uniformly distributed points on T), e.g., see [53], [102]. For example, the method of Davis =-=[50]-=- takes an n-tuple of points {z1, . . . , zn} on the unit circle, computes an image polygon using the SchwarzChristoffel formula with these parameters (and the known angles) and compares the side lengt... |

23 | Stability and homotopy of a subset of the medial axis
- Chazal, Lieutier
- 2004
(Show Context)
Citation Context ...orithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], =-=[32]-=-, [33], [36], [46], [65], [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint se... |

23 |
The Best of the 20th Century: Editors Name Top 10 Algorithms
- Cipra
- 2000
(Show Context)
Citation Context ...ror at most on a Whitney square Qj. Cɛ 2 diam(Qj) , We shall use the fast multipole algorithm of Rokhlin and Greengard [79]. This method was named one of the top ten algorithms of the 20th century in =-=[44]-=-. The basic idea is that we have n empty regions where want to compute a series expansion, each of which is influenced by the n regions where the data is supported. This is n 280 CHRISTOPHER J. BISHO... |

23 |
Computing the visibility polygon from a convex set and related problems.
- Ghosh, S
- 1991
(Show Context)
Citation Context ...inking two points inside a simple polygon is super-quadratic if one takes into account the number of bits that must be maintained, even though the problem takes only linear time in infinite precision =-=[76]-=-, [85]. A similar analysis for conformal mapping would be interesting and perhaps useful, but we will be content with the infinite precision assumption here to allow us to introduce the algorithm as s... |

21 |
The definitions and exceptional sets for quasiconformal mappings.
- Gehring
- 1960
(Show Context)
Citation Context ...e first two, see [2] and for a discussion of the third and a generalization to metric spaces see [83] and its references. In Euclidean space the equivalence of the three definitions is due to Gehring =-=[74]-=-, [75], [75]. A composition of a K1-quasiconformal map with a K2-quasiconformal map is (K1K2)quasiconformal. Thus the distance used in Theorem 2 satisfies the triangle inequality. A closed curve in th... |

20 |
Lectures on quasiconformal mappings. The Wadsworth & Brooks/Cole Mathematics Series
- Ahlfors
- 1987
(Show Context)
Citation Context ... admissible ρ for Γ. is called the extremal length of the path family. The reciprocal of the modulus These are important conformal invariants whose basic properties are discussed in many sources such =-=[2]-=-. A generalized quadrilateral Q is a Jordan domain in the plane with four specified boundary points x1, x2, x3, x4 (in counterclockwise order). We define the modulus of Q, MQ(x1, x2, x3, x4) (or just ... |

20 |
A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane
- Daripa
- 1992
(Show Context)
Citation Context ...p of D to itself extends continuously to the boundary. We shall discuss these boundary values in more detail below. Numerical computation of quasiconformal maps with given dilatation is considered in =-=[47]-=-, [48], [49]. A.7. Quasi-isometries: Quasiconformal maps are a generalization of biLipschitz maps, i.e., maps which satisfy 1 K ≤ |f(x) − f(y)| |x − y| ≤ K. From the metric definition it is clear that... |

19 | Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture
- Bishop
(Show Context)
Citation Context ... notation) but still seems the easiest way to verify the proof. This paper is the culmination of a series of papers which have studied hyperbolic geometry and its relation to conformal mappings [14], =-=[15]-=-, [16], [18], [17]. Along the way, many people have contributed helpful comments, advice and encouragement including Raphy Coifman, Tobin Driscoll, David Epstein, John Garnett, Peter Jones, Al Marden,... |

19 |
A fast algorithm to solve the beltrami equation with applications to quasiconformal mappings
- Daripa
- 1993
(Show Context)
Citation Context ... to itself extends continuously to the boundary. We shall discuss these boundary values in more detail below. Numerical computation of quasiconformal maps with given dilatation is considered in [47], =-=[48]-=-, [49]. A.7. Quasi-isometries: Quasiconformal maps are a generalization of biLipschitz maps, i.e., maps which satisfy 1 K ≤ |f(x) − f(y)| |x − y| ≤ K. From the metric definition it is clear that any K... |

18 |
Average bending of convex pleated planes in hyperbolic three-space
- Bridgeman
- 1998
(Show Context)
Citation Context ....34 CHRISTOPHER J. BISHOP there is an upper bound ∑ j αj ≤ B(s) which only depends on s. See [16], [57] for some variations of this idea. Estimates of B are also closely tied to results of Bridgeman =-=[25]-=-, [26] on bending of surfaces in hyperbolic spaces. Here we shall give a simple conceptual proof without an explicit estimate. Lemma 9. There is a C < ∞ so that B(s) ≤ Ce 3s . Proof. Suppose Ω is norm... |

18 |
Rough isometries and pharmonic functions with finite Dirichlet integral
- Holopainen
- 1994
(Show Context)
Citation Context ...such that 1 ρ(x, y) − B ≤ ρ(f(x), f(y)) ≤ Aρ(x, y) + B. A This says f is biLipschitz for the hyperbolic metric at large scales. A quasi-isometry is also called a rough isometry in some sources, e.g., =-=[90]-=-, [153]. We will say f is a quasi-isometry with constant ɛ if we can take A = 1 + ɛ and B = ɛ. In [60] Epstein, Marden and Markovic show that any K-quasiconformal selfmap of the disk is a quasi-isomet... |

18 |
Earthquakes are analytic
- Kerckhoff
- 1985
(Show Context)
Citation Context ...s of the circle. Homeomorphisms of this form are called “earthquakes” (the terminology is due to Thurston) and there is an extensive literature about such maps, e.g., [71], [152], [108], [142], [99], =-=[100]-=-, [24], [126]. The boundary of our domains are moving in what is called a “holomorphic motion”. There is an extensive theory holomorphic motions e.g., [4] and its references. Complex scaling allows us... |

17 | Divergence groups have the Bowen property
- Bishop
(Show Context)
Citation Context ...to the notation) but still seems the easiest way to verify the proof. This paper is the culmination of a series of papers which have studied hyperbolic geometry and its relation to conformal mappings =-=[14]-=-, [15], [16], [18], [17]. Along the way, many people have contributed helpful comments, advice and encouragement including Raphy Coifman, Tobin Driscoll, David Epstein, John Garnett, Peter Jones, Al M... |

17 | Stability and finiteness properties of medial axis and skeleton
- Chazal, Soufflet
(Show Context)
Citation Context ...s for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], =-=[33]-=-, [36], [46], [65], [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (ca... |

17 | The c∞-convergence of hexagonal disk packings to the riemann map, Acta Mathematica 180
- He, Schramm
- 1998
(Show Context)
Citation Context ...om the correct answer given a starting point arbitrarily close to it. The example is taken from [92]. small enough the mapping between the packings is an approximation to the Riemann map [125], [81], =-=[82]-=-, [135], [136], [137]. A polynomial time algorithm for computing conformal mappings is described in [133] using a polynomial time algorithm for finding circle packings, but no details are provided abo... |

16 |
From the boundary of the convex core to the conformal boundary. Geom. Dedicata 96
- Bridgeman, Canary
- 2003
(Show Context)
Citation Context ...arest retraction map R is C-Lipschitz from ρΩ to ρS for some C < ∞ (e.g., see [16]). It is easy to prove this for some C; the sharp estimate of C = 2 is given in [60] and earlier results are given in =-=[27]-=-, [29], [57]. Now suppose Ω is a finitely bent domain. Then the dome S of Ω is a finite union of geodesic faces. On the interior of each face the retraction map has a well defined inverse and the imag... |

15 |
The logarithmic spiral: a counterexample to the K = 2 conjecture
- Epstein, Markovic
(Show Context)
Citation Context ...ngle αs/t and C2, of angle α(1 − s/t). On32 CHRISTOPHER J. BISHOP Figure 25. An approximate logarithmic spiral with t = 0, .2, .4, .6, .8, 1. Logarithmic spirals were used by Epstein and Markovic in =-=[62]-=- to disprove Thurston’s K = 2 conjecture. The showed that (in a precise sense) certain spirals have too much gray. Figure 26. The domain from Figure 22 with t = 0, .2, .4, .6, .8, 1.CONFORMAL MAPPING... |

14 |
On the hausdorff dimension of some sets in euclidean space
- Erdös
- 1946
(Show Context)
Citation Context ...the distance to the sea), wildfire set (think of a fire started simultaneously along the boundary which burns inward at a constant rate). The earliest reference I am aware of is a 1945 paper of Erdös =-=[64]-=-, where he proves the medial axis (he calls it “M2”) of a planar domain has Hausdorff dimension 1. In some parts of the literature the medial axis is confused with the set of centers of maximal disks ... |

13 | A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides
- Banjai, Trefethen
(Show Context)
Citation Context ...nd v = {v0, . . . , vn} are the vertices of the target polygon. The method works in practice in many cases but is known to sometimes diverge even locally [92]. See Figure 60. Davis’ method is used in =-=[9]-=- by Banjai and Trefethen to give a O(n) method for finding the prevertices that is practical for tens of thousands of vertices (the bound, however is an average case analysis, not a uniform estimate f... |

13 | An explicit constant for Sullivan’s convex hull theorem
- Bishop
(Show Context)
Citation Context ...st of the paper. The first idea is to consider the so called “iota-map”, ι : P → T to obtain an n-tuple w = ι(v) ⊂ T which is only a bounded dQC-distance K from the true prevertices (it is known from =-=[16]-=- that we can take K ≤ 7.82). The definition of this map and the proof that it has the desired approximation properties are motivated by results from hyperbolic 3-dimensional geometry, but we can give ... |

13 |
Complex earthquakes and deformations of the unit disk
- Epstein, Marden, et al.
(Show Context)
Citation Context ...π. In order to simplify the discussion here, we simply omit this case (with the correct interpretations the results above still hold in this case; this is discussed in complete detail in Section 5 of =-=[61]-=-). Explicit estimates of the constant in the Sullivan-Epstein-Marden theorem are given elsewhere in the literature. For example, it is proven in [16] that one can take K = 7.82. The estimates K ≈ 80 a... |

13 |
Computer Vision, Descriptive Geometry, and Classical Mechanics
- Hoffmann
- 1991
(Show Context)
Citation Context ...ions to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], [80], =-=[87]-=-, [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding Voron... |

13 | On the bit complexity of minimum link paths: Superquadratic algorithms for problems solvable in linear time.
- Kahan, Snoeyink
- 1999
(Show Context)
Citation Context ...be maintained throughout the calculations to achieve the desired output accuracy. When such considerations are taken into account, a linear time algorithm need not remain linear time. For example, in =-=[97]-=- it is shown that the problem of finding the minimal number of straight segments linking two points inside a simple polygon is super-quadratic if one takes into account the number of bits that must be... |

12 |
The modulus of a plane condenser
- Bagby
- 1967
(Show Context)
Citation Context ...us for x = {x1, x2, x3, x4} ⊂ T, since ModD = 2M, MD(x) ≃ 1 log |cr(x)|, π |cr(x)| ≫ 1, π MD(x) ≃ , | log |cr(x)|| |cr(x)| ≪ 1, Another elegant connection between modulus and cross ratios is given in =-=[7]-=- where Bagby shows that conformal modulus for a ring domain is given by minimizing an integral involving logarithms of cross ratios. A.6. Quasiconformal mappings. Quasiconformal mappings are a general... |

12 |
Symmetrization of rings in space,
- Gehring
- 1961
(Show Context)
Citation Context ...t two, see [2] and for a discussion of the third and a generalization to metric spaces see [83] and its references. In Euclidean space the equivalence of the three definitions is due to Gehring [74], =-=[75]-=-, [75]. A composition of a K1-quasiconformal map with a K2-quasiconformal map is (K1K2)quasiconformal. Thus the distance used in Theorem 2 satisfies the triangle inequality. A closed curve in the plan... |

11 | Linear one-sided stability of MAT for weakly injective 3D domain
- Choi, Seidel
- 2002
(Show Context)
Citation Context ...r science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the medial axis (e.g., [38], [39], [40], =-=[41]-=-, [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generat... |

11 | The accuracy of numerical conformal mapping methods: a survey of examples and results
- DeLillo
- 1994
(Show Context)
Citation Context ...compute conformal maps using the fast multipole method to solve an integral equation arising from the Kerzman-Stein formula. For surveys of different numerical conformal mapping techniques see, e.g., =-=[51]-=-, [70], [84], [95], [117], [144], [148], [151]. A circle packing of a domain is a collection of disjoint (except for tangencies) disks in the domain. The Andreev-Thurston theorem say that given such a... |

11 |
Bounded analytic functions, volume 96 of Pure and Applied Mathematics.
- Garnett
- 1981
(Show Context)
Citation Context ...ize. See Figure 29. Carleson squares are named after40 CHRISTOPHER J. BISHOP Lennart Carleson who used them in his solution of the corona problem and they are now ubiquitous in function theory [30], =-=[72]-=-. Q T(Q) Figure 29. A decomposition of a Carleson square into dyadic Whitney boxes. [72] Dyadic Carleson squares form a tree under intersection of the interiors. Each square has a unique parent and tw... |

10 |
Symmetry sets and medial axes in two and three dimensions. The mathematics of surfaces
- Giblin
- 2000
(Show Context)
Citation Context ...finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include [8], [28], [55], [66], =-=[77]-=-, [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the th... |

10 |
On the Skeleton of Simple CSG Objects,"
- Dutta, Hoffman
- 1993
(Show Context)
Citation Context ...s like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], [80], [87], [88], =-=[89]-=-, [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding Voronoi diagram d... |

10 | Computation of Conformal Maps by Modified Schwarz-Christoffel Transformations, Partial fulfillment of the requirements for the degree of
- Howell
- 1990
(Show Context)
Citation Context ...refethen, Jack Snoeyink and Steve Vavasis. Many thanks to them and the others who helped me reach the results described here. Two papers in particular helped motivate me to write this paper; [54] and =-=[92]-=-. The first is by Tobin Driscoll and Steve Vavasis and describes their CRDT algorithm for conformal mappings; this is very closely related to medial axes, domes and ι although they use the language of... |

10 |
Handbook of conformal mapping with computer-aided visualization
- Ivanov, Trubetskov
- 1995
(Show Context)
Citation Context ...maps using the fast multipole method to solve an integral equation arising from the Kerzman-Stein formula. For surveys of different numerical conformal mapping techniques see, e.g., [51], [70], [84], =-=[95]-=-, [117], [144], [148], [151]. A circle packing of a domain is a collection of disjoint (except for tangencies) disks in the domain. The Andreev-Thurston theorem say that given such a packing one can f... |

10 | A linear-time randomized algorithm for the bounded Voronoi diagram of a simple polygon
- Klein, Lingas
- 1996
(Show Context)
Citation Context ...celebrated result of Chazelle [34] that one can cut interior of P into trapezoids with vertical sides in linear time (this is equivalent to triangulating the polygon in linear time). Klein and Lingas =-=[101]-=- showed how to use Chazelle’s result to cut a polygon into “pseudo-normal histograms”; Chin, Snoeyink and Wang then show how to cut these into monotone histograms. The next step is to show that the Vo... |

9 | The conformal boundary and the boundary of the convex core
- Canary
(Show Context)
Citation Context ...retraction map R is C-Lipschitz from ρΩ to ρS for some C < ∞ (e.g., see [16]). It is easy to prove this for some C; the sharp estimate of C = 2 is given in [60] and earlier results are given in [27], =-=[29]-=-, [57]. Now suppose Ω is a finitely bent domain. Then the dome S of Ω is a finite union of geodesic faces. On the interior of each face the retraction map has a well defined inverse and the images of ... |

9 | An efficient and novel numerical method for quasiconformal mappings of doubly connected domains
- Daripa, Masha
- 1998
(Show Context)
Citation Context ...self extends continuously to the boundary. We shall discuss these boundary values in more detail below. Numerical computation of quasiconformal maps with given dilatation is considered in [47], [48], =-=[49]-=-. A.7. Quasi-isometries: Quasiconformal maps are a generalization of biLipschitz maps, i.e., maps which satisfy 1 K ≤ |f(x) − f(y)| |x − y| ≤ K. From the metric definition it is clear that any K-biLip... |

9 | On the convergence of circle packings to the Riemann map
- He, Schramm
- 1996
(Show Context)
Citation Context ...rge from the correct answer given a starting point arbitrarily close to it. The example is taken from [92]. small enough the mapping between the packings is an approximation to the Riemann map [125], =-=[81]-=-, [82], [135], [136], [137]. A polynomial time algorithm for computing conformal mappings is described in [133] using a polynomial time algorithm for finding circle packings, but no details are provid... |

9 | Algorithm 785: A software package for computing Schwarz–Christoffel conformal transformation for doubly connected polygonal regions
- HU
- 1998
(Show Context)
Citation Context ... Schwarz in 1869 [127], [128]. For other references and a brief history see Section 1.2 of [53]. It is also possible to formulate it with other base domains, such as an infinite strip (see [53]). See =-=[94]-=- for a version involving doubly connected polygonal regions. There are also versions for domains other than polygons, e.g., circular arc polygons as in [93], [114]. In this case, we get a simple formu... |

9 |
Computational conformal mapping
- Kythe
- 1998
(Show Context)
Citation Context ...e formula seems circular. However, there are various iterative methods for finding the points z starting from an initial guess (often taken to be n uniformly distributed points on T), e.g., see [53], =-=[102]-=-. For example, the method of Davis [50] takes an n-tuple of points {z1, . . . , zn} on the unit circle, computes an image polygon using the SchwarzChristoffel formula with these parameters (and the kn... |

8 | Hyperbolic hausdorff distance for medial axis transform
- Choi, Seidel
(Show Context)
Citation Context ...omputer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the medial axis (e.g., [38], [39], =-=[40]-=-, [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh g... |

8 |
Automated Interrogation and Adaptive Subdivision of Shape Using Medial Axis Transform
- Gursoy, Patrikalakis
- 1991
(Show Context)
Citation Context ...plications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], =-=[80]-=-, [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding... |

8 |
Numerical conformal mapping of circular arc polygons.
- Howell
- 1993
(Show Context)
Citation Context ..., such as an infinite strip (see [53]). See [94] for a version involving doubly connected polygonal regions. There are also versions for domains other than polygons, e.g., circular arc polygons as in =-=[93]-=-, [114]. In this case, we get a simple formula for the Schwarzian derivative of the conformal map, but it involves unknown parameters with no obvious geometric interpretation. One particular case of t... |

7 |
Global theorems for symmetry sets of smooth curves and polygons
- Banchoff, Giblin
- 1987
(Show Context)
Citation Context ...ar domain always has σ-finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include =-=[8]-=-, [28], [55], [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A fe... |

7 | Stability analysis of medial axis transform under relative hausdorff distance
- Choi, Lee
- 2000
(Show Context)
Citation Context ...nce literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the medial axis (e.g., [38], [39], [40], [41], =-=[42]-=-, [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A... |

7 |
Schwarz-Christoffel mapping, volume 8
- Driscoll, Trefethen
- 2002
(Show Context)
Citation Context ...l formula gives a formula for the Riemann map of the disk onto a polygonal region Ω: if the interior angles of P are απ = {α1π, . . . , αnπ}, then f(z) = A + C ∫ z n ∏ k=1 (1 − w ) αk−1 dw. See e.g., =-=[53]-=-, [114], [145]. On the half-plane the formula is f(z) = A + C ∫ n−1 zk ∏ (w − zk) αk−1 dw. k=1 The formula was discovered independently by Christoffel in 1867 [43] and Schwarz in 1869 [127], [128]. Fo... |

7 |
Stable computation of the 2D medial axis transform
- Evans, AE, et al.
- 1998
(Show Context)
Citation Context ... and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], =-=[65]-=-, [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the c... |

7 |
An effective Riemann Mapping Theorem, Theoretical Computer Science 219
- Hertling
- 1999
(Show Context)
Citation Context ...n an ɛ-grid stopped by the oracle). This is related to other notions of the computability of conformal maps, such as constructibility in the sense of Brouwder and Errett Bishop, e.g., see [20], [35], =-=[86]-=-, [158].s100 CHRISTOPHER J. BISHOP Möbius transformations are the only 1-1, onto holomorphic maps of the Riemann sphere to itself. In the complex plane we write these maps as z → (az + b)/(cz + d). Ev... |

6 |
Earthquakes on Riemann surfaces and on measured geodesic laminations
- Bonahon
(Show Context)
Citation Context ...e circle. Homeomorphisms of this form are called “earthquakes” (the terminology is due to Thurston) and there is an extensive literature about such maps, e.g., [71], [152], [108], [142], [99], [100], =-=[24]-=-, [126]. The boundary of our domains are moving in what is called a “holomorphic motion”. There is an extensive theory holomorphic motions e.g., [4] and its references. Complex scaling allows us to co... |

6 |
Complex angle scaling
- Epstein, Marden, et al.
- 2003
(Show Context)
Citation Context ...ut also dilates by a factor of esα , i.e., it pulls points towards one vertex and away from the other. These “complex angle scalings” have been considered in detail by Epstein, Marden and Markovic in =-=[59]-=- and form the basis of one of their proofs of the Sullivan-Epstein-Marden theorem. See Figure 59 for some perturbations of a domain by complex angle scalings. If we set t = 0 and only scale by a facto... |

6 |
Skeletons and central sets
- Fremlin
- 1997
(Show Context)
Citation Context ...edial axis (he calls it “M2”) of a planar domain has Hausdorff dimension 1. In some parts of the literature the medial axis is confused with the set of centers of maximal disks in Ω, which, following =-=[69]-=-, we will call the central set of Ω. For polygons the two sets are the same, but in general they are not (e.g., the parabolic region Ω = {(x, y) : y > x2 } contains a maximal disk which is only tangen... |

6 |
Earthquake curves
- Gardiner, Hu, et al.
(Show Context)
Citation Context ...e boundary maps are homeomorphisms of the circle. Homeomorphisms of this form are called “earthquakes” (the terminology is due to Thurston) and there is an extensive literature about such maps, e.g., =-=[71]-=-, [152], [108], [142], [99], [100], [24], [126]. The boundary of our domains are moving in what is called a “holomorphic motion”. There is an extensive theory holomorphic motions e.g., [4] and its ref... |

4 | Bounds for the CRDT algorithm
- Bishop
- 2008
(Show Context)
Citation Context ...luing crescents to the root disk according to the tree structure. Moreover, one can prove that the ι : ∂Ω → ∂D still has a quasiconformal extension to the interiors with a constant independent of P , =-=[17]-=-. Since ∂Ω contains the vertices of P , we can apply ι to get points on the unit circle. This is the initial guess of the CRDT algorithm (although Driscoll and Vavasis describe the map differently). S... |

4 |
NageL Shape descriptors using weighted symmetric axis features. Pattern Recognition
- Blum, N
- 1978
(Show Context)
Citation Context ...nclude [8], [28], [55], [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], =-=[23]-=-. A few papers consider the theory of the medial axis (e.g., [38], [39], [40], [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern rec... |

4 |
Sul problema della tempurature stazonaire e la rappresetazione di una data superficie
- Christoffle
(Show Context)
Citation Context ... n ∏ k=1 (1 − w ) αk−1 dw. See e.g., [53], [114], [145]. On the half-plane the formula is f(z) = A + C ∫ n−1 zk ∏ (w − zk) αk−1 dw. k=1 The formula was discovered independently by Christoffel in 1867 =-=[43]-=- and Schwarz in 1869 [127], [128]. For other references and a brief history see Section 1.2 of [53]. It is also possible to formulate it with other base domains, such as an infinite strip (see [53]). ... |

4 |
Applied and computational complex analysis. Vol. 3, Pure and Applied Mathematics (New York
- Henrici
- 1986
(Show Context)
Citation Context ...ormal maps using the fast multipole method to solve an integral equation arising from the Kerzman-Stein formula. For surveys of different numerical conformal mapping techniques see, e.g., [51], [70], =-=[84]-=-, [95], [117], [144], [148], [151]. A circle packing of a domain is a collection of disjoint (except for tangencies) disks in the domain. The Andreev-Thurston theorem say that given such a packing one... |

4 |
Inspection and feature extraction of marine propellers
- Jinkerson, Abrams, et al.
- 1993
(Show Context)
Citation Context ... pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], [80], [87], [88], [89], =-=[96]-=-, [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding Voronoi diagram divides... |

3 |
Rigid Motion in
- Bel, Martin, et al.
(Show Context)
Citation Context ...aps, e.g., [71], [152], [108], [142], [99], [100], [24], [126]. The boundary of our domains are moving in what is called a “holomorphic motion”. There is an extensive theory holomorphic motions e.g., =-=[4]-=- and its references. Complex scaling allows us to connect the disk to a domain by many different paths (corresponding to different paths in the plane connecting 0 and 1). Is is ever advantageous to no... |

3 | A central set of dimension 2
- Bishop, Hakobyan
(Show Context)
Citation Context ...al disk which is only tangent at the origin). More dramatically, the medial axis of a planar domain always has σ-finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, =-=[19]-=-. Other papers in the mathematical literature which deal with the medial axis include [8], [28], [55], [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis i... |

3 |
A constructive Riemann mapping theorem
- Cheng
- 1973
(Show Context)
Citation Context ...walk on an ɛ-grid stopped by the oracle). This is related to other notions of the computability of conformal maps, such as constructibility in the sense of Brouwder and Errett Bishop, e.g., see [20], =-=[35]-=-, [86], [158].s100 CHRISTOPHER J. BISHOP Möbius transformations are the only 1-1, onto holomorphic maps of the Riemann sphere to itself. In the complex plane we write these maps as z → (az + b)/(cz + ... |

3 |
O'Shea "The Bitangent Sphere Problem
- Giblin, B
- 1990
(Show Context)
Citation Context ... 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include [8], [28], [55], [66], [77], =-=[78]-=-, [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory o... |

2 | A fast mapping theorem for polygons
- Bishop
- 2007
(Show Context)
Citation Context ...Ω → ∂D. The initial approximation by a union of disks is unnecessary, but convenient for various reasons (the ι map for a polygon can be computed directly, using the medial axis of the polygon. e.g., =-=[18]-=-. Besides being a good rough approximation to the Riemann map, the ι map has other desirable properties: it has a simple geometric definition, it is natural withCONFORMAL MAPPING IN LINEAR TIME 3 Fig... |

2 |
The medial axis transform for 2d regions
- Chiang, Hoffmann
- 1982
(Show Context)
Citation Context ...computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], =-=[36]-=-, [46], [65], [73], [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called s... |

2 |
Medial axis tranform distance and its applications 2000
- Choi, Han, et al.
- 2000
(Show Context)
Citation Context ... the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers consider the theory of the medial axis (e.g., [38], =-=[39]-=-, [40], [41], [42], [132], [154]), but most deal with algorithms for computing it and with applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and ... |

1 |
On computational complexity of Riemann mapping. 2005. preprint
- Binder, Braverman, et al.
(Show Context)
Citation Context ...nformal maps via circle packings is available from Ken Stephenson [134]. An alternate approach to the computational complexity of conformal mapping is considered by Binder, Braverman and Yampolsky in =-=[13]-=-. They consider domains with complicated boundaries (such a fractals) and assume an oracle is given that will decide whether a given point is within ɛ of the boundary. The complexity of a domain is de... |

1 |
Average curvature of convex curves in H 2
- Bridgeman
- 1998
(Show Context)
Citation Context ...HRISTOPHER J. BISHOP there is an upper bound ∑ j αj ≤ B(s) which only depends on s. See [16], [57] for some variations of this idea. Estimates of B are also closely tied to results of Bridgeman [25], =-=[26]-=- on bending of surfaces in hyperbolic spaces. Here we shall give a simple conceptual proof without an explicit estimate. Lemma 9. There is a C < ∞ so that B(s) ≤ Ce 3s . Proof. Suppose Ω is normalized... |

1 |
Über die reducktion der positiven quadratischen formen mit drei unbestimmten ganzen zahlen
- Dirichlet
(Show Context)
Citation Context ...ee Figure 16. Thus the medial axis can be computed by using algorithms for computing generalized Voronoi diagrams. Voronoi diagrams were defined by Voronoi in [149], but go back at least to Dirichlet =-=[52]-=- (indeed, in the theory of Kleinian groupsCONFORMAL MAPPING IN LINEAR TIME 21 the Voronoi cells of an orbit are called Dirichlet fundamental domains). For more about Voronoi diagrams see e.g., [5], [... |

1 |
The existence of bitangent spheres
- Duan, Rees
- 1989
(Show Context)
Citation Context ...lways has σ-finite 1-dimensional measure [69], but the central set can have Hausdorff dimension 2, [19]. Other papers in the mathematical literature which deal with the medial axis include [8], [28], =-=[55]-=-, [66], [77], [78], [91], [109], [110], [141]. In the computer science literature the medial axis is credited to Blum who introduced it to describe biological shapes [21], [22], [23]. A few papers con... |

1 |
Convex regions in c and their domes
- Epstein, Marden, et al.
(Show Context)
Citation Context ...rom Ω to its dome with the same boundary values by Theorem 46. This implies Theorem 7, e.g., see [14]. Moreover, R is quasiconformal in some special cases; e.g., Epstein, Marden and Markovic prove in =-=[58]-=- that for Euclidean convex domains the retraction map is 2-quasiconformal. The nearest retraction map R is C-Lipschitz from ρΩ to ρS for some C < ∞ (e.g., see [16]). It is easy to prove this for some ... |

1 |
Squelettisation et anamorphose dans l’étude de la dynamique des déformations des structures: application à l’analyse de la motricité gastrique
- Gaudeau, Boiron, et al.
- 1979
(Show Context)
Citation Context ...ith applications to areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], =-=[73]-=-, [80], [87], [88], [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresp... |

1 |
Geometric Approaches to Mesh Generation, volume 75
- Hoffmann
- 1995
(Show Context)
Citation Context ...o areas like pattern recognition, robotic motion, control of cutting tools, sphere packing and mesh generation. A sample of such papers includes: [31], [32], [33], [36], [46], [65], [73], [80], [87], =-=[88]-=-, [89], [96], [103], [104], [105], [106], [107], [118], [119], [120], [124], [130], [131], [150], [155], [157]. Given a finite collection of disjoint sets (called sites), the corresponding Voronoi dia... |