NDLI: Mappings and meshes

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On topological changes in the delaunay triangulation of moving points

The point-set embeddability problem for plane graphs

Protein molecules in 3D: complex behavior from simple local geometry plus weak interactions

Higher-order Erdös: Szekeres theorems

How to walk your dog in the mountains with no magic leash

Approximating low-dimensional coverage problems

Mappings and meshes

Cubic plane graphs on a given point set

Linear-size approximations to the vietoris-rips filtration

Self-improving algorithms for coordinate-wise maxima

Tessellation analysis of the cosmic web

Stabbing horizontal segments with vertical rays

Periodic body-and-bar frameworks

Disjoint edges in complete topological graphs

Configuration space visualization

Geometric packing under non-uniform constraints

Algorithmic complexity of finding cross-cycles in flag complexes

Add isotropic Gaussian kernels at own risk: more and more resilient modes in higher dimensions

Tracing compressed curves in triangulated surfaces

Improved range searching lower bounds

Multinerves and helly numbers of acyclic families

Alexander duality for functions: the persistent behavior of land and water and shore

Finding the maximal empty disk containing a query point

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

On sub-determinants and the diameter of polyhedra

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

Surface patches from unorganized space curves

Computing cartograms with optimal complexity

Counting crossing-free structures

On the expected complexity of voronoi diagrams on terrains

Testing graph isotopies on surfaces

An output-sensitive algorithm for computing projections of resultant polytopes

Duals of orphan-free anisotropic voronoi diagrams are embedded meshes

Certified computation of planar morse-smale complexes

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

A theorem of bárány revisited and extended

Optimal area-sensitive bounds for polytope approximation

The 2-page crossing number of $K_{n}$

Lang's universal molecule algorithm

Approximate bregman near neighbors in sublinear time: beyond the triangle inequality

A fast and simple surface reconstruction algorithm

On Klee's measure problem for grounded boxes

On laplacians of random complexes

Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn's technique

Stability of Delaunay-type structures for manifolds: [extended abstract]

A deterministic o(m log m) time algorithm for the reeb graph

Approximating Tverberg points in linear time for any fixed dimension

Improved pointer machine and I/O lower bounds for simplex range reporting and related problems

Object-image correspondence for curves under central and parallel projections

String graphs and incomparability graphs

The sticky geometry of the cosmic web

Mappings and meshes

Content Provider	ACM Digital Library
Author	Bishop, Christopher J.
Abstract	In my talk I will attempt to draw some connections between complex analysis and computational geometry, particularly between conformal mappings, hyperbolic geometry, the medial axis and optimal meshing. The Riemann mapping theorem says that there is a conformal (angle preserving) map of the unit disk, DISK, to the interior Ω of any simple n-gon. How much work is needed to compute this map? [MR2671015] We can compute the conformal map f: DISK -> Ω to within error ε in time O(n ⋅ log 1/ε log log 1/ε). The proof uses an iteration that converges quadratically to f, but it needs a good initial guess that is close to the correct answer. The medial axis allows us to quickly construct a map Ω -> DISK that is close to Riemann's map with precise estimates. This gives our starting point, and the time estimates in the theorem depend on time needed to compute the medial axis (linear by work of Chin, Snoeyink and Wang). Conversely, the proof of Theorem 1 uses ideas from analysis that may be of interest in CG. For example, the dome of a planar domain Ω is the surface in $R_{+}^{3}$ given by the upper envelope of all hemispheres whose base on $R^{2}$ is a medial axis disk of Ω. Domes arise in the study of 3-manifolds, and a fundamental result of Dennis Sullivan about convex sets in hyperbolic 3-space lies at the heart of Theorem 1. The proof of Theorem 1 also introduces the thick-thin decomposition of a polygon, inspired by the thick-thin decomposition of a hyperbolic manifold. The thin parts correspond to "long, narrow" pieces of the polygon (but not in the obvious way; the precise definition uses a conformal invariant called extremal length). This decomposition plays in important role in proving: [MR2671014] Any simple n-gon has a O(n) quadrilateral mesh with all angles ≤ $120^{O}$ and all new angles ≥ $60^{O}.$ The sharp upper bound is due to Bern and Eppstein; the lower bound is the novel part here. The result is also true for PSLGs with at most n edges and vertices, although simple examples show the O(n) must be replaced by $O(n^{2}):$ [Bishop-Quad] Every PSLG has an $O(n^{2})$ quadrilateral mesh with all angles ≤ $120^{O}$ and all new angles ≥ $60^{O}.$ Only O(n/ε) angles satisfy $\|θ-90^{O}\|$ > ε, for any ε >0.
Starting Page	199
Ending Page	200
Page Count	2
File Format	PDF
ISBN	9781450312998
DOI	10.1145/2261250.2261279
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-06-17
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Hyperbolic geometry Nonobtuse triangulation Acute triangulation Quadrilateral meshes Conformal maps Delaunay triangulation Medial axis Schwarz-christoffel formula
Content Type	Text
Resource Type	Article

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Conformal Mapping in Linear Time

Conformal Mapping in Linear Time

Bounds for the CRDT Conformal Mapping Algorithm

Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation

Computation of conformal maps by modified schwarz-christoffel transformations (1990).

(j) The Schwarz Reflection Principle. (k) The Schwarz-Christoffel formula. (l) Hyperbolic geometry and Schwarz’s Lemmas. (m) The Area Theorem and distortion estimates. (n) The Hayman-Wu Theorem.

Revisiting the crowding phenomenon in Schwarz-Christoffel mapping

Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings

A new Voronoi-based surface reconstruction algorithm

Mappings and meshes

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