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Time and place: Tu-Th 9:50-11:10 Physics P-124
The course will deal with a new method for numerically approximating conformal maps in two dimensions which combines aspects of classical function theory, 3-dimensional hyperbolic geometry and computational geometry.
Given a simply polygon P, the Riemann mapping theorem implies there is a conformal map of the unit disk to the interior of the polygon. There is actually an explicit formula for this map, called the Schwarz-Christoffle formula. This formula takes two types of parameters: the angles of the polygon and the locations of the conformal prevertices (the points on the unit circle which map to the vertices of the ploygon by the conformal map). Unfortunately, the conformal prevertices are unknown until the conformal map is known so the Schwarz-Christoffel formula seems a bit circular. In practice, one takes an initial guess for the prevertices and then improves the guess by some iteration. However, so far as I know, there is no simply iteration which is guarented to converge to the correct prevertices (and some methods used in practice are known to diverge in some cases).
We will focus on a recent theorem on mine which says that given a simple polygon with n sides, one can compute apprxomimate prevertices in time O(n) which are accurate with estimates which are independent of the number of sides n and the geometry of the polygon P. ``Accuracy'' is precisely defined in terms of the best quasiconformal selfmap of the disk which maps the approximate prevertices to the true prevertices. The result depends on two other results. One is a theorem of Dennis Sullivan (with other proofs by Al Marden, David Epstein and myself) on the boundaries of convex bodies in hyperbolic 3-space. The other is a a theorem in computational geometry that an object called the medial axis of the polygon can be computed in time O(n). The medial axis is the set of point in the interior which are equidistant from at least two boundary points and is closely related to other objects in computational geometry such as Voronoi diagrams and Delaunay triangulations. The medial axis has many applications as a compact description of the shape of an object (pattern recognition, robotic motion, mesh generation), but our application to conformal mappings seems new.
In the course I will try to give several proofs of Sullivan's theorem and a discussion of the best known constant in this theorem. I will also discuss, so far as I am able, the theorem of Chin-Snoeyink-Wang that the medial axis of a simply n-gon can be computed in time O(n). I will also show how combining these gives the result on approximating the prevertices of an n-gon in time O(n). I would also like to discuss actual implementations of this and other methods for computing conformal mappings numerically and actually draw some conformal maps on a computer. Along the way we will describe numerous open problems of both theoretical and experimental flavors.
Here are a few papers on my own that we will discuss in the class. I will try to cover the contents of most of these in detail.
Divergence groups have Bowen's property
An explicit constant for Sullivan's convex hull theorem
Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture
A fast approximation for the Riemann map
Lecture tranperencies for `Convex hulls and Klienian groups postscript pdf
Lecture tranperencies for `A fast approximation to the Riemann map' postscript pdf Here are some color pictures of domes of some simple domains: example 1, example 2, example 3, example 4, example 5, example 6, example 7, example 8.
Here are some webistes of other mathematicians and computer scientists whose work we will also discuss.
Jack Snoeyink . We will be very interested in his paper with F. Chin and C. Wang Finding the medial axis of a simple polygon in linear time .
Vladimir Markovic .
He has several papers with Al Marden and David Epstein on convex hulls in
hyperbolic 3-space which are relevant to the course. Among these
Complex earthquakes and deformations of the unit disk ,
Complex angle scaling ,
The logarithmic spiral: a counterexample to the K=2 conjecture ,
Quasiconformal homeomorphisms and the convex hull boundary .
Toby Driscoll has a program ``SCToolbox'' which runs with MATLAB to compute the conformal mapping on to a simple polygon. The polygon can be specified by either a list of vertices or by clicking in a graphical interface to ``draw'' the polygon by hand. The package can be downloaded from SC-toolbox webpage .
My method for approximating conformal maps is closely related to a method of Toby Driscoll and Steve Vavasis . Their algorithim is called the CRDT algorithm (Cross Ratios of Delaunay Triangulations) and is described in their paper Numerical conformal mapping using cross-ratios and Delaunay triangulations (this is a pdf file from Driscoll's website)..
David Eppstein is a computer scientist at UC Irvine. His webpage contains many interesting papers on computational geometry and he has a webpage giving various applications including the medial axis .
Here are some other webpages and papers dealing with the medial axis.
Mathematical theory of the medial axis transform by
Choi, Choi and Moon
Hyperbolic Hausdorff distance for the medial axis transform by S.W. Choi and H.-P. Seidel
Shape simplification base3d on the medial axis transformation by R. Tam and W. Heidrich
Stability analysis of the medial axis transform by S.W. Choi and S.w. Lee
This is an essay by Lloyd Trefethen on the definition of numerical analysis
He is also the author (with T. Driscoll) of a monograph on the Schwarz-Christoffel mapping . He also has a recent paper on applying Schwarz-Christoffel to polygons with thousands of sides using the fast Multipole method
This is about a modified version of the medial axis transform called the chordal axis transform .