Computing Teichm\"{u}ller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.Comment: 28 pages, 6 figure

The approximate conformal mapping of a disk onto domain with an acute angle

The method of boundary curve reparametrization is applied to construction of the approximate analytical conformal mapping of the unit disk onto an arbitrary given finite domain with a boundary smooth at every point but fininte number of acute angle points. The method is based on both the Fredholm equation solution and spline-interpolation. This approach consists of approximate solution of a linear system with unknown Fourier coefficients and construction of correction splines. The approximate mapping function has the form of a Cauchy integral. The method presentation is supported by demonstration of some examples. This method is applicable to the case of multiply connected domains with boundary angle points.Comment: Accepted for publication in IJACM 11.01.2

Riemannin kuvauslause

Tutkielmassa todistetaan Riemannin kuvauslause, joka on yksi funktioteorian perustuloksista. Sen mukaan jokainen yhdesti yhtenäinen alue D voidaan kuvata yksikkökiekoksi konformikuvauksella, kunhan D ei ole koko kompleksitaso. Esitettävä todistus edustaa niin sanottua Koebe–Montel-lähestymistapaa, jossa haluttu konformikuvaus saadaan erään funktiojonon raja-arvona. Tutkielmaa varten kehitettiin uudenlaisia havainnekuvia, ja kuvien tuottamiseen käytetty ohjelmakoodi on asetettu julkisesti saataville. Oheistuotteena syntyi Youtuben ensimmäinen visualisaatiovideo Riemannin kuvauslauseen geometrisesta tulkinnasta. Tutkielman päälähteenä on Bruce Palkan ”An Introduction to Complex Function Theory”