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Mathematics, Differential Geometry, math.DG,Mathematics, Complex Variables, math.CV
Abstract:
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.
