136 research outputs found
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Approximation of conformal mappings using conformally equivalent triangular lattices
Consider discrete conformal maps defined on the basis of two conformally
equivalent triangle meshes, that is edge lengths are related by scale factors
associated to the vertices. Given a smooth conformal map , we show that it
can be approximated by such discrete conformal maps . In
particular, let be an infinite regular triangulation of the plane with
congruent triangles and only acute angles (i.e.\ ). We scale this
tiling by and approximate a compact subset of the domain of
with a portion of it. For small enough we prove that there exists a
conformally equivalent triangle mesh whose scale factors are given by
on the boundary. Furthermore we show that the corresponding discrete
conformal maps converge to uniformly in with error of
order .Comment: 14 pages, 3 figures; v2 typos corrected, revised introduction, some
proofs extende
Computing Teichm\"{u}ller Maps between Polygons
By the Riemann-mapping theorem, one can bijectively map the interior of an
-gon to that of another -gon conformally. However, (the boundary
extension of) this mapping need not necessarily map the vertices of to
those . 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 . 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 of the dilatation of the extremal map, our
method uses 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
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