1,713 research outputs found
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
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
Maximal Area Triangles in a Convex Polygon
The widely known linear time algorithm for computing the maximum area
triangle in a convex polygon was found incorrect recently by Keikha et.
al.(arXiv:1705.11035). We present an alternative algorithm in this paper.
Comparing to the only previously known correct solution, ours is much simpler
and more efficient. More importantly, our new approach is powerful in solving
related problems
- …