2,232 research outputs found
Reconstruction of Weakly Simple Polygons from their Edges
Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most epsilon, for any epsilon > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n log n) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are directed, and the counterclockwise traversal of a polygon should follow the orientation.
We study related problems for the case that the union of the n input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT
In this paper, we develop a shape optimization-based algorithm for the
electrical impedance tomography (EIT) problem of determining a piecewise
constant conductivity on a polygonal partition from boundary measurements. The
key tool is to use a distributed shape derivative of a suitable cost functional
with respect to movements of the partition. Numerical simulations showing the
robustness and accuracy of the method are presented for simulated test cases in
two dimensions
Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study
We construct an order reconstruction (OR)-type Landau-de Gennes critical
point on a square domain of edge length , motivated by the well order
reconstruction solution numerically reported by Kralj and Majumdar. The OR
critical point is distinguished by an uniaxial cross with negative scalar order
parameter along the square diagonals. The OR critical point is defined in terms
of a saddle-type critical point of an associated scalar variational problem.
The OR-type critical point is globally stable for small and undergoes
a supercritical pitchfork bifurcation in the associated scalar variational
setting. We consider generalizations of the OR-type critical point to a regular
hexagon, accompanied by numerical estimates of stability criteria of such
critical points on both a square and a hexagon in terms of material-dependent
constants.Comment: 29 pages, 12 figure
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