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Untangling Planar Curves

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the...

Recognizing Weakly Simple Polygons

if and only if f is flat foldable. Therefore, it is also NP-hard to decide whether a polyhedron is weakly simple. 1 2 3 4 5 6 7 8 Hugo A. Akitaya , Greg Aloupis, Jeff Erickson, and Csaba D. T?th ... . Marshall Bern and Barry Hayes . The complexity of flat origami . In Proc. 7th ACM-SIAM Sympos. on Discrete Algorithms , pages 175 - 183 , 1996 . Hsien-Chih Chang , Jeff Erickson , and Chao Xu . Detecting

Efficiently Hex-Meshing Things with Topology

A topological quadrilateral mesh \(Q\) of a connected surface in \(\mathbb {R}^3\) can be extended to a topological hexahedral mesh of the interior domain \(\varOmega \) if and only if \(Q\) has an even number of quadrilaterals and no odd cycle in \(Q\) bounds a surface inside \(\varOmega \). Moreover, if such a mesh exists, the required number of hexahedra is within a constant...

Tracing Compressed Curves in Triangulated Surfaces

A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. A normal curve is a finite set of disjoint normal paths and normal cycles. We describe an algorithm to “trace” a normal curve in \(O(\min \{ X, n^2\log X \})\) time, where \(n\) is the complexity of the surface...

Optimally Cutting a Surface into a Disk

]. See http://www.cs.uiuc.edu/ jeffe/pubs/schema.html for the most recent version of this paper. Jeff Erickson was partially supported by a Sloan Fellowship, NSF CAREER Award CCR-0093348, and NSF ITR Grant