1,759 research outputs found
Deconstructing Approximate Offsets
We consider the offset-deconstruction problem: Given a polygonal shape Q with
n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance,
as the Minkowski sum of another polygonal shape P with a disk of fixed radius?
If it does, we also seek a preferably simple-looking solution P; then, P's
offset constitutes an accurate, vertex-reduced, and smoothened approximation of
Q. We give an O(n log n)-time exact decision algorithm that handles any
polygonal shape, assuming the real-RAM model of computation. A variant of the
algorithm, which we have implemented using CGAL, is based on rational
arithmetic and answers the same deconstruction problem up to an uncertainty
parameter \delta; its running time additionally depends on \delta. If the input
shape is found to be approximable, this algorithm also computes an approximate
solution for the problem. It also allows us to solve parameter-optimization
problems induced by the offset-deconstruction problem. For convex shapes, the
complexity of the exact decision algorithm drops to O(n), which is also the
time required to compute a solution P with at most one more vertex than a
vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011,
submitted to DC
Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
Let T be a triangulation of a simple polygon. A flip in T is the operation of
removing one diagonal of T and adding a different one such that the resulting
graph is again a triangulation. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
Angles of Arc-Polygons and Lombardi Drawings of Cacti
We characterize the triples of interior angles that are possible in
non-self-crossing triangles with circular-arc sides, and we prove that a given
cyclic sequence of angles can be realized by a non-self-crossing polygon with
circular-arc sides whenever all angles are at most pi. As a consequence of
these results, we prove that every cactus has a planar Lombardi drawing (a
drawing with edges depicted as circular arcs, meeting at equal angles at each
vertex) for its natural embedding in which every cycle of the cactus is a face
of the drawing. However, there exist planar embeddings of cacti that do not
have planar Lombardi drawings.Comment: 12 pages, 8 figures. To be published in Proc. 33rd Canadian
Conference on Computational Geometry, 202
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