195 research outputs found
A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions
Chazelle [FOCS\u2789] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same
Ear-clipping Based Algorithms of Generating High-quality Polygon Triangulation
A basic and an improved ear clipping based algorithm for triangulating simple
polygons and polygons with holes are presented. In the basic version, the ear
with smallest interior angle is always selected to be cut in order to create
fewer sliver triangles. To reduce sliver triangles in further, a bound of angle
is set to determine whether a newly formed triangle has sharp angles, and edge
swapping is accepted when the triangle is sharp. To apply the two algorithms on
polygons with holes, "Bridge" edges are created to transform a polygon with
holes to a degenerate polygon which can be triangulated by the two algorithms.
Applications show that the basic algorithm can avoid creating sliver triangles
and obtain better triangulations than the traditional ear clipping algorithm,
and the improved algorithm can in further reduce sliver triangles effectively.
Both of the algorithms run in O(n2) time and O(n) space.Comment: Proceedings of the 2012 International Conference on Information
Technology and Software Engineering Lecture Notes in Electrical Engineering
Volume 212, 2013, pp 979-98
Shortest Path in a Polygon using Sublinear Space
\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}
\newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due
to Tetsuo Asano, showing how to compute the shortest path in a polygon, given
in a read only memory, using sublinear space and subquadratic time.
Specifically, given a simple polygon \Polygon with vertices in a read
only memory, and additional working memory of size \Space, the new algorithm
computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected
time. This requires several new tools, which we believe to be of independent
interest
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Deterministic Linear Time Constrained Triangulation using Simplified Earcut
Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to: (1) deleting all the triangles it intersects; (2) filling the so generated hole with two polygons that have the wanted segment as shared edge; (3) triangulate each polygon separately. In this paper we prove that these polygons are such that all their convex vertices but two can be used to form triangles in an earcut fashion, without the need to check whether other polygon points are located within each ear. The fact that any simple polygon contains at least three convex vertices guarantees the existence of a valid ear to cut, ensuring convergence. Not only this translates to an optimal deterministic linear time triangulation algorithm, but such algorithm is also trivial to implement. We formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art
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