SESSION NOVEL ALGORIHMS AND APPLICATIONS + ...

42 Int'l Conf. Foundations of Computer Science | FCS'11 |
Minimum Pseudo-Triangulation Using Convex Hull
Layers
F. Taherkhani 1 , A. Nourollah 1,2
1 Department of Computer Engineering & IT, Islamic Azad University, Qazvin, Iran
2 Department of Electrical & Computer Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran
Abstract - Pseudo-triangulation is regarded as one of the
most commonly used problems in computational geometry. In
this paper we consider the problem of minimum pseudotriangulation
of a given set of points S in the plane using
convex hull layers and we propose two new methods that will
lead to the production of minimum pseudo-triangulation. This
means that the number of pseudo-triangles created in
minimum pseudo-triangulation is exactly n-2 pseudo-triangles
and the minimum number of edges needed is 2n-3.
Keywords: pseudo-triangulation, reflex chain, convex hull
layers, visibility
1 Introduction
The names pseudo-triangle and pseudo-triangulation
were coined by Pocchiola and Vegter in 1993. For polygons,
pseudo-triangulations has been already expressed in the
computational geometry’s literature in the early 1990’s, under
the name of geodesic triangulations [1]. The geodesic path
between two points of a polygon is the shortest path from one
to the other in polygon. Pseudo-triangulations of a simple
polygon are also called geodesic triangulations, because they
arise by inserting non-crossing geodesic paths in polygon.
A pseudo-triangle is a planar polygon with exactly three
convex vertices, called corners and three reflex chains of
edges join the corners. Let S be a set of n points in general
position in the plane. A pseudo-triangulation for S is a
partition of the convex hull of S into pseudo-triangles whose
vertex set is S [2].
In 2000, Streinu [3] has shown that there are strong links
between minimally rigid graphs and minimum pseudotriangulations.
In addition, she proved that the minimum
number of edges needed to obtain a pseudo-triangulation is
2n-3 and thus, by Euler's polyhedron theorem, the number of
pseudo-triangles in a minimum pseudo-triangulation is n-2,
which does not depend on the structure of the point set but
only on its size [4]. Every vertex of a minimum pseudotriangulation
is pointed. A vertex is pointed if it has an
incident angle greater than .
Pseudo-triangulations are received considerable attention in
computational geometry. This is mainly due to their
applications in rigidity theory, robot arm motion planning,
visibility, ray-shooting, kinetic collision detection and
guarding polygons [5-8].
With respect to the fact that some of the interesting geometric
and combinatorial properties of pseudo-triangulations have
been recently discovered, but many main open questions still
remain [2]. In this paper we consider the problem of minimum
pseudo-triangulation of a set S of n points in the plane and we
show that the generation of convex hull layers for set points
and their pseudo-triangulation, using two new methods,
minimizes pseudo-triangulation.
The rest of this paper has been organized as follows: In
section 2 some basic definitions are presented. Section 3
describes how to create convex hull layers. In section 4
determine for all vertices in convex hull layers visible
vertices and finally in section 5 we propose two new methods
of pseudo-triangulation on created convex hull layers to attain
minimum pseudo-triangulation.
2 Initial definitions
A simple polygon is called a convex polygon when all the
internal angles are less than . According to this definition,
the set of points S on a plane is called convex if and only if in
exchange for both the points p,q∈ S, the line segment pq
completely lies inside S (pq ⊆ S).
The most applicable structure in robatic geometry is convex
hull. Convex hull of the given points p0,…, pn-1 is the smallest
convex set on the plane which contains the points.
Let three points p1(x1, y1), p2(x2, y2) and p3(x3, y3) are given in
the plane. Hence matrix A is defined as follows:
(1)