59.17 An Elementary Theorem for Simple Polygons

1975 ◽  
Vol 59 (410) ◽  
pp. 266 ◽  
Author(s):  
A. S. Jones
Keyword(s):  
Simple Polygons ◽  
Elementary Theorem

scholarly journals II.—General Theorems on Determinants

1880 ◽  
Vol 29 (1) ◽  
pp. 47-54
Author(s):  
Thomas Muir
Keyword(s):  
The Other ◽  
Elementary Theorem ◽  
A Minor

The rows of a determinant of the nth order having been separated into two sets, one containing the first p rows and the other the rest, if each minor of the pth degree formed from the first set be multiplied by a minor, called its complementary, formed from the second set, and the result have its sign chosen in accordance with a certain law, it is well known as an elementary theorem that the aggregate of the products thus obtained is equal to the original determinant.

An Efficient Algorithm to Determine the Intersection of Simple Polygons for Robot Path Planning

1996 ◽  
Author(s):  
S. Miao ◽  
D. Howard
Keyword(s):  
Path Planning ◽  
Efficient Algorithm ◽  
Robot Path Planning ◽  
Simple Polygons ◽  
Clockwise Direction ◽  
Robot Path

Abstract This paper presents an efficient algorithm for determining the intersection of two simple polygons. The proposed algorithm is based on the idea of searching for the vertices of the intersection polygon vertex by vertex along the boundary in a clockwise direction. This method finds the intersection polygon vertices and their order in one pass. The algorithm almost eliminates the need for testing whether candidate vertices are inside both polygons and the sorting stage is no longer needed.

Morphing Simple Polygons

2000 ◽  
Vol 24 (1) ◽  
pp. 1-34 ◽  
Author(s):  
L. Guibas ◽  
J. Hershberger ◽  
S. Suri
Keyword(s):  
Simple Polygons

scholarly journals Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

2017 ◽  
Vol 27 (01n02) ◽  
pp. 121-158 ◽  
Author(s):  
Martin Nöllenburg ◽  
Roman Prutkin ◽  
Ignaz Rutter
Keyword(s):  
Routing Protocol ◽  
Graph Drawing ◽  
Geographic Routing ◽  
Np Hard ◽  
Line Drawings ◽  
Simple Polygons ◽  
Straight Line ◽  
Starting Point ◽  
Minimum Decomposition ◽  
Geographic Routing Protocol

A greedily routable region (GRR) is a closed subset of [Formula: see text], in which any destination point can be reached from any starting point by always moving in the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygonal regions with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles and even for trees, but can be solved optimally for trees in polynomial time, if we allow only certain types of GRR contacts. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

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