Difference between revisions of "Polygon"

Revision as of 21:28, 24 April 2008

A polygon is a closed planar figure consisting of straight line segments.

In their most general form, polygons are an ordered set of vertices, $\{A_1, A_2, \ldots, A_n\}$ , $n \geq 3$ , with edges $\{\overline{A_1A_2}, \overline{A_2A_3}, \ldots, \overline{A_nA_1}\}$ joining consecutive vertices. Most frequently, one deals with simple polygons in which no two edges are allowed to intersect. (In fact, the adjective "simple" is almost always omitted, so the term "polygon" should be understood to mean "simple polygon" unless otherwise noted.)

A degenerate polygon is one in which some vertex lies on an edge joining two other vertices. This can happen in one of two ways: either the vertices $A_{i - 1},A_i$ and $A_{i+1}$ can be colinear or the vertices $A_i$ and $A_{i + 1}$ can overlap (fail to be distinct). In either of these cases, our polygon of $n$ vertices will appear to have $n - 1$ or fewer -- it will have "degenerated" from an $n$ -gon to an $(n - 1)$ -gon. (In the case of triangles, this will result in either a line segment or a point.)