Difference between revisions of "Polygon"
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A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segments]]. | A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segments]]. | ||
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| + | In their most general form, polygons are an ordered [[set]] of [[vertex|vertices]], <math>\{A_1, A_2, \ldots, A_n\}</math>, <math>n \geq 3</math>, with [[edge]]s <math>\{\overline{A_1A_2}, \overline{A_2A_3}, \ldots, \overline{A_nA_1}\}</math> 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 <math>A_{i - 1},A_i</math> and <math>A_{i+1}</math> can be colinear or the vertices <math>A_i</math> and <math>A_{i + 1}</math> can overlap (fail to be distinct). In either of these cases, our polygon of <math>n</math> vertices will appear to have <math>n - 1</math> or fewer -- it will have "degenerated" from an <math>n</math>-gon to an <math>(n - 1)</math>-gon. (In the case of triangles, this will result in either a line segment or a point.) | ||
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| + | The other adjectives most commonly attached to polygons are [[convex]] and [[concave]]. | ||
== See also == | == See also == | ||
Revision as of 10:06, 29 July 2006
A polygon is a closed planar figure consisting of straight line segments.
In their most general form, polygons are an ordered set of vertices, 
, 
, with edges 
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 
and 
can be colinear or the vertices 
and 
can overlap (fail to be distinct). In either of these cases, our polygon of 
vertices will appear to have 
or fewer -- it will have "degenerated" from an -gon to an 
-gon. (In the case of triangles, this will result in either a line segment or a point.)
The other adjectives most commonly attached to polygons are convex and concave.
See also
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