Polygon

Builds on Euclidean plane Collinear points

Definition

Let $n \ge 3$ be an integer. A polygon with $n$ vertices in the Euclidean plane is a cyclic sequence of points $P_1, \ldots, P_n$ , with indices taken modulo $n$ , such that:

$P_i \ne P_{i+1}$ for every $i$ .
$P_{i-1}$ , $P_i$ , $P_{i+1}$ are not collinear for every $i$ .

The points $P_i$ are its vertices and the segments $P_i P_{i+1}$ are its edges. The polygon is simple if two edges meet only when they are consecutive, and then only in their common vertex.

Remarks

By the Jordan curve theorem, a simple polygon bounds one bounded region of the plane, its interior. Many authors use the word polygon for this closed region rather than for the boundary. A polygon with $3$ vertices is a triangle, and one with $4$ vertices is a quadrilateral. A simple polygon is convex if its interior is a convex set.

Example: $(0, 0), (1, 0), (1, 1), (0, 1)$ is a simple polygon, the unit square. Consecutive vertices differ, and no three consecutive ones lie on a line.

Non-example: $(0, 0), (1, 0), (2, 0), (1, 1)$ fails clause 2. The three consecutive vertices $(0, 0), (1, 0), (2, 0)$ are collinear, so the figure is a triangle with an extra point on one edge.

Reading path

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Collinear points concept
Euclidean plane object
Polygon definition

Tag	`AXS7YYF1YY`
URI	`m2://AXS7YYF1YY`
Kind	definition
Writing	W0 stub
MSC	51M04, 52B05
Defines	Polygon
Uses	Euclidean plane, Collinear points

Polygon

Definition

Remarks

Reading path

Further reading

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