Geometry: Polygons: Defining a Polygon

Curves

When points or lines are arranged in some formation, rarely does it result in a recognizable geometric figure. Well-known shapes like squares and triangles are actually only subsets of larger groups of geometric figures and other collections of points in space.

One of the easiest and most common collections of points in space is a curve. A curve can be any continuous arrangement of points, straight or curved, in space. A curve can be defined as the trace of the motion of a point in space. So a curve is like a path through space by which a point could travel. For our purposes, we'll only consider curves that lie in a plane. A curve is continuous, meaning that there aren't any gaps or holes in the curve; any point on a curve can be reached from another point on the curve without leaving the curve. A dotted line, for example, is not a curve. Here are some examples of curves below.

A curve whose starting point is also its endpoint is called a closed curve. The reason for this is that such a curve encloses a region in the plane. A simple closed curve is an even more specific kind of curve: one that is closed, and doesn't intersect itself. The region enclosed by a simple closed curve is not divided by any part of the curve. Closed curves sometimes intersect themselves, but not simple closed curves. Below are some closed curves and simple closed curves.

Figure %: Some closed curves and simple closed curves

Polygons

A polygon is one type of simple closed curve. A polygon is the union of three or more line segments whose endpoints meet. The segments are called the sides of the polygon. The points at which the segments meet (always the endpoints of the segments) are called vertices. Segments that share a vertex are called adjacent sides. Vertices next to each other are called consecutive vertices. A segment whose endpoints are nonadjacent vertices is called a diagonal. See the picture below.

Figure %: A polygon and its characteristics

A polygon is named for its vertices, but the vertices must be listed in order. It doesn't matter which direction the order goes, as long as consecutive vertices are next to each other in the name. The first and last letters in the name, of course, are consecutive vertices, but won't be listed next to each other. For example, the polygon above could be called BCDEFA, or EDCBAF, or some other name incorporating the six vertices in order.