A closed chain of *n* line segments (*p _{i}*,

2 Polygons

A polygon
*P* with no two non-consecutive edges intersecting. There is a
well-defined bounded interior and unbounded exterior for a simple polygon,
where the interior is surrounded by edges. When referring to
*P*,
the convention is to include the interior of
*P*.

A polygon *P* is convex if and only if for any
pair of points *x*, *y* in *P* the line segment between
*x* and *y* lies entirely in *P*.

A good sub-polygon of a simple polygon *P*, denoted by *GSP*,
is a sub-polygon whose
boundary differs from that of *P* by at most one edge. This edge,
if it exists, is called the cutting edge.

A simple polygon containing precisely two ears and one mouth.

The convex hull *CH*(*P*) of a polygon
*P* is the
smallest convex polygon that contains *P*.

A vertex *p _{i}* is concave if a left turn is made at

A vertex *p _{i}* is convex if a right turn is made at

A vertex *p _{i}* of simple polygon

The one edge of a good sub-polygon *GSP*
that is not in simple polygon
*P* (where *GSP* is a good sub-polygon of *P*).

A line segment lying entirely inside polygon *P* and
joining two non-consecutive vertices *p _{i}* and

Given a triangulated simple polygon, the dual-tree is the graph generated by plotting a vertex at each triangle and edges joining vertices in adjacent triangles (triangles which share a diagonal).

A principal vertex *p _{i}* of a simple polygon

A proper ear of a good sub-polygon *GSP* is an ear of *GSP* which is not an end-point of the cutting edge of *P*.

A vertex in a graph with only 1 edge incident to it.

A principal vertex *p _{i}* of a simple polygon

A triangulation of a simple polygon consists of *n*-3
non-intersecting diagonals or
*n*-2 triangles where *n* is the number of vertices in
the simple polygon.

Given a convex polygons *P*, a line of support *l* is a line
intersecting *P* and such that the interior of *P* lies to one side of
*l*.

This concept is comparable to that of a tangent line.

If two points *p* and *q* (belonging to *P*) admit parallel
lines of support, then they form an anti-podal pair.

Two distinct parallel lines of support always determine at least one anti-podal
pair. Depending on how the lines intersect the polygon, three cases arise:

- Vertex-vertex anti-podal pair
- Vertex-edge anti-podal pair
- Edge-edge anti-podal pair

**Case 1**
Occurs when the lines of support intersect the polygon at two
vertices only, as illustrated. The vertices shown as black dots form an anti-podal
pair.

**Case 2**
Occurs when one line of support intersects the polygon at an
edge while the other line of support is tangent at a vertex only. Note that the
existence of such lines of support automatically implies the existence of two
distinct vertex-vertex anti-podal pairs.

**Case 3**
Occurs only when the lines of support intersect the polygon at
parallel edges. In this case, the lines of support also determine four distinct
vertex-vertex anti-podal pairs.

Except for convex polygons, every simple polygon has at least one mouth.

Except for triangles every simple polygon has at least two non-overlapping ears.

A simple closed curve *C* in the plane divides the plane into exactly two
domains, an inside and an outside.