 ## Definitions ### Polygon

A closed chain of n line segments (pi, pi+1) for 0 <= i <= n-1 and (pn-1, p0) , where n >= 3. The polygon P is represented by its vertices P = (p0, p1,..., pn-1). 2 Polygons

### Simple Polygon

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. ### Convex Polygon

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. ### Good Sub-Polygon

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. ### Anthropomorphic Polygon

A simple polygon containing precisely two ears and one mouth. ### Convex Hull

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

A vertex pi is concave if a left turn is made at pi while going from pi-1 to pi+1 where the interior of the simple polygon P is to the right (ie. P is being traversed in clockwise order). ### Convex Vertex

A vertex pi is convex if a right turn is made at pi while going from pi-1 to pi+1 where the interior of the simple polygon P is to the right (ie. P is being traversed in clockwise order). ### Principal Vertex

A vertex pi of simple polygon P is called a principal vertex if the diagonal (pi-1, pi+1) intersects the boundary of P only at pi-1 and pi+1. ### Cutting Edge

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). ### Diagonal

A line segment lying entirely inside polygon P and joining two non-consecutive vertices pi and pj ### Dual-Tree

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). ### Ear

A principal vertex pi of a simple polygon P is called an ear if the diagonal (pi-1, pi+1) that bridges pi lies entirely in P. We say that two ears pi and pj are non-overlapping if the interior of triangle (pi-1, pi, pi+1) does not intersect the interior of triangle (pj-1, pj, pj+1). ### Proper Ear

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. # Leaf

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

A principal vertex pi of a simple polygon P is called a mouth if the diagonal (pi-1, pi+1) is an external diagonal, i.e., the interior of (pi-1, pi+1) lies in the exterior of P. ### Triangulation

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. ### Lines of Support

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.

### Anti-Podal pairs

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:

1. Vertex-vertex anti-podal pair
2. Vertex-edge anti-podal pair
3. 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. ### One-Mouth Theorem

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

### Two-Ears Theorem

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

### Jordan Curve Theorem

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