In this thesis, the notions of the visibility in a simple polygon are extended, the polygon hierarchy according to its visibility properties is established and it is shown that polygons having such properties can be easily triangulated.
First, after introducing the convex chain visible polygon and the reflex chain visible polygon we show that whether or not a polygon is visible from a given convex or reflex chain can be determined in linear-time, respectively. Also, linear-time algorithms for finding all convex or reflex chains, if any, from which a given polygon is visible are presented.
Second, new classes of polygons, intersection free polygons and generalized intersection free polygons are introduced and linear-time algorithms for determining whether or not a polygon is an intersection free polygon or a generalized intersection free polygon from a given point are presented. Also, it is shown that many polygon classes previously known with some visibility properties are subclasses of intersection free polygons or generalized intersection free polygons. Based on our definitions of polygon classes, the hierarchy of some polygon classes is suggested.
Finally, the problem of polygon triangulations is considered. Linear-time algorithms for triangulating convex chain visible polygons, reflex chain visible polygons, intersection free polygons and generalized intersection free polygons are given. Also, we present an improved algorithm for triangulating point visible polygons without decomposition, which is simpler to implement and easier to understand.