Polygon triangulation is one of the oldest problems in geometry, whose study led to the discovery of the Catalan numbers [161, 183]. In general, polygon triangulation is not unique, as there are multiple ways to perform it. This makes the task of decomposing a polygon into triangles particularly challenging, especially for concave polygons. However, when considering only convex polygons—where all diagonals lie entirely within the polygon—the number of possible triangulations becomes limited and depends solely on the number of vertices, not the shape of the polygon. Polygon triangulation has wide applications in various fields, including terrain modeling, point location in space, visibility problems in art galleries, robotics, and mesh generation [57]. For this reason, triangulation is regarded as one of the fundamental subroutines in many geometric algorithms. For example, algorithms that fill the interior of a polygon often begin by triangulating it and then progressively color the resulting individual triangles from this decomposition. In the remainder of this chapter, we consider simple polygons defined by a finite set of line segments forming a simple closed path \( \partial {\mathcal{P}} \) . It will be shown that any such polygon \({\mathcal{P}}\) with n vertices can be triangulated into exactly \({\text{n}} - {2}\) triangles using \({\text{n}} - {3}\) diagonals [50].