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Polygon decomposition is applied in several areas: [1] Pattern recognition techniques extract information from an object in order to describe, identify or classify it. An established strategy for recognising a general polygonal object is to decompose it into simpler components, then identify the components and their interrelationships and use this information to determine the shape of the object.
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) P into a set of triangles, [1] i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is P. Triangulations may be viewed as special cases of planar straight-line graphs.
One can recursively decompose the given polygon into triangles, allowing some triangles of the subdivision to have area larger than 1/2. Both the area and the counts of points used in Pick's formula add together in the same way as each other, so the truth of Pick's formula for general polygons follows from its truth for triangles.
Of particular interest to rectilinear polygons are problems of decomposing a given rectilinear polygon to simple units - usually rectangles or squares. There are several types of decomposition problems: In covering problems, the goal is to find a smallest set of units (squares or rectangles) whose union is equal to the polygon. The units may ...
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles—polygons that, like triangles, have exactly three convex vertices. As in point set triangulations ...
It was generalized to report all ways to decompose a simple polygon into two monotone chains (possibly monotone in different directions.) [3] Point in polygon queries with respect to a monotone polygon may be answered in logarithmic time after linear time preprocessing (to find the leftmost and rightmost vertices). [1]
The area of a regular decagon of side length a is given by: [3] = = + In terms of the apothem r (see also inscribed figure), the area is: = = In terms of the circumradius R, the area is:
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