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The subdivision of the polygon into triangles forms a planar graph, and Euler's formula + = gives an equation that applies to the number of vertices, edges, and faces of any planar graph. The vertices are just the grid points of the polygon; there are = + of them. The faces are the triangles of the subdivision, and the single region of the ...
Given a rectilinear polygon (whose sides meet at right angles) R in the plane, a set S of points in R, and a set of identical squares, the goal is to find the largest number of non-overlapping squares that can be packed in points of S. Suppose that, for each point p in S, we put a square centered at p. Let G S be the intersection graph of these ...
In geometry, the Wallace–Bolyai–Gerwien theorem, [1] named after William Wallace, Farkas Bolyai and P. Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations.
An ear of a polygon is defined as a triangle formed by three consecutive vertices ,, of the polygon, such that its edge lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. [1]
For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set of size k, and is given by
The convex hull of a simple polygon is divided by the polygon into pieces, one of which is the polygon itself and the rest are pockets bounded by a piece of the polygon boundary and a single hull edge. Although many algorithms have been published for the problem of constructing the convex hull of a simple polygon, nearly half of them are ...
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]
In 2019 this was strengthened by Amit Wolecki who showed that for each such polygon, the number of pairs of points which do not illuminate each other is finite. [ 6 ] The first polygonal Tokarsky Unilluminable room with 4 sides, 1995.