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Concept
Shoelace formula
Summary
The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces. It has applications in surveying and forestry, among other areas.
The formula was described by Albrecht Ludwig Friedrich Meister (1724–1788) in 1769 and is based on the trapezoid formula which was described by Carl Friedrich Gauss and C.G.J. Jacobi. The triangle form of the area formula can be considered to be a special case of Green's theorem.
The area formula can also be applied to self-overlapping polygons since the meaning of area is still clear even though self-overlapping polygons are not generally simple. Furthermore, a self-overlapping polygon can have multiple "interpretations" but the Shoelace formula can be used to show that the polygon's area is the same regardless of the interpretation.
Given: A planar simple polygon with a positively oriented (counter clock wise) sequence of points in a Cartesian coordinate system.
For the simplicity of the formulas below it is convenient to set .
The formulas:
The area of the given polygon can be expressed by a variety of formulas, which are connected by simple operations (see below):
If the polygon is negatively oriented, then the result of the formulas is negative. In any case is the sought area of the polygon.
The trapezoid formula sums up a sequence of oriented areas of trapezoids with as one of its four edges (see below):
The triangle formula sums up the oriented areas of triangles :
The determinant formulas are the base of the popular shoelace formula, which is a scheme, that optimizes the calculation of the sum of the 2×2-Determinants by hand:
A particularly concise statement of the formula can be given in terms of the exterior algebra.
Official source
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