Wikipedia / Shoelace formula
Wikipedia / Shoelace formula
The shoelace formula, 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.
The polygon area formulas
Given: A planar simple polygon with a positively oriented in a Cartesian coordinate system..
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 is the sought area of the polygon.
Trapezoid formula
of trapezoids
Triangle formula
:
Shoelace formula
The triangle formula is the base of the popular shoelace formula
Sometimes this determinant is transposed (written vertically, in two columns), as shown in the diagram.
Other formulas
Exterior algebra
A particularly concise statement of the formula can be given in terms of the exterior algebra be the consecutive vertices of the polygon. The Cartesian coordinate expansion of the outer product.
Example
The advantage of the shoelace form: Only 6 columns have to be written for calculating the 5 determinants with 10 columns.
Deriving the formulas
Trapezoid formula
with its oriented area
. In the first case the trapezoid is called negative in the second case positive. The negative trapezoids delete those parts of positive trapezoids, which are outside the polygon. In case of a convex
Triangle form, determinant form
above), one gets the determinant form Because one half of the i-th determinant is the oriented area of the triangle this version of the area formula is called triangle form.
Other formulae
Alternatively, this is a special case of Green's theorem with one function set to 0 and the other set to x, such that the area is the integral of xdy along the boundary.
Manipulations of a polygon
one should first shift the indices.
Hence:
.
, which can be positive or negative.
, which can be positive or negative.
Example:
With the above notation of the shoelace scheme one gets for the oriented area of the
blue
green
red
blue polygon minus
blue polygon plus
Generalization
In higher dimensions the area of a polygon can be calculated from its vertices using the exterior algebra form of the Shoelace formula (e.g. in 3d, the sum of successive cross products(when the vertices are not coplanar this computes the vector area enclosed by the loop, i.e. the projected area or "shadow" in the plane in which it is greatest).
This formulation can also be generalized to calculate the volume of an n-dimensional polytope from the coordinates of its vertices, or more accurately, from its hypersurface mesh. For example, the volume of a 3-dimensional polyhedron can be found by triangulating its surface mesh and summing the signed volumes of the tetrahedrawhere the sum is over the faces and care has to be taken to order the vertices consistently (all clockwise or anticlockwise viewed from outside the polyhedron). Alternatively, an expression in terms of the face areas and surface normals may be derived using the divergence theorem (see Polyhedron § Volume).
See also
External links
References