# POLYGON_INTEGRALS Arbitrary Moments of a Polygon

POLYGON_INTEGRALS is a Python library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

We suppose that POLY is a planar polygon with N vertices X, Y, listed in counterclockwise order.

For nonnegative integers P and Q, the (unnormalized) moment of order (P,Q) for POLY is defined by:

```        Nu(P,Q) = Integral ( x, y in POLY ) x^p y^q dx dy
```
In particular, Nu(0,0) is the area of POLY.

Simple formulas are available for low orders:

```        Nu(0,0) = 1/2 (1<=i<=N) X(i-1)Y(i)-X(i)Y(i-1)
Nu(1,0) = 1/6 (1<=i<=N) ( X(i-1)Y(i)-X(i)Y(i-1) ) * (X(i-1)+X(i))
Nu(0,1) = 1/6 (1<=i<=N) ( X(i-1)Y(i)-X(i)Y(i-1) ) * (Y(i-1)+Y(i))
Nu(2,0) = 1/12 (1<=i<=N) ( X(i-1)Y(i)-X(i)Y(i-1) ) * (X(i-1)^2+X(i-1)X(i)+X(i)^2)
Nu(1,1) = 1/24 (1<=i<=N) ( X(i-1)Y(i)-X(i)Y(i-1) ) * (2X(i-1)Y(i-1)+X(i-1)Y(i)+X(i)Y(i-1)+2X(i)Y(i))
Nu(0,2) = 1/12 (1<=i<=N) ( X(i-1)Y(i)-X(i)Y(i-1) ) * (Y(i-1)^2+Y(i-1)Y(i)+Y(i)^2)
```

The normalized moment of order (P,Q) for POLY is defined by:

```        Alpha(P,Q) = Integral ( x, y in POLY ) x^p y^q dx dy / Area ( Poly )
= Nu(P,Q) / Nu(0,0)
```
In particular, Alpha(0,0) is 1.

The central moment of order (P,Q) for POLY is defined by:

```        x* = Alpha(1,0)
y* = Alpha(0,1)
Mu(P,Q) = Integral ( x, y in POLY ) (x-x*)^p (y-y*)^q dx dy / Area ( Poly )
```

Simple formulas are available for low orders:

```        Mu(0,0) = 1
Mu(1,0) = 0
Mu(0,1) = 0
Mu(2,0) = Alpha(2,0) - Alpha(1,0)^2
Mu(1,1) = Alpha(1,1) - Alpha(1,0) * Alpha(0,1)
Mu(0,2) = Alpha(0,2) - Alpha(0,1)^2
```

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

POLYGON_INTEGRALS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

BALL_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

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POLYGON_GRID, a Python library which generates a grid of points over the interior of a polygon in 2D.

POLYGON_MONTE_CARLO, a Python library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

POLYGON_PROPERTIES, a Python library which computes properties of an arbitrary polygon in the plane, defined by a sequence of vertices, including interior angles, area, centroid, containment of a point, convexity, diameter, distance to a point, inradius, lattice area, nearest point in set, outradius, uniform sampling.

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PYRAMID_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

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SPHERE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TOMS112, a Python library which determines whether a point is contained in a polygon, by Moshe Shimrat. This is a version of ACM TOMS algorithm 112.

TRIANGLE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

### Reference:

1. SF Bockman,
Generalizing the Formula for Areas of Polygons to Moments,
American Mathematical Society Monthly,
Volume 96, Number 2, February 1989, pages 131-132.
2. Carsten Steger,
On the calculation of arbitrary moments of polygons,
Technical Report FGBV-96-05,
Forschungsgruppe Bildverstehen, Informatik IX,
Technische Universitaet Muenchen, October 1996.

### Examples and Tests:

You can go up one level to the Python source codes.

Last revised on 23 June 2015.