# polygon_integrals

polygon_integrals, an Octave code 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
```

### Languages:

polygon_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

ball_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3d.

circle_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit circle in 2d.

cube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit cube in 3d.

disk01_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2d.

hyperball_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in m dimensions.

hypercube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hypercube in m dimensions.

hypersphere_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in m dimensions.

line_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the length of the unit line in 1d.

simplex_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit simplex in m dimensions.

### 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.

### Source Code:

Last revised on 02 November 2022.