POLYGON_INTEGRALS is a FORTRAN90 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
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
POLYGON_INTEGRALS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
BALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.
CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.
CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.
DISK01_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.
HYPERBALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.
HYPERCUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.
HYPERSPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.
LINE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.
POLYGON_MONTE_CARLO, a FORTRAN90 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 FORTRAN90 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.
POLYGON_TRIANGULATE, a FORTRAN90 library which triangulates a possibly nonconvex polygon, and which can use gnuplot to display the external edges and internal diagonals of the triangulation.
PYRAMID_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.
SIMPLEX_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.
SPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.
SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.
TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.
TRIANGLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.
WEDGE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.
You can go up one level to the FORTRAN90 source codes.