POLYGON_INTEGRALS is a C 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 dyIn 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 C library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.
CIRCLE_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.
CUBE_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.
DISK01_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.
HYPERBALL_INTEGRALS, a C 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 C 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 C 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 C 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 C 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 C 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 C 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 C library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.
SIMPLEX_INTEGRALS, a C 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 C library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.
SQUARE_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.
TETRAHEDRON_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.
TRIANGLE_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.
WEDGE_INTEGRALS, a C 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 C source codes.