polygon_integrals


polygon_integrals, a Python 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
      

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.

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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.


Last revised on 29 January 2020.