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<html>
<head>
<title>
POLYGON_INTEGRALS - Arbitrary Moments of a Polygon
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POLYGON_INTEGRALS <br> Arbitrary Moments of a Polygon
</h1>
<hr>
<p>
<b>POLYGON_INTEGRALS</b>
is a C++ library which
returns the exact value of the integral of any monomial
over the interior of a polygon in 2D.
</p>
<p>
We suppose that POLY is a planar polygon with N vertices X, Y, listed
in counterclockwise order.
</p>
<p>
For nonnegative integers P and Q, the (unnormalized) moment of order (P,Q)
for POLY is defined by:
<pre>
Nu(P,Q) = Integral ( x, y in POLY ) x^p y^q dx dy
</pre>
In particular, Nu(0,0) is the area of POLY.
</p>
<p>
Simple formulas are available for low orders:
<pre>
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)
</pre>
</p>
<p>
The normalized moment of order (P,Q) for POLY is defined by:
<pre>
Alpha(P,Q) = Integral ( x, y in POLY ) x^p y^q dx dy / Area ( Poly )
= Nu(P,Q) / Nu(0,0)
</pre>
In particular, Alpha(0,0) is 1.
</p>
<p>
The central moment of order (P,Q) for POLY is defined by:
<pre>
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 )
</pre>
</p>
<p>
Simple formulas are available for low orders:
<pre>
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
</pre>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this
web page are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>POLYGON_INTEGRALS</b> is available in
<a href = "../../c_src/polygon_integrals/polygon_integrals.html">a C version</a> and
<a href = "../../cpp_src/polygon_integrals/polygon_integrals.html">a C++ version</a> and
<a href = "../../f77_src/polygon_integrals/polygon_integrals.html">a FORTRAN77 version</a> and
<a href = "../../f_src/polygon_integrals/polygon_integrals.html">a FORTRAN90 version</a> and
<a href = "../../m_src/polygon_integrals/polygon_integrals.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/ball_integrals/ball_integrals.html">
BALL_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit ball in 3D.
</p>
<p>
<a href = "../../cpp_src/circle_integrals/circle_integrals.html">
CIRCLE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the surface of the unit circle in 2D.
</p>
<p>
<a href = "../../cpp_src/cube_integrals/cube_integrals.html">
CUBE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit cube in 3D.
</p>
<p>
<a href = "../../cpp_src/disk_integrals/disk_integrals.html">
DISK_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit disk in 2D.
</p>
<p>
<a href = "../../cpp_src/hyperball_integrals/hyperball_integrals.html">
HYPERBALL_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit hyperball in M dimensions.
</p>
<p>
<a href = "../../cpp_src/hypercube_integrals/hypercube_integrals.html">
HYPERCUBE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../cpp_src/hypersphere_integrals/hypersphere_integrals.html">
HYPERSPHERE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the surface of the unit hypersphere in M dimensions.
</p>
<p>
<a href = "../../cpp_src/line_integrals/line_integrals.html">
LINE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the length of the unit line in 1D.
</p>
<p>
<a href = "../../cpp_src/polygon_monte_carlo/polygon_monte_carlo.html">
POLYGON_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of a polygon in 2D.
</p>
<p>
<a href = "../../cpp_src/polygon_properties/polygon_properties.html">
POLYGON_PROPERTIES</a>,
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.
</p>
<p>
<a href = "../../cpp_src/polygon_triangulate/polygon_triangulate.html">
POLYGON_TRIANGULATE</a>,
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.
</p>
<p>
<a href = "../../cpp_src/pyramid_integrals/pyramid_integrals.html">
PYRAMID_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/simplex_integrals/simplex_integrals.html">
SIMPLEX_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/sphere_integrals/sphere_integrals.html">
SPHERE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../cpp_src/square_integrals/square_integrals.html">
SQUARE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit square in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_integrals/triangle_integrals.html">
TRIANGLE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_integrals/wedge_integrals.html">
WEDGE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
SF Bockman,<br>
Generalizing the Formula for Areas of Polygons to Moments,<br>
American Mathematical Society Monthly,<br>
Volume 96, Number 2, February 1989, pages 131-132.
</li>
<li>
Carsten Steger,<br>
On the calculation of arbitrary moments of polygons,<br>
Technical Report FGBV-96-05,<br>
Forschungsgruppe Bildverstehen, Informatik IX,<br>
Technische Universitaet Muenchen, October 1996.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "polygon_integrals.cpp">polygon_integrals.cpp</a>, the source code.
</li>
<li>
<a href = "polygon_integrals.hpp">polygon_integrals.hpp</a>, the include file.
</li>
<li>
<a href = "polygon_integrals.sh">polygon_integrals.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "polygon_integrals_prb.cpp">polygon_integrals_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "polygon_integrals_prb.sh">polygon_integrals_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "polygon_integrals_prb_output.txt">polygon_integrals_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MOMENT</b> computes an unnormalized moment of a polygon.
</li>
<li>
<b>MOMENT_CENTRAL</b> computes central moments of a polygon.
</li>
<li>
<b>MOMENT_NORMALIZED</b> computes a normalized moment of a polygon.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 03 October 2012.
</i>
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