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<html>
<head>
<title>
SIMPLEX_COORDINATES - Coordinates of Regular Simplex in M Dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SIMPLEX_COORDINATES <br> Coordinates of Regular Simplex in M Dimensions
</h1>
<hr>
<p>
<b>SIMPLEX_COORDINATES</b>
is a C++ library which
computes the Cartesian coordinates of the vertices of a regular
simplex in M dimensions.
</p>
<p>
Note that the unit simplex, formed by the origin and the M unit
coordinate axes, is not a regular simplex, because some sides have
length 1 while other sides have length sqrt(2).
</p>
<p>
There is a straightforward but tedious method for computing these
coordinates, coded in SIMPLEX_COORDINATES1, based on the idea of
selecting the first coordinate to be (1,0,0,...0), and noting
that the dot product with vectors 2 through N+1 must be -1/N,
which means the first row and first column of the coordinate matrix
are done. We can then move to entry (2,2), assume the coordinates
below it are 0, and set its value by requiring that the sum of the
squares of the (2,1) and (2,2) entries must be 1. Setting the (2,2)
entry then allows us to determine the rest of row 2 by the same
dot product criterion, and we proceed in this way til we have
completed the matrix.
</p>
<p>
A simpler algorithm, in SIMPLEX_COORDINATES2, notes that the identity
matrix will almost work for the first N vertices. Choose the last
vertex by assuming all its entries are equal to some constant A, and
that its distance from any other vertex must be sqrt ( 2 ). This
determines that (A-1)^2 + (N-1)*A^2 = 2, from which we get the value of
A as (1-sqrt(N+1))/N. To clean things up, we compute the centroid C
of these vertices, and recenter the simplex around the origin.
Then we determine the distance S of one vertex to the origin, and
rescale so that this becomes 1. The coding is simpler, and there
is much less chance for the accumulation of numerical error. Plus
I thought of this one myself.
</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>SIMPLEX_COORDINATES</b> is available in
<a href = "../../c_src/simplex_coordinates/simplex_coordinates.html">a C version</a> and
<a href = "../../cpp_src/simplex_coordinates/simplex_coordinates.html">a C++ version</a> and
<a href = "../../f77_src/simplex_coordinates/simplex_coordinates.html">a FORTRAN77 version</a> and
<a href = "../../f_src/simplex_coordinates/simplex_coordinates.html">a FORTRAN90 version</a> and
<a href = "../../m_src/simplex_coordinates/simplex_coordinates.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/asa299/asa299.html">
ASA299</a>,
a C++ library which
computes the lattice points in an M-dimensional simplex;
this is Applied Statistics Algorithm 299;
</p>
<p>
<a href = "../../cpp_src/geometry/geometry.html">
GEOMETRY</a>,
a C++ library which
performs geometric calculations in 2, 3 and M-dimensional space.
</p>
<p>
<a href = "../../cpp_src/random_data/random_data.html">
RANDOM_DATA</a>,
a C++ library which
generates sample points for
various probability distributions, spatial dimensions, and geometries,
including the M-dimensional simplex.
</p>
<p>
<a href = "../../f77_src/simpack/simpack.html">
SIMPACK</a>,
a FORTRAN77 library which
approximates the integral of a function over an M-dimensional simplex.
</p>
<p>
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a C++ library which
defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron,
or general M-dimensional simplex.
</p>
<p>
<a href = "../../cpp_src/simplex_grid/simplex_grid.html">
SIMPLEX_GRID</a>,
a C++ library which
generates a regular grid of points
over the interior of an arbitrary simplex in M dimensions.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "simplex_coordinates.cpp">simplex_coordinates.cpp</a>, the source code.
</li>
<li>
<a href = "simplex_coordinates.hpp">simplex_coordinates.hpp</a>,
the include file.
</li>
<li>
<a href = "simplex_coordinates.sh">simplex_coordinates.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "simplex_coordinates_prb.cpp">simplex_coordinates_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "simplex_coordinates_prb.sh">simplex_coordinates_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "simplex_coordinates_prb_output.txt">simplex_coordinates_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8MAT_DET</b> computes the determinant of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>SIMPLEX_COORDINATES1</b> computes the Cartesian coordinates of simplex vertices.
</li>
<li>
<b>SIMPLEX_COORDINATES2</b> computes the Cartesian coordinates of simplex vertices.
</li>
<li>
<b>SIMPLEX_VOLUME</b> computes the volume of a simplex.
</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 20 September 2010.
</i>
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