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<html>
<head>
<title>
SPHERE_STEREOGRAPH - Stereographic Mapping Between Sphere and Plane
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_STEREOGRAPH <br>
Stereographic Mapping Between Sphere and Plane
</h1>
<hr>
<p>
<b>SPHERE_STEREOGRAPH</b>
is a C++ program which
implements the standard stereographic mapping between the unit sphere and
the plane Z=1, as well as a generalization of this mapping.
</p>
<p>
The stereographic projection preserves angles, and is a conformal mapping.
This implies, for instance, that the Delaunay triangulation of a sphere
maps to a corresponding Delaunay triangulation of the plane.
</p>
<p>
Circles on the sphere that do not pass through the focus will be
projected to circles on the plane. Circles on the sphere that do pass
through the focus will be projected to straight lines on the plane.
</p>
<h3 align = "center">
The Standard Projection
</h3>
<p>
We start with a sphere of radius 1 and center <b>C</b> = (0,0,0).
</p>
<p>
A plane is chosen, tangent to the sphere, at a point of tangency
<b>T</b> which we take to be the "north pole", <b>T</b> = (0,0,1).
We use a focus point <b>F</b>, which we take to be the "south pole",
(0,0,-1)
</p>
<p>
For any point <b>P</b> on the sphere, the stereographic projection <b>Q</b>
of the point is defined by drawing the line from <b>F</b> through <b>P</b>,
and computing <b>Q</b> as the intersection of this line with the plane.
</p>
<p>
For any point <b>Q</b> on the plane, the stereographic inverse projection
<b>P</b> of the point is defined by drawing the line from <b>F</b>
through <b>Q</b>, and
computing <b>P</b> as the intersection of this line with the sphere.
</p>
<p>
The function <b>sphere_stereograph</b> carries out the standard projection,
and <b>sphere_stereograph_inverse</b> does the inverse.
</p>
<h3 align = "center">
Projection with arbitrary center and focus
</h3>
<p>
One way to generalize the projection is to allow the center <b>C</b> and
focus point <b>F</b> to be arbitrary. If we assume the point of tangency
is antipodal to <b>F</b> then <b>T</b> = 2*<b>C</b>-<b>F</b>. Once these
points are defined, the stereographic projection relative to <b>F</b>,
<b>C</b>, and <b>T</b> can be set up in the same way as before.
</p>
<p>
The function <b>sphere_stereograph2</b> carries out the generalized
projection, and <b>sphere_stereograph2_inverse</b> does the inverse.
</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>SPHERE_STEREOGRAPH</b> is available in
<a href = "../../c_src/sphere_stereograph/sphere_stereograph.html">a C version</a> and
<a href = "../../cpp_src/sphere_stereograph/sphere_stereograph.html">a C++ version</a> and
<a href = "../../f77_src/sphere_stereograph/sphere_stereograph.html">a FORTRAN77 version</a> and
<a href = "../../f_src/sphere_stereograph/sphere_stereograph.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_stereograph/sphere_stereograph.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a C++ library which
provides a number of ways of generating grids of points, or of
points and lines, or of points and lines and faces, over the unit sphere.
</p>
<p>
<a href = "../../m_src/sphere_stereograph_display/sphere_stereograph_display.html">
SPHERE_STEREOGRAPH_DISPLAY</a>,
a MATLAB library which
computes and displays the results of several stereographic projections
between a sphere and a plane.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
C F Marcus,<br>
The stereographic projection in vector notation,<br>
Mathematics Magazine,<br>
Volume 39, Number 2, March 1966, pages 100-102.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_stereograph.cpp">sphere_stereograph.cpp</a>, the source code.
</li>
<li>
<a href = "sphere_stereograph.hpp">sphere_stereograph.hpp</a>, the include file.
</li>
<li>
<a href = "sphere_stereograph.sh">sphere_stereograph.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_stereograph_prb.cpp">sphere_stereograph_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "sphere_stereograph_prb.sh">sphere_stereograph_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "sphere_stereograph_prb_output.txt">sphere_stereograph_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>PLANE_NORMAL_BASIS_3D</b> finds two perpendicular vectors in a plane in 3D.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8MAT_UNIFORM_01_NEW</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8VEC_ANY_NORMAL</b> returns some normal vector to V1.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_CROSS_PRODUCT_3D</b> computes the cross product of two R8VEC's in 3D.
</li>
<li>
<b>R8VEC_NORM</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_NORM_AFFINE</b> returns the affine L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_NORMAL_01</b> samples the standard normal probability distribution.
</li>
<li>
<b>R8VEC_TRANSPOSE_PRINT</b> prints a vector on one line.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> fills a double precision vector with pseudorandom values.
</li>
<li>
<b>R8VEC_UNIFORM_01_NEW</b> returns a new unit pseudorandom R8VEC.
</li>
<li>
<b>R8VEC_ZERO</b> zeroes an R8VEC.
</li>
<li>
<b>SPHERE_STEREOGRAPH</b> computes the stereographic image of points on a sphere.
</li>
<li>
<b>SPHERE_STEREOGRAPH_INVERSE</b> computes stereographic preimages of points.
</li>
<li>
<b>SPHERE_STEREOGRAPH2</b> computes the stereographic image of points on a sphere.
</li>
<li>
<b>SPHERE_STEREOGRAPH2_INVERSE</b> computes stereographic preimages of points.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>UNIFORM_ON_SPHERE01_MAP</b> maps uniform points onto the unit sphere.
</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 11 November 2010.
</i>
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