SPHERE_STEREOGRAPH Stereographic Mapping Between Sphere and Plane

SPHERE_STEREOGRAPH is a MATLAB program which implements the standard stereographic mapping between the unit sphere and the plane Z=1, as well as a generalization of this mapping.

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.

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.

The Standard Projection

A plane is chosen, tangent to the sphere, at a point of tangency T which we take to be the "north pole", T = (0,0,1). We use a focus point F, which we take to be the "south pole", (0,0,-1)

For any point P on the sphere, the stereographic projection Q of the point is defined by drawing the line from F through P, and computing Q as the intersection of this line with the plane.

For any point Q on the plane, the stereographic inverse projection P of the point is defined by drawing the line from F through Q, and computing P as the intersection of this line with the sphere.

The function sphere_stereograph carries out the standard projection, and sphere_stereograph_inverse does the inverse.

Projection with arbitrary center and focus

One way to generalize the projection is to allow the center C and focus point F to be arbitrary. If we assume the point of tangency is antipodal to F then T = 2*C-F. Once these points are defined, the stereographic projection relative to F, C, and T can be set up in the same way as before.

The function sphere_stereograph2 carries out the generalized projection, and sphere_stereograph2_inverse does the inverse.

Languages:

SPHERE_STEREOGRAPH is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

GEOMETRY, a MATLAB library which performs geometric calculations in 2, 3 and M dimensional space.

HYPERSPHERE, a MATLAB library which carries out various operations for an M-dimensional hypersphere, including converting between Cartesian and spherical coordinates, stereographic projection, sampling the surface of the sphere, and computing the surface area and volume.

SPHERE_GRID, a MATLAB 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.

SPHERE_STEREOGRAPH_DISPLAY, a MATLAB library which computes and displays the results of several stereographic projections between a sphere and a plane.

SPHERE_XYZ_DISPLAY, a MATLAB program which reads XYZ information defining points in 3D, and displays a unit sphere and the points in the MATLAB 3D graphics window.

Reference:

1. C F Marcus,
The stereographic projection in vector notation,
Mathematics Magazine,
Volume 39, Number 2, March 1966, pages 100-102.

Examples and Tests:

You can go up one level to the MATLAB source codes.

Last revised on 11 November 2010.