sphere_stereograph

sphere_stereograph, a MATLAB code 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

We start with a sphere of radius 1 and center C = (0,0,0).

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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

sphere_stereograph is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

geometry, a MATLAB code which performs geometric calculations in 2, 3 and M dimensional space.

sphere_grid, a MATLAB code 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_test

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

Reference:

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

Source Code:

• plane_normal_basis_3d.m, finds two perpendicular vectors in a plane in normal form in 3D;
• r8mat_transpose_print.m, prints the transpose of an R8MAT;
• r8mat_transpose_print_some.m, prints some of the transpose of an R8MAT;
• r8vec_any_normal.m, returns a unit normal vector to a given vector;
• r8vec_cross_product_3d.m, computes the cross product of two R8VEC's in 3D;
• r8vec_normal_01.m, returns unit pseudonormal R8VEC.
• sphere_stereograph.m maps points on the unit sphere to the plane Z = 1 using the standard projection.
• sphere_stereograph_inverse.m maps points on the plane Z = 1 to the unit sphere using the standard projection.
• sphere_stereograph2.m stereographic mapping using a sphere centered at C, with focus point F and antipodal tangent point T=2*C-F.
• sphere_stereograph2_inverse.m inverse of stereographic mapping using a sphere centered at C, with focus point F and antipodal tangent point T=2*C-F.
• uniform_on_sphere01_map.m returns uniformly random points on the unit sphere.