Stereographic Projection

Clear[xd, xlen, projectF, grid2d, grid3d, xball  ];

xd = 0.2;
xlen = 2;

projectF = ( With[{xx = 1 + #.#}, {#[[1]]/xx, #[[2]]/xx, 1 - 1/xx}] &);

grid2d = Table[{Hue[0], Line[{{x, y}, {x, y + xd}}], GrayLevel[ 0.5 ],
    Line[{{x, y}, {x + xd, y}}]}, {x, -xlen, xlen, xd}, {y, -xlen,
     xlen, xd}];

Graphics[grid2d]

grid3d = grid2d /. Line[xpts_] :> Line[(({#[[1]],#[[2]],0}) &) /@ xpts];
Graphics3D[grid3d]

xball = grid2d /. Line[xpts_] :> Line[projectF /@ xpts];

Graphics3D[xball]

Graphics3D[{grid3d, xball}]

• Suppose you have a ball sitting on a elaborately beautifully tiled floor.
• Now, suppose the top of the ball we call it North Pole.
• Now, Let there be a straight line from the north pole to a point on the floor.
• Call this point P.
• This line will intersect the ball somewhere, let's call the point of intersection Q.
• Now, every point P on the floor will then have a corresponding point Q on the ball.
• This process, of making a image from the floor to the ball is called Stereographic Projection.

more examples:

Notes

The images on this page is generated by the following Mathematica packages:

• “PlaneTiling.m” available at Plane Tiling Mathematica Package.
• “Transform2DPlot.m” available at Geometric Transformation on the Plane.
• Notebook: sphere_proj.nb
• sphere_proj_illus.nb