6-sphere coordinates

In mathematics, 6-sphere coordinates are the coordinate system created by inverting the Cartesian coordinates across the unit sphere. They are so named because the loci where one coordinate is constant form spheres tangent to the origin from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative).

The three coordinates are

u = \frac{x}{x^2+y^2+z^2},\quad v = \frac{y}{x^2+y^2+z^2},\quad w = \frac{z}{x^2+y^2+z^2}.

Since inversion is its own inverse, the equations for x, y, and z in terms of u, v, and w are similar:

x = \frac{u}{u^2+v^2+w^2},\quad y = \frac{v}{u^2+v^2+w^2},\quad z = \frac{w}{u^2+v^2+w^2}.

This coordinate system is $R$ -separable for the 3-variable Laplace equation.

References

Moon, P. and Spencer, D. E. 6-sphere Coordinates. Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 122–123, 1988.
Weisstein, Eric W., "6-sphere coordinates", MathWorld.
Six-Sphere Coordinates by Michael Schreiber, the Wolfram Demonstrations Project.