Six-sphere coordinates
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In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere x^2+y^2+z^2=1. They are so named because the loci where one coordinate is constant form
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
s tangent to the
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from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). They have nothing whatsoever to do with the 6-sphere, which is an object of considerable interest in its own right. The three coordinates are :u = \frac,\quad v = \frac,\quad w = \frac. Since inversion is its own inverse, the equations for ''x'', ''y'', and ''z'' in terms of ''u'', ''v'', and ''w'' are similar: :x = \frac,\quad y = \frac,\quad z = \frac. This coordinate system is R-separable for the 3-variable Laplace equation.


See also

*
Multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
(for 1-dimensional version) *
Y-Δ transform The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like t ...
(unrelated, but similar formula for comparison) * 6-sphere


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. *
Six-Sphere Coordinates
by Michael Schreiber, the
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. Three-dimensional coordinate systems Inversive geometry Orthogonal coordinate systems {{geometry-stub