In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 6-sphere coordinates are a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
for three-dimensional space obtained by
inverting the 3D
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
across the
unit 2-sphere . They are so named because the
loci where one coordinate is constant form
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s tangent to the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
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
:
Since inversion is its own
inverse, the equations for ''x'', ''y'', and ''z'' in terms of ''u'', ''v'', and ''w'' are similar:
:
This coordinate system is
-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 Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
(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 th ...
(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 Coordinatesby Michael Schreiber, the
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Three-dimensional coordinate systems
Inversive geometry
Orthogonal coordinate systems
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