Editing Inversive geometry (section)

== In three dimensions ==
[[File:Inv-kugel.svg|300px|thumb|Inversion of a sphere at the red sphere]]
[[File:Inv-ellipsoid.svg|300px|thumb|Inversion of a spheroid (at the red sphere)]]
[[File:Inv-hyperboloid.svg|300px|thumb|Inversion of a hyperboloid of one sheet]]

Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point ''P'' in 3D with respect to a reference sphere centered at a point ''O'' with radius ''R'' is a point ''P'' ' on the ray with direction ''OP'' such that <math>OP \cdot OP^{\prime} = ||OP|| \cdot ||OP^{\prime}|| = R^2</math>. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center ''O'' of the reference sphere, then it inverts to a plane. Any plane passing through ''O'', inverts to a sphere touching at ''O''. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through ''O'' it inverts into a line. This reduces to the 2D case when the secant plane passes through ''O'', but is a true 3D phenomenon if the secant plane does not pass through ''O''.

=== Examples in three dimensions ===

==== Sphere ====
The simplest surface (besides a plane) is the sphere. The first picture shows a non trivial inversion (the center of the sphere is not the center of inversion) of a sphere together with two orthogonal intersecting pencils of circles.

==== Cylinder, cone, torus ====
The inversion of a cylinder, cone, or torus results in a [[Dupin cyclide]].

==== Spheroid ====
A spheroid is a [[surface of revolution]] and contains a pencil of circles which is mapped onto a pencil of circles (see picture). The inverse image of a spheroid is a surface of degree 4.

==== Hyperboloid of one sheet ====
A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion.

=== Stereographic projection as the inversion of a sphere ===
[[File:Inv-stereogr-proj.svg|250px|thumb|Stereographic projection as an inversion of a sphere]]
A [[stereographic projection]] usually projects a sphere from a point  <math>N</math> (north pole) of the sphere onto the tangent plane at the opposite point <math>S</math> (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation <math>x^2+y^2+z^2 = -z</math> (alternately written <math>x^2+y^2+(z+\tfrac{1}{2})^2=\tfrac{1}{4}</math>; center <math>(0,0,-0.5)</math>, radius <math>0.5</math>, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point <math>S=(0,0,-1)</math>. The lines through the center of inversion (point <math>N</math>) are mapped onto themselves. They are the projection lines of the stereographic projection.

=== 6-sphere coordinates ===
The [[6-sphere coordinates]] are a coordinate system for three-dimensional space obtained by inverting the [[Cartesian coordinates]].