Editing Inversive geometry (section)

=== Inverse of a point ===

[[File:Inversion illustration1.svg|thumb|''P''{{'}} is the inverse of ''P'' with respect to the circle.]]

To invert a number in arithmetic usually means to take its [[Multiplicative inverse|reciprocal]]. A closely related idea in geometry is that of "inverting" a point. In the [[plane (geometry)|plane]], the '''inverse''' of a point ''P'' with respect to a ''reference circle (Ø)'' with center ''O'' and radius ''r'' is a point ''P''{{'}}, lying on the ray from ''O'' through ''P'' such that

:<math>OP \cdot OP^{\prime} = r^2.</math>

This is called '''circle inversion''' or '''plane inversion'''.  The inversion taking any point ''P'' (other than ''O'') to its image ''P''{{'}} also takes ''P''{{'}}  back to ''P'', so the result of applying the same inversion twice is the identity transformation which makes it a [[self-inversion]] (i.e. an involution).<ref>{{harvtxt|Altshiller-Court|1952|p=230}}</ref><ref>{{harvtxt|Kay|1969|p=264}}</ref> To make the inversion a [[total function]] that is also defined for ''O'', it is necessary to introduce a [[point at infinity]], a single point placed on all the lines, and extend the inversion, by definition, to interchange the center ''O'' and this point at infinity.

It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is ''invariant'' under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.

==== Compass and straightedge construction ====
[[File:Inversion in circle.svg|thumb|To construct the inverse ''P{{'}}'' of a point ''P'' outside a circle ''Ø'':  Let ''r'' be the radius of ''Ø''. Right triangles ''OPN'' and ''ONP{{'}}'' are similar. ''OP'' is to ''r'' as ''r'' is to ''OP{{'}}''.]]

===== Point outside circle =====
To [[Compass and straightedge constructions|construct]] the inverse ''P{{'}}'' of a point ''P'' outside a circle ''Ø'':
* Draw the segment from ''O'' (center of circle ''Ø'') to ''P''.
* Let ''M'' be the midpoint of ''OP''. (Not shown)
* Draw the circle ''c'' with center ''M'' going through ''P''. (Not labeled. It's the blue circle)
* Let ''N'' and ''N{{'}}'' be the points where ''Ø'' and ''c'' intersect.
* Draw segment ''NN{{'}}''.
* ''P{{'}}'' is where ''OP'' and ''NN{{'}}'' intersect.

===== Point inside circle =====
To construct the inverse ''P'' of a point ''P{{'}}'' inside a circle ''Ø'':
* Draw ray ''r'' from ''O'' (center of circle ''Ø'') through ''P{{'}}''. (Not labeled, it's the horizontal line)
* Draw line ''s'' through ''P{{'}}'' perpendicular to ''r''. (Not labeled. It's the vertical line)
* Let ''N'' be one of the points where ''Ø'' and ''s'' intersect.
* Draw the segment ''ON''.
* Draw line ''t'' through ''N'' perpendicular to ''ON''.
*  ''P'' is where ray ''r'' and line ''t'' intersect.