Editing Inversive geometry (section)

== Inversion in a circle {{anchor|Circle}}==

[[File:Inversion of lambda Mandelbrot set with different translations.gif|thumb|right|Inversion of lambda [[Mandelbrot set]] with different translations]]

=== 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.

=== Dutta's construction ===
There is a construction of the inverse point to ''A'' with respect to a circle ''Ø'' that is ''independent'' of whether ''A'' is inside or outside ''Ø''.<ref name=SD>Dutta, Surajit (2014) [http://forumgeom.fau.edu/FG2014volume14/FG201422index.html A simple property of isosceles triangles with applications] {{Webarchive|url=https://web.archive.org/web/20180421201921/http://forumgeom.fau.edu/FG2014volume14/FG201422index.html |date=2018-04-21 }}, [[Forum Geometricorum]] 14: 237–240</ref>

Consider a circle ''Ø'' with center ''O'' and a point ''A'' which may lie inside or outside the circle ''Ø''.
* Take the intersection point ''C'' of the ray ''OA'' with the circle ''Ø''.
* Connect the point ''C'' with an arbitrary point ''B'' on the circle ''Ø'' (different from ''C'' and from the point on ''Ø'' antipodal to ''C'')
* Let ''h'' be the reflection of ray ''BA'' in line ''BC''. Then ''h'' cuts ray ''OC'' in a point ''A{{'}}''. ''A{{'}}'' is the inverse point of ''A'' with respect to circle ''Ø''.<ref name=SD/>{{rp|§ 3.2}}

=== Properties ===
<gallery>
File:Inversion illustration2.svg|The inverse, with respect to the red circle, of a circle going through ''O'' (blue) is a line not going through ''O'' (green), and vice versa.
File:Inversion illustration3.svg|The inverse, with respect to the red circle, of a circle ''not'' going through ''O'' (blue) is a circle not going through ''O'' (green), and vice versa.
File:Inversion.gif|Inversion with respect to a circle does not map the center of the circle to the center of its image
</gallery>

The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points.  The following properties make circle inversion useful.
*A circle that passes through the center ''O'' of the reference circle inverts to a line not passing through ''O'', but parallel to the tangent to the original circle at ''O'', and vice versa; whereas a line passing through ''O'' is inverted into itself (but not pointwise invariant).<ref name="Kay 1969 265">{{harvtxt|Kay|1969|p=265}}</ref>
*A circle not passing through ''O'' inverts to a circle not passing through ''O''. If the circle meets the reference circle, these invariant points of intersection are also on the inverse circle. A circle (or line) is unchanged by inversion [[if and only if]] it is [[Orthogonal circles|orthogonal]] to the reference circle at the points of intersection.<ref name="Kay 1969 265"/>

Additional properties include:
*If a circle ''q'' passes through two distinct points A and A' which are inverses with respect to a circle ''k'', then the circles ''k'' and ''q'' are orthogonal.
*If the circles ''k'' and ''q'' are orthogonal, then a straight line passing through the center O of ''k'' and intersecting ''q'', does so at inverse points with respect to ''k''.
*Given a triangle OAB in which O is the center of a circle ''k'', and points A' and B' inverses of A and B with respect to ''k'', then
:: <math> \angle OAB = \angle OB'A' \ \text{ and }\ \angle OBA = \angle OA'B'.</math>
*The points of intersection of two circles ''p'' and ''q'' orthogonal to a circle ''k'', are inverses with respect to ''k''.
*If M and M' are inverse points with respect to a circle ''k'' on two curves m and m', also inverses with respect to ''k'', then the tangents to m and m' at the points M and M' are either perpendicular to the straight line MM' or form with this line an isosceles triangle with base MM'.
*Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.<ref>{{harvtxt|Kay|1969|p=269}}</ref>

=== Examples in two dimensions ===
[[File:circle_inversion_examples.svg|thumb|upright|link={{filepath:circle_inversion_examples.svg}}|Examples of inversion of circles A to J with respect to the red circle at O. Circles A to F, which pass through O, map to straight lines. Circles G to J, which do not, map to other circles. The reference circle and line L map to themselves. Circles intersect their inverses, if any, on the reference circle. In [{{filepath:circle_inversion_examples.svg}} the SVG file,] click or hover over a circle to highlight it.]]
* Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center
* Inversion of a circle is another circle; or it is a line if the original circle contains the center 
* Inversion of a parabola is a [[cardioid]]
* Inversion of hyperbola is a [[lemniscate of Bernoulli]]

=== Application ===
For a circle not passing through the center of inversion, the center of the circle being inverted and the center of its image under inversion are [[collinear]] with the center of the reference circle. This fact can be used to prove that the [[Euler line]] of the [[intouch triangle]] of a triangle coincides with its OI line. The proof roughly goes as below:

Invert with respect to the [[incircle]] of triangle ''ABC''. The [[medial triangle]] of the intouch triangle is inverted into triangle ''ABC'', meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ''ABC'' are [[collinear]].

Any two non-intersecting circles may be inverted into [[concentric]] circles. Then the [[inversive distance]] (usually denoted δ) is defined as the [[natural logarithm]] of the ratio of the radii of the two concentric circles.

In addition, any two non-intersecting circles may be inverted into [[congruence (geometry)|congruent]] circles, using circle of inversion centered at a point on the [[circle of antisimilitude]].

The [[Peaucellier–Lipkin linkage]] is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.

==== Pole and polar ====
{{main article|pole and polar}}
[[File:Pole and polar.svg|thumb|right|The polar line ''q'' to a point '''Q''' with respect to a circle of radius ''r'' centered on the point '''O'''.  The point '''P''' is the [[inversive geometry#Circle inversion|inversion point]] of '''Q'''; the polar is the line through '''P''' that is perpendicular to the line containing '''O''', '''P''' and '''Q'''.]]

If point ''R'' is the inverse of point ''P'' then the lines [[perpendicular]]  to the line ''PR'' through one of the points is the [[pole and polar|polar]] of the other point (the [[pole and polar|pole]]).

Poles and polars have several useful properties:

* If a point '''P''' lies on a line ''l'', then the pole '''L''' of the line ''l'' lies on the polar ''p'' of point '''P'''.
* If a point '''P''' moves along a line ''l'', its polar ''p'' rotates about the pole '''L''' of the line ''l''.
* If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points.
* If a point lies on the circle, its polar is the tangent through this point.
* If a point '''P''' lies on its own polar line, then '''P''' is on the circle.
* Each line has exactly one pole.