Inversive geometry

P' is the inverse of P with respect to the circle.

To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the 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

${\displaystyle OP\cdot OP^{\prime }=r^{2}.}$

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 on all the points of the plane other than O (self-inversion).[2][3] To make inversion an involution 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, the nearer a point to the center, the further away its transformation, and vice versa.

To 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.

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.

There is a construction of the inverse point to A with respect to a circle P that is independent of whether A is inside or outside P.[4]

Consider a circle P with center O and a point A which may lie inside or outside the circle P.

• Take the intersection point C of the ray OA with the circle P.
• Connect the point C with an arbitrary point B on the circle P (different from 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 P.[4]^{: § 3.2}

• 
  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.
• 
  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.
• 
  Inversion with respect to a circle does not map the center of the circle to the center of its image

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).[5]
• 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 to the reference circle at the points of intersection.[6]

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

${\displaystyle \angle OAB=\angle OB'A'\ {\text{ and }}\ \angle OBA=\angle OA'B'.}$

• 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.[7]

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 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

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 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.

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 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 polar of the other point (the 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.

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.

The inversion of a cylinder, cone, or torus results in a Dupin cyclide.

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.

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 an inversion of a sphere

A stereographic projection usually projects a sphere from a point ${\displaystyle N}$ (north pole) of the sphere onto the tangent plane at the opposite point ${\displaystyle S}$ (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 ${\displaystyle x^{2}+y^{2}+z^{2}=-z}$ (alternately written ${\displaystyle x^{2}+y^{2}+(z+{\tfrac {1}{2}})^{2}={\tfrac {1}{4}}}$; center ${\displaystyle (0,0,-0.5)}$, radius ${\displaystyle 0.5}$, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point ${\displaystyle S=(0,0,-1)}$. The lines through the center of inversion (point ${\displaystyle N}$) are mapped onto themselves. They are the projection lines of the stereographic projection.

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

The combination of two inversions in concentric circles results in a similarity, homothetic transformation, or dilation characterized by the ratio of the circle radii.

${\displaystyle x\mapsto R^{2}{\frac {x}{|x|^{2}}}=y\mapsto T^{2}{\frac {y}{|y|^{2}}}=\left({\frac {T}{R}}\right)^{2}x.}$

When a point in the plane is interpreted as a complex number ${\displaystyle z=x+iy,}$ with complex conjugate ${\displaystyle {\bar {z}}=x-iy,}$ then the reciprocal of z is

${\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{|z|^{2}}}.}$

Consequently, the algebraic form of the inversion in a unit circle is given by ${\displaystyle z\mapsto w}$ where:

${\displaystyle w={\frac {1}{\bar {z}}}={\overline {\left({\frac {1}{z}}\right)}}}$.

Reciprocation is key in transformation theory as a generator of the Möbius group. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the conjugation mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are analytic functions of the whole plane and so are necessarily conformal.

Consider, in the complex plane, the circle of radius ${\displaystyle r}$ around the point ${\displaystyle a}$

${\displaystyle (z-a)(z-a)^{*}=r^{2}}$

where without loss of generality, ${\displaystyle a\in \mathbb {R} .}$ Using the definition of inversion

${\displaystyle w={\frac {1}{z^{*}}}}$

it is straightforward to show that ${\displaystyle w}$ obeys the equation

${\displaystyle ww^{*}-{\frac {a}{(a^{2}-r^{2})}}(w+w^{*})+{\frac {a^{2}}{(a^{2}-r^{2})^{2}}}={\frac {r^{2}}{(a^{2}-r^{2})^{2}}}}$

and hence that ${\displaystyle w}$ describes the circle of center ${\textstyle {\frac {a}{a^{2}-r^{2}}}}$ and radius ${\textstyle {\frac {r}{|a^{2}-r^{2}|}}.}$

When ${\displaystyle a\to r,}$ the circle transforms into the line parallel to the imaginary axis ${\displaystyle w+w^{*}={\tfrac {1}{a}}.}$

For ${\displaystyle a\not \in \mathbb {R} }$ and ${\displaystyle aa^{*}\neq r^{2}}$ the result for ${\displaystyle w}$ is

${\displaystyle {\begin{aligned}&ww^{*}-{\frac {aw+a^{*}w^{*}}{(a^{*}a-r^{2})}}+{\frac {aa^{*}}{(aa^{*}-r^{2})^{2}}}={\frac {r^{2}}{(aa^{*}-r^{2})^{2}}}\\[4pt]\Longleftrightarrow {}&\left(w-{\frac {a^{*}}{aa^{*}-r^{2}}}\right)\left(w^{*}-{\frac {a}{a^{*}a-r^{2}}}\right)=\left({\frac {r}{\left|aa^{*}-r^{2}\right|}}\right)^{2}\end{aligned}}}$

showing that the ${\displaystyle w}$ describes the circle of center ${\textstyle {\frac {a}{(aa^{*}-r^{2})}}}$ and radius ${\textstyle {\frac {r}{\left|a^{*}a-r^{2}\right|}}}$.

When ${\displaystyle a^{*}a\to r^{2},}$ the equation for ${\displaystyle w}$ becomes

${\displaystyle {\begin{aligned}&aw+a^{*}w^{*}=1\Longleftrightarrow 2\operatorname {Re} \{aw\}=1\Longleftrightarrow \operatorname {Re} \{a\}\operatorname {Re} \{w\}-\operatorname {Im} \{a\}\operatorname {Im} \{w\}={\frac {1}{2}}\\[4pt]\Longleftrightarrow {}&\operatorname {Im} \{w\}={\frac {\operatorname {Re} \{a\}}{\operatorname {Im} \{a\}}}\cdot \operatorname {Re} \{w\}-{\frac {1}{2\cdot \operatorname {Im} \{a\}}}.\end{aligned}}}$

As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 . In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami, Cayley, and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872. Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.

For example, Smogorzhevsky[11] develops several theorems of inversive geometry before beginning Lobachevskian geometry.