Inversive geometry

In three dimensions

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 ${\displaystyle OP\cdot O{P}^{\mathrm{\prime }}=||OP||\cdot ||O{P}^{\mathrm{\prime }}||=R^2}$. 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

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 ${\displaystyle x^2+y^2+(z+\frac{1}{2}{)}^2=\frac{1}{4}}$; 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.

6-sphere coordinates

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

Axiomatics and generalization

One of the first to consider foundations of inversive geometry was Mario Pieri in 1911 and 1912. Edward Kasner wrote his thesis on "Invariant theory of the inversion group".

More recently the mathematical structure of inversive geometry has been interpreted as an incidence structure where the generalized circles are called "blocks": In incidence geometry, any affine plane together with a single point at infinity forms a Möbius plane, also known as an inversive plane. The point at infinity is added to all the lines. These Möbius planes can be described axiomatically and exist in both finite and infinite versions.

A model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere.

Invariant

The cross-ratio between 4 points ${\displaystyle x,y,z,w}$ is invariant under an inversion. In particular if O is the centre of the inversion and ${\displaystyle r_1}$ and ${\displaystyle r_2}$ are distances to the ends of a line L, then length of the line ${\displaystyle d}$ will become ${\displaystyle d/(r_1{r}_2)}$ under an inversion with radius 1. The invariant is:

${\displaystyle I=\frac{|x-y||w-z|}{|x-w||y-z|}}$

Relation to Erlangen program

According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein's Erlangen program, an outgrowth of certain models of hyperbolic geometry.

Dilation

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

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

Reciprocation

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

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

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

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

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.

Transforming circles into circles

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

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

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

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

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

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

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

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

For ${\displaystyle a\notin \mathbb{R}}$ and ${\displaystyle a{a}^{\ast }\ne r^2}$ the result for ${\displaystyle w}$ is

${\displaystyle \begin{array}{cc}& w{w}^{\ast }-\frac{aw+a^{\ast }{w}^{\ast }}{(a^{\ast }a-r^2)}+\frac{a{a}^{\ast }}{(a{a}^{\ast }-r^2{)}^2}=\frac{r^2}{(a{a}^{\ast }-r^2{)}^2}\\ ⟺& \left(w-\frac{a^{\ast }}{a{a}^{\ast }-r^2}\right)\left(w^{\ast }-\frac{a}{a^{\ast }a-r^2}\right)={\left(\frac{r}{\left|a{a}^{\ast }-r^2\right|}\right)}^2\end{array}}$

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

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

${\displaystyle \begin{array}{cc}& aw+a^{\ast }{w}^{\ast }=1⟺2\mathrm{Re}\{aw\}=1⟺\mathrm{Re}\{a\}\mathrm{Re}\{w\}-\mathrm{Im}\{a\}\mathrm{Im}\{w\}=\frac{1}{2}\\ ⟺& \mathrm{Im}\{w\}=\frac{\mathrm{Re}\{a\}}{\mathrm{Im}\{a\}}\cdot \mathrm{Re}\{w\}-\frac{1}{2\cdot \mathrm{Im}\{a\}}.\end{array}}$

Higher geometry

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 develops several theorems of inversive geometry before beginning Lobachevskian geometry.

In higher dimensions

In a real n-dimensional Euclidean space, an inversion in the sphere of radius r centered at the point ${\displaystyle O=(o_1,...,o_n)}$ is a map of an arbitrary point ${\displaystyle P=(p_1,...,p_n)}$ found by inverting the length of the displacement vector ${\displaystyle P-O}$ and multiplying by ${\displaystyle r^2}$:

${\displaystyle \begin{array}{cc}P& \mapsto P^{\prime }=O+\frac{r^2(P-O)}{\Vert P-O{\Vert }^2},\\ p_j& \mapsto p_j^{\prime }=o_j+\frac{r^2(p_j-o_j)}{\sum _k(p_k-o_k{)}^2}.\end{array}}$

The transformation by inversion in hyperplanes or hyperspheres in Eⁿ can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or homothety about the hyperspheres' center.

When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an (n−2)-flat, successive reflections produce a rotation where every point of the (n−2)-flat is a fixed point of each reflection and thus of the composition.

Any combination of reflections, translations, and rotations is called an isometry. Any combination of reflections, dilations, translations, and rotations is a similarity.

All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry.

The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an n-sphere as the base space. The transformations of inversive geometry are often referred to as Möbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of an n-sphere.

Anticonformal mapping property

The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves oriented angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if J is the Jacobian, then ${\displaystyle J\cdot J^T=kI}$ and ${\displaystyle det(J)=-\sqrt{k}.}$ Computing the Jacobian in the case z_i = x_i/‖x‖², where ‖x‖² = x₁² + ... + x_n² gives JJ^T = kI, with k = 1/‖x‖⁴ⁿ, and additionally det(J) is negative; hence the inversive map is anticonformal.

In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking z to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a homography is conformal while an anti-homography is anticonformal.

Hyperbolic geometry

The (n − 1)-sphere with equation

${\displaystyle x_1^2+\cdots +x_n^2+2a_1{x}_1+\cdots +2a_n{x}_n+c=0}$

will have a positive radius if a₁² + ... + a_n² is greater than c, and on inversion gives the sphere

${\displaystyle x_1^2+\cdots +x_n^2+2\frac{a_1}{c}{x}_1+\cdots +2\frac{a_n}{c}{x}_n+\frac{1}{c}=0.}$

Hence, it will be invariant under inversion if and only if c = 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (n − 1)-spheres with equation

${\displaystyle x_1^2+\cdots +x_n^2+2a_1{x}_1+\cdots +2a_n{x}_n+1=0,}$

which are invariant under inversion, orthogonal to the unit sphere, and have centers outside of the sphere. These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disk model of hyperbolic geometry.

Since inversion in the unit sphere leaves the spheres orthogonal to it invariant, the inversion maps the points inside the unit sphere to the outside and vice versa. This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice versa; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere. These reflections generate the group of isometries of the model, which tells us that the isometries are conformal. Hence, the angle between two curves in the model is the same as the angle between two curves in the hyperbolic space.