Inversive geometry

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In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (loosely speaking, a circle with infinite radius). Many difficult problems in geometry become much more tractable when an inversion is applied.

The concept of inversion can be generalized to higher-dimensional spaces.

Circle inversion

Inverse of a point

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 of center O and radius r is a point P', lying on the ray from O through P such that

$OP\times OP^{\prime} =r^2.$

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.[1][2] 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.

Properties

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 reference circle at O, and vice versa; whereas a line passing through O is inverted into itself (but not pointwise invariant).[3]
• 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.[4]

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
$\angle OAB = \angle OB'A' \ \text{ and }\ \angle OBA = \angle OA'B'.$
• 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.[5]

Application

Note that the center of a circle (not through the center of inversion) 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 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.

Inversions 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 ' such that $\scriptstyle OP \times OP^{\prime} =R^2$ and the points P and P ' are on the same ray starting at O. 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 not 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.

Stereographic projection is a special case of sphere inversion. Consider a sphere B of radius 1 and a plane P touching B at the South Pole S of B. Then P is the stereographic projection of B with respect to the North Pole N of B. Consider a sphere B2 of radius 2 centered at N. The inversion with respect to B2 transforms B into its stereographic projection P.

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.[6] Edward Kasner wrote his thesis on "Invariant theory of the inversion group".[7]

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.

Relation to Erlangen program

According to Coxeter,[8] 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

Dilations

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

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

Reciprocation

When a point in the plane is interpreted as a complex number $z=x+iy \,$, with complex conjugate $\bar{z}=x-iy$, then the reciprocal of z is $\scriptstyle \frac{1}{z} = \frac{\bar{z}}{|z|^2}$. Consequently, the algebraic form of the inversion in a unit circle is given by $z \mapsto w$ where:

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

Reciprocation is key in transformation theory as a generator of the Mobius 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 Mobius 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 Mobius group since they are non-conformal (see below). Mobius group elements are analytic functions of the whole plane and so are necessarily conformal.

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[9] develops several theorems of inversive geometry before beginning Lobachevskian geometry.

Inversion in higher dimensions

In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:

$x_i\mapsto \frac{r^2 x_i}{\sum_j x_j^2}$

where r is the radius of the inversion.

In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle.

As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is simply a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.

A remarkable fact about higher-dimensional conformal maps is that they arise strictly from inversions in n-spheres or hyperplanes and Euclidean motions: see Liouville's theorem (conformal mappings).

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 $\scriptstyle J \cdot J^T = k I$ and $\scriptstyle \det(J) = -\sqrt{k}.$ Computing the Jacobian in the case zi = xi/||x||2, where ||x||2 = x12 + ... + xn2 gives JJT = kI, with k = 1/||x||4, 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.

Inversive geometry and hyperbolic geometry

The (n − 1)-sphere with equation

$x_1^2 + \cdots + x_n^2 + 2a_1x_1 + \cdots + 2a_nx_n + c = 0$

will have a positive radius so long as a12 + ... + an2 is greater than c, and on inversion gives the sphere

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

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

$x_1^2 + \cdots + x_n^2 + 2a_1x_1 + \cdots + 2a_nx_n + 1 = 0,$

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 disc 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.

Notes

1. ^ Altshiller-Court (1925, p. 230)
2. ^ Kay (1969, p. 264)
3. ^ Kay (1969, p. 265)
4. ^ Kay (1969, p. 265)
5. ^ Kay (1969, p. 269)
6. ^ M. Pieri (1911,12) "Nuovi principia di geometria della inversion", Giornal di Matematiche di Battaglini 49:49–96 & 50:106–140
7. ^ Kasner, E. (1900). "The Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface". Transactions of the American Mathematical Society 1 (4): 430–498. doi:10.1090/S0002-9947-1900-1500550-1. JSTOR 1986367.
8. ^ Coxeter 1969, pp. 77–95
9. ^ A.S. Smogorzhevsky (1982) Lobachevskian Geometry, Mir Publishers, Moscow