Inversive geometry

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In geometry, inversive geometry is the study of transformations that map circles into circles, where by a circle one may also mean a line (a circle with infinite radius).

Contents

[edit] Circle inversion

[edit] Inverse of a point

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

In the plane, the inverse of a point P with respect to a circle of center O and radius R is a point P' such that P and P' are on the same ray going from O, and OP times OP ' equals the radius squared,

OP\times OP'=R^2.

This circle with respect to which inversion is performed will be called the reference circle.

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 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.
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.
A procedure to construct the inverse P' of a point P outside a circle C. Let r be the radius of C. Since the triangles OPN and OP'N are similar, OP is to r as r is to OP'.
A procedure to construct the inverse P' of a point P outside a circle C. Let r be the radius of C. Since the triangles OPN and OP'N are similar, OP is to r as r is to OP'.

It follows from the definition that the inverse of a point inside the reference circle is outside the reference circle and vice-versa. A point on the circle stays in the same place under inversion. The center of the circle gets transformed to the point at infinity, which is transformed back to the center of the circle. In summary, the closer a point is to the center, the further away its transformation is, and vise-versa. This inversive relationship between points P and P' is the reasoning for this transformation's name.

[edit] Properties

One may invert a set of points with respect to a circle by inverting each of the points which make it up. The following properties are what make circle inversion important.

  • A line not passing through the center of the reference circle is inverted into a circle passing through the center of the reference circle, and vice versa; whereas a line passing through the center of the reference circle is inverted into itself.
  • A circle not passing through the center of the reference circle is inverted into a circle not passing through the center of the reference circle. The circle (or line) after inversion stays as before if and only if it is orthogonal to the reference circle at their points of intersection.

[edit] Application

Note that the center of a circle being inverted and the center of the circle as result of inversion are collinear with the center of the reference circle. This fact could be useful in proving 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.

[edit] 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 OP\times OP'=R^2 and the points P and P' are on the same ray going from 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.

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.

[edit] Transformation theory

According to Coxeter[1], the transformation by inversion in circle was invented by L. J. 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

[edit] 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 \to R^2 \frac {x} {|x|^2} = y \to T^2 \frac {y} {|y|^2} = ( \frac {T} {R} )^2 \ x .

[edit] Reciprocation

When a point in the plane is interpreted as a complex number z = x + iy = r exp(ai), with complex conjugate z* = x - iy, then the reciprocal of z is z*/|z|2. Consequently the algebraic form of the inversion in a unit circle is

w = 1/z* = (1/z)* .

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.

[edit] 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 Arthur Cayley, Felix Klein, and Henri Poincaré. 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 the group of mappings of some space characterized by a group invariant, a measure, like distance or angle.

[edit] 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).

[edit] Inversion of an algebraic curve

We may invert a plane algebraic curve given by a single polynomial equation f(x, y) = 0 by setting

u = \frac{x}{x^2+y^2},\ v=\frac{y}{x^2+y^2}.

Clearing denominators, we have the polynomial equations ux2 + uy2 − x = 0,vx2 + vy2 − y = 0, and eliminating x and y from the system of three equations in four unknowns consisting of these two equations and f (for instance, by using resultants) we can readily find the equation of the curve inverted in the unit circle. Now x=u/(u^2+v^2),\ y=v/(u^2+v^2) and applying the transformation again leads back to the original curve.

For example, applying the above transformation to the lemniscate

(x^2 + y^2)^2 = a^2 (x^2 - y^2)\,

gives us

a^2(u^2-v^2) = 1,\,

the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply it to the Fermat curve xn + yn = 1, where n is odd, we obtain

(u^2+v^2)^n = u^n+v^n.\,

Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.

[edit] 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 J \cdot J^T = k I and \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.

[edit] Inversive geometry and hyperbolic geometry

The (n − 1)-sphere with equation

x_1^2 + \cdots + x_n^2 + 2a_1x + \cdots + 2a_nx + 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 + \cdots + 2\frac{a_n}{c}x + \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 + \cdots + 2a_nx + 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.

[edit] References

  1. ^ H.S.M. Coxeter (1961) Introduction to Geometry, Chapter 6: Circles and Spheres (pp.77-95), John Wiley & Sons.

[edit] External links