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In geometry, an inversion is a particular type of transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius).

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

In the plane, the inverse of a point P in 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,

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

The inverse in 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 in 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'.

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.

One may invert a set of points in respect to a circle by inverting each of the points which make it up. The following properties is what makes 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.

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.

In addition, two dimensional inversion can be extended to 3-dimensional by making use of a sphere instead.

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 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. Indeed, 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 in respect to the North Pole N of B. Consider a sphere B₂ of radius 2 centered at N. The inversion in respect to B₂ transforms B into its stereographic projection P.

In the spirit of the Erlangen program, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversions, which in coordinate form, basically are conjugate to

where r is the radius of the inversion.

In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle. In the complex plane this corresponds to taking the reciprocal of the conjugate.

As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.

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

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 and applying the transformation again leads back to the original curve.

For example, applying the above transformation to the lemniscate

(x² + y²)² = a²(x² − y²)

gives us

a²(u² − v²) = 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

(u² + v²)ⁿ = uⁿ + vⁿ.

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

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 and 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||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.

The (n − 1)-sphere with equation

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

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

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.

• David E. Blair (2000) Inversion Theory and Conformal Mapping, American Mathematical Society, ISBN 0-8218-2636-0.

• Inversion: Reflection in a Circle at cut-the-knot
• Wilson Stother's inversive geometry page
• IMO Compendium Training Materials practice problems on how to use inversion for math olympiad problems
• Eric W. Weisstein, Inversion at MathWorld.