It is natural to call inversion the function that associates to each real number greater than zero its inverse \(x\rightarrow\frac{1}{x}\).

This function sends 1 to 1: 1 is a fixed point of the inversion. The inversion sends the half-line that represents the numbers bigger than zero to itself, switching the red half-line to the blue segment:

The product of the distance to the origin of any point and its inverse is (always) 1.

If we want to extend this concept to the plane or space, we can define a inversion with centre at a point \(O\), applying to each half-line of origin \(O\) the idea described above. The inverse of a point \(P\) diferent of \(O\) is the point \(P'\) of the half-line \(OP\), such that \(\overline{OP}.\overline{OP'}=1\).

In the plane, all the points of the unit circle centered at \(O\) become fixed by this transformation: this circle is called the inversion circle. The inversion exchange the blue and violet regions:

Similarly, we have in the space an inversion spherical surface of radius 1 and centre \(O\), consisting of fixed points for the inversion.

Remark: It is possible to consider, more generally, inversion circles of radius different form 1. If \(C\) is a circle with centre \(O\) and radius \(r\), the inverse of a point \(P\) different from \(O\), with respect to \(C\), is the point \(P'\) of the half-line \(OP\), such that \(\overline{OP}.\overline{OP'}=r^{2}\). (Analogously, we can define inversion in the space, relative to a any spherical surface.)

Some properties of inversion in the plane

Any line that does not pass through the inversion centre \(O\) is sent to a circle passing through the inversion centre (the only point of that circle which is not inverse to a point of the line is \(O\)). If the line passes through \(O\), \(O\) determines in the line two open half-lines (with extreme \(O\)). Each one of those half-lines is sent by the inversion to itself, exchanging the part which is within the circle with the part which is outside.

Any circle that does not pass through the inversion center \(O\) is sent into a circle. In the case of the circle passing through \(O\), their points different from \(O\) will have as inverse (all) the points of a line.

The inversion preserves angles. In particular, the inversion preserves tangency and orthogonality. There are plans to insert here the proofs of this and other properties of the inversion mentioned above. For those who are familiar with the properties of complex numbers and analytic functions, some of the properties can be deduced in a very simple and elegant way from an alternative definition of inversion.