I have been revisiting math from my highschool through undergrad. I picked Courant’s excellent What is Mathematics?

The flow is well so far. However, in one of the chapters he introduces inversion - mostly w.r.t. the circle. This seemed a bit abrupt especially since I was never introduced to this concept. I’m aware of mapping planes to spheres for topological purposes etc.

What I’m try to understand is why was this investigated and who started it? Some reference (Googling) point to Apollinius of Perga and then jump straight to someone in 1911.

The formula for circle inversion:

$OP \cdot OP’ = r^2$

Where $O$ is the center and $P$ is a point inside the circle and $P’$ the inverse.

This formula seems rather contrived and doesn’t make intuitive sense to me. Why is this formula used and who investigated inversion and for what purpose? Any interesting references?


1 Answer 1


The wiki page and links seem good except maybe on history. For that see e.g. Coolidge (1940, pp. 278-282):

§2. Circular transformations.

The point transformations of the plane which carry points on a line or circle into points on a line or circle, are surely the simplest and most interesting plane transformations, after the linear ones. The simplest of these is circular inversion (...) $$ OP.OP'=r^2,\qquad x'=\frac{r^2x}{x^2+y^2},\qquad y'=\frac{r^2y}{x^2+y^2}. $$ We saw (pp. 64-65) that this transformation was first mentioned by Pappus, who knew that it carried a line or circle into a line or circle. It has been discovered subsequently by several writers, the first being perhaps Steiner (in) an unpublished manuscript (...)

It is truly astonishing what can be accomplished by the use of this transformation. If two circles intersect they can be inverted into two straight lines. If they touch they can be inverted into two parallel lines. If they have no common point we can find two intersecting circles cutting both of them orthogonally. Hence we may invert the first two into two circles that cut two intersecting lines orthogonally, i.e. into two concentric circles. The properties of systems of circles tangent to two given circles are most easily found by means of these transformations.

(Thus the relation with Apollonius’s problem.)

Circular inversion is but the simplest of all the transformations that carry a line or circle into a line or circle. The first writer to undertake a systematic study of transformations so defined, and by purely geometric means, was Möbius in his Theorie der Kreisverwandschaft which appeared in 1855 (:)

Möbius' Theorem 4]  Every one-to-one transformation of the Euclidean plane that carries points of a line or circle into points of a line or circle can be factored into reflections and inversions and is conformal.

(...) That enables us to write the general analytic form for the transformation $$ z'=\frac{\alpha z+\beta}{\gamma z+\delta}\qquad\text{(2a)},\qquad z'=\frac{\alpha \bar z+\beta}{\gamma \bar z+\delta}\qquad\text{(2b)},\qquad (\alpha\delta-\beta\gamma\ne0). $$

(i.e. one of two types, where $z=x+iy$; etc.) Earlier Coolidge had written in (1916, p. 22):

This transformation is usually credited to Plücker. See his Analytisch-geometrische Aphorismen, Crelle, vol. xi, 1836. It was rediscovered a decade later by Sir William Thompson, Principe des images électriques, Liouville, vol. x, 1845. The most recent view, however, seems to be that the method was found some time previous by Steiner.

  • $\begingroup$ Conicidentally, I ordered the book you’ve linked above last week and is yet to arrive. Glad I did. This is a very useful answer. Appreciate you taking the time writing this out. $\endgroup$
    – PhD
    Oct 18, 2018 at 1:54

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