Infinity in Inversive Geometry

Inversive geometry is a field of study centered around the inversion - inversive transformation. Unlike, say projective geometry which is an axiomatic theory in its own right, inversive geometry is an outgrowth of the common Euclidean geometry augmented with an extra point - the point at infinity - to avoid the exceptional situation associated with the center of inversion.

Given a point O and a positive real number R. With every point P ≠ O, the inversion with center O and radius R associates another point P' that satisfies two conditions:

  1. O, P, and P' are collinear. In other words, P' lies on the line OP. Furthermore, P and P' are stipulated to lie on the same side from O.
  2. |OP| × |OP'| = R², where vertical bar denote the length of a segment in-between. (This is a common convention which is not adhered systematically at this site.)

Every point P ≠ O has a unique image P' under so defined transformation. Point O place an exceptional role for other reason as well. For example, an image of a circle under an inversion is a circle, except the circles that pass through O. The circles that contain O are mapped on straight lines, instead. Conversely, straight lines are mapped onto circles that contain O - the center of inversion.

Hmm. It would have been nice to be able to claim that but this is not quite so. As a matter of fact, the images of the straight lines thought of as sets of points are circles minus O, for no point in the common plane maps onto the center of inversion.

It may be a minuscule gap in a circle that misses a point but, being an exception, causes a theoretical inconvenience. And the solution? The solution is to fill the gap in the only natural way, i.e., by letting the center of inversion to be included in every circle that is said to pass through it. Making circles "complete" we now face the responsibility of finding a point that maps onto the center of inversion. Since |OP| = 0, for P = O, no finite P' may satisfy the second part of the definition: |OP| × |OP'| = R², for a finite R. At this point we invoke the mysterious but otherwise forbidden product, 0 × ∞ which elsewhere remains undefined. It would not be very consistent to declare that in case of inversion, 0 × ∞ = R² if several inversive transformation (with different radii) are employed at the same time. But, for a single one, accepting |OP| × |OP'| = R² even when P = O does not entail contradictions.

The Euclidean plane, as the play field for an inversion, augmented with a point at infinity is known as the inversive plane. In the inversive plane parallel lines meet at the point at infinity, the fact that has numerous applications, the most famous of which is the demonstration of Steiner's porism.

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