Steiner's Porism in finite Miquelian M\"obius planes

TL;DR

This paper explores Steiner's Porism within finite Miquelian M"obius planes, establishing conditions for Steiner chains, introducing the concept of capacitance, and comparing finite and classical Euclidean cases.

Contribution

It introduces a finite version of Steiner's Porism, formulates conditions on chain length using quadratic residues, and generalizes results via the invariant notion of capacitance.

Findings

Finite Steiner's Porism for concentric circles proved.

Conditions on chain length derived using quadratic residues.

Capacitance introduced as an invariant for arbitrary circle pairs.

Abstract

We investigate Steiner's Porism in finite Miquelian M\"obius planes constructed over the pair of finite fields $GF(p^m)$ and $GF(p^{2m})$, for $p$ an odd prime and $m \geq 1$. Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner's Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in $GF(p^m)$. These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under M\"obius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.