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

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.