Exotic Steiner Chains in Miquelian M\"obius Planes

Abstract

In the Euclidean plane, two intersecting circles or two circles which are tangent to each other clearly do not carry a finite Steiner chain. However, in this paper we will show that such exotic Steiner chains exist in finite Miquelian M\"obius planes of odd order. We state and prove explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles $C_1$ and $C_2$ for the existence, length, and number of Steiner chains carried by $C_1$ and $C_2$.