Apollonius Meets Computer Algebra

    The circles of Apollonius problem is a classic geometry problem dating from
Greek antiquity.  Although there are various generalizations and special cases,
the main problem was to find the circle(s) tangent to three given circles in
the plane.  Here we are concerned with higher dimensional generalizations.
First, to find equations for a sphere tangent to four given spheres in
three-space.  Secondly, to similarly find equations for a hypersphere tangent
to n+1 given hyperspheres in n-dimensional space.  Thirdly, to replace some of
these spheres with ellipsoids.

   The three-dimensional problem is motivated by work in trying to compute the
medial axis or surface (a sort of skeleton) of the space around molecules, with
eventual medical research applications.

   We will discuss computer algebra solutions based on Groebner bases and on
Dixon resultants.


Robert H. Lewis
Department of Mathematics
Fordham University
New York

Stephen Bridgett
Queen's University
Belfast