  Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to
be in contact if $|p-q|^2 = (r-s)^2$. Using Lie's line-sphere correspondence,
we show that if $F$ is a field in which $-1$ is not a square, then there is an
isomorphism between the set of spheres in $F^3$ and the set of lines in a
suitably constructed Heisenberg group that is embedded in $(F[i])^3$; under
this isomorphism, contact between spheres translates to incidences between
lines.
  In the past decade there has been significant progress in understanding the
incidence geometry of lines in three space. The contact-incidence isomorphism
allows us to translate statements about the incidence geometry of lines into
statements about the contact geometry of spheres. This leads to new bounds for
Erd\H{o}s' repeated distances problem in $F^3$, and improved bounds for the
number of point-sphere incidences in three dimensions. These new bounds are
sharp for certain ranges of parameters.
