  We investigate theoretically the elliptical shapes of soft colloids in freely
standing smectic C films, that have been reported recently. The colloids favour
parallel alignment of the liquid crystal molecules at their surfaces and, for
sufficiently strong anchoring, will generate a pair of defects at the poles of
the colloidal particles. The elastic free energy of the liquid crystal matrix
will, in turn, affect the shape of the colloids. In this study we will focus on
elliptical soft colloids and determine how their equilibrium shapes depend on
the elastic constants of the liquid crystal, the anchoring strength, the
surface tension and the size of the colloids. A shape diagram is obtained
analytically, by minimizing the Frank elastic free energy, in the limit of
small eccentricities. The analytical results are verified, and generalized to
arbitrary eccentricities, by numerical minimization of an appropriate Landau
free energy. The latter is required for an adequate description of the
topological defects when the liquid crystal correlation length is comparable to
the size of the colloidal particles.
