  In this paper we define a family of nonlinear, stationary, interpolatory
subdivision schemes with the capability of reproducing conic shapes including
polynomials upto second order. Linear, non-stationary, subdivision schemes do
also achieve this goal, but different conic sections require different
refinement rules to guarantee exact reproduction. On the other hand, with our
construction, exact reproduction of different conic shapes can be achieved
using exactly the same nonlinear scheme. Convergence, stability, approximation
and shape preservation properties of the new schemes are analyzed. In addition,
the conditions to obtain $\mathcal{C}^1$ limit functions are also studied.
