  Evolution of planar curves under a nonlocal geometric equation is
investigated. It models the simultaneous contraction and growth of carbonate
particles called ooids in geosciences. Using classical ODE results and a
bijective mapping we demonstrate that the steady parameters associated with the
physical environment determine a unique, time-invariant, compact shape among
smooth, convex curves embedded in $R^2$. It is also revealed that any
time-invariant solution possesses $D_2$ symmetry. The model predictions
remarkably agree with ooid shapes observed in nature.
