  We propose a geometrically motivated mathematical model which reveals the key
features of coastal and fluvial fragment shape evolution from the earliest
stages of the abrasion. Our \textit{collisional polygon model} governs the
evolution through an ordinary differential equation (ODE) that determines the
rounding rate of initially sharp corners in the function of the size reduction.
As an approximation, the basic structure of our model adopts the concept of
Bloore's partial differential equation (PDE) in terms of the curvature
dependent local collisional frequency. We tested our model under various
conditions and made comparisons with the predictions of Bloore's PDE. Moreover,
we applied the model to discover and quantify the mathematical conditions
corresponding to typical and special shape evolution. By further extending our
model to investigate the self-dual and mixed cases, we outline a possible
explanation of the long-term preservation of initial pebble shape
characteristics.
