Discrete differential geometry (DDG) explores discrete counterparts of classical differential geometry, such that the classical smooth theory arises in the limit of refinement of the discrete one. The tools of DDG offer a variety of exciting applications—ranging from geometry processing and computational geometry over computer graphics to physical simulations—you may want to see our publications to get a feel.
Discrete differential geometry can be said to have arisen from the observation that when a notion from smooth geometry (such as the notion of a minimal surface) is discretized properly, the discrete objects are not merely approximations of the smooth ones, but have special properties of their own which make them form in some sense a coherent entity by themselves. The discrete theory would seem to be the more fundamental one: The smooth theory can always be recovered as a limit, while there seems to be no natural way to predict from the smooth theory which discretizations will have the nicest properties.
— Oberwolfach Report 12/2006.