A new approach to subdivision based on the evolution of surfaces under
curvature motion is presented. Such an evolution can be understood
as a natural geometric filter process where time corresponds to the
filter width. Thus, subdivision can be interpreted as the application
of a "geometric" filter on an initial surface. The concrete scheme
is a model of such a filtering based on a successively improved
spatial approximation starting with some initial coarse mesh and
leading to a smooth limit surface.
In every subdivision step the underlying grid is refined by some
regular refinement rule and a linear finite element problem is either
solved exactly or, especially on fine grid levels, one confines to a
small number of smoothing steps within the corresponding iterative
linear solver. The approach closely connects subdivision to surface
fairing concerning the geometric smoothing and to cascadic multigrid
methods with respect to the actual numerical procedure. The derived
method does not distinguish between different valences of nodes nor
between different mesh refinement types. Furthermore, the method
comes along with a new approach for the theoretical treatment of
subdivision.
