Approximation Flows in Shape Manifolds

We consider manifolds of curves and surfaces which are controlled by certain systems of shape parameters. These systems may be given by the control points of a spline curve, the coefficients of an implicit equation, or other parameters controlling the shape. Each system of shape parameters corresponds to a chart of the manifold. In order to fit a curve or surface from such a manifold to given unorganized point data, we define an evolution process which takes an initial solution and modifies it in order to adapt it to the data. We show that this evolution defines a flow on the shape manifold. Consequently, the result of the evolution is independent of the particular choice of the shape parameters / of the chart. §