Matching a surface to a given sectional curve

We describe an algorithm to position a rigid surface so as to make its cross-section by a given plane match a given curve in that plane, a problem relevant to model-based medical imaging. After building an atlas of cross-sections of the surface and searching it for a best position to start from, each iteration of the algorithm (1) determines a vector field along the intersection curve that will improve its matching with the target curve, and (2) computes and applies a small displacement of the surface whose effect on the intersection will approximate best the required vector field. Computations use least-square techniques, an exponential formula for Lie groups of transformations, and generic properties of cross-sections. Experiments with an implementation are reported and theoretical tools for justifying and improving the algorithm, some of them based on Catastrophe Theory, are outlined.