Fitting a surface to one of its sectional planar curves using adaptive trees in spaces of curves

We motivate from interventional medical imaging some geometric problems where one should identify how a curve could be generated from a known surface, asking for the pose of the surface from the outline of an orthogonal projection of it, or the position of a sectional plane from an isometric instance of the sectional curve. We describe a distance between curves that is efficiently implementable and use it to build stochastic trees in metric spaces towards subsets of curves of interest that either have strong convergence properties or permit quick searches on curves, and propose a real time application on a simplified version of such problem.