This article presents a new method to find the best spatial registration between a rigid curve and a rigid surface. We show how to locally exploit the knowledge of differential properties computed on both the curve and the surface to constrain the rigid matching problem. It is in fact possible to write a compatibility equation between a curve point and a surface point, which constrains completely the 6 parameters of the sought rigid displacement. This requires the local computation of third order differential quantities and leads to an algebraic equation of degree 16. A second approach consists in considering pairs of curve and surface points. It is then possible to use only first order differential constraints to compute locally the parameters of the rigid displacement. Each approach leads to a different matching algorithm. Although computationally more expensive, the second approach is more robust and can be accelerated with a preprocessing of the surface data. The paper presents the mathematical details of both approaches, algorithms and a preliminary experimental study on both synthetic and real 3D medical data. To our knowledge, it is the first method which takes full advantage of local differential computations to register a curve on a surface.