The centres of the osculating circles (red balls) of all curves (black) on a given surface passing through a given point, which have a common tangent at this point (blue), are located on a circle (red). The osculating circles (green) form a sphere. Here, the surface is a elliptic paraboloid; some of its level curves are shown.

Johannes Kepler University Linz, Institute for Applied Geometry, Altenberger Str.69, Austria-4040 Linz