Singularities of motions constrained by contacting surfaces

We study motions where a surface in the moving system always touches a surface in the fixed system. It is shown that a singularity in configuration space occurs exactly at positions, where the two surfaces share normal curvature and conjugate direction to some tangent direction at the common point of contact. The case of rolling surfaces is also investigated and applied to the construction of connecting developable surfaces and pipe surfaces. Moreover, an application of the results to the geometry of five-axis sculptured surface machining is given.