Geometrical characteristics of some typical spatial constraints

The invariants of some typical constraint ruled surfaces are extensively studied by means of the moving frame method of differential geometry. Constraint ruled surfaces correspond to combinations of certain kinematic pairs, such as the R–C, H–C, C′–C and C′–P–C. In this paper, the sufficient and necessary conditions are given for a ruled surface to be one of the above constraint ruled surfaces. Then, the contact order between two ruled surfaces along a common ruling is discussed, and contact conditions are derived for a ruled surface to contact a constraint ruled surface along their common ruling to a given order. As a result, some equations are derived for the lines of a moving body to lie on one of the above constraint ruled surfaces in several infinitesimally separated positions.