Abstract: In the analytical mechanics, when two rigid surfaces with smooth shape keep contact at one point without friction or in the pure rolling state, two conditions are considered to deduce the ideal constraints for the contact. In the frictionless case, the possible infinitesimal displacements of the two contact points on the two surfaces are equal in the normal direction. In the pure rolling case, the relative velocity of the contact point is zero. In this paper, starting from the geometric conditions of taking the same position in space and in the tangent state, the preconditions are more clear and rigorous.