This paper studies the statics and the instantaneous kinematics of a rigid body constrained by one to six contacts with a rigid static environment. These properties are analyzed under the frictionless assumption by modeling each contact with a kinematic chain that, instantaneously, is statically and kinematically equivalent to the contact and studying the resulting parallel chain using the Grassmann-Cayley algebra. This algebra provides a complete interpretation of screw theory, in which twist and wrench spaces are expressed by means of the concept of extensor and its inherent duality reflects the reciprocity condition between possible twists and admissible wrenches of partially constrained rigid bodies. Moreover, its join and meet operators are used to compute sum and intersections of the twist and wrench spaces resulting from serial and parallel composition of motion constraints. In particular, it has an explicit formula for the meet operator that gives closed-form expressions of twist and wrench spaces of rigid bodies in contact. The Grassmann-Cayley algebra permits us to work at the symbolic level, that is, in a coordinate-free manner and therefore provides a deeper insight into the kinestatics of rigid body interactions.