This work proposes a theoretical foundation for a general spatial geometric mechanism-environment contact model. In the proposed model the curvature of the environment in the vicinity of the contact is approximated by a number of spherical surfaces with known radii of curvature that constrain/define the movement of the body. We show how the modeled body-environment contact and curvature constraints can be transformed into conditions on spatial velocity and acceleration (i.e. first and second order effects) of certain points of the moving body that can be incorporated in the kinematic task for designing spatial mechanisms. Further, we explore the exact synthesis of a spatial six degrees-of-freedom TPS kinematic chain which end-effector maintains contact with objects in the environment and varies orientation in the vicinity of a contact location. It is discussed how the higher order motion constraints allow for the introduction of kinematic task variations in the vicinity of a contact, resulting in different behaviors of the designed spatial mechanism. The theoretical foundation presented in this paper is crucial in gaining understanding of the constraints in describing mechanism-environment interactions in the vicinity of a contact and is a new contribution.