The sense of touch and the capability to analyze potential contacts is important to many interactions of robots, such as planning exploration, handling objects, or avoiding collisions based on sensing of the environment. It is a pleasant surprise that the mathematics of touching and contact can be developed along the same algebraic lines as that of linear systems theory. In this paper we exhibit the relevant spectral transform, delta functions and sampling theorems. We do this mainly for a piecewise representation of the geometrical object boundary by Monge patches, i.e. in a representation by (umbral) functions. For this representation, the analogy with the linear systems theory is obvious, and a source of inspiration for the treatment of geometric contact using a spectrum of directions.