This paper introduces mathematical models of \sensors\ for mobile robots based on visibility. Serving a purpose similar to the pinhole camera model for computer vision, the introduced models are expected to provide a useful, idealized characterization of task-relevant information that can be inferred from their outputs or observations. Possible tasks include navigation, localization and mapping when a mobile robot is deployed in an unknown environment. These models allow direct comparisons to be made between traditional depth sensors, highlighting cases in which touch sensing may be interchangeable with time of flight or vision sensors, and characterizing unique advantages provided by touch sensing. The models include contact detection, compression, load bearing, and deflection. The results could serve as a basic building block for innovative touch sensor designs for mobile robot sensor fusion systems.