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This paper discusses how local measurements of three-dimensional positions and surface normals recorded by a set of tactile sensors may be used to identify and locate objects, from among a set of known objects. The objects are modeled as polyhedra having up to six degrees of freedom relative to the sensors. We show that inconsistent hypotheses about pairings between sensed points and object surfaces can be discarded efficiently by using local constraints on: distances between faces, angles between face normals, and angles (relative to the surface normals) of vectors between sensed points. We show by simulation that the number of hypotheses consistent with these constraints is small. We also show how to recover the position and orientation of the object from the sense data.