Spatial reasoning with incomplete information on relative positioning

This paper describes a probabilistic method of inferring the position of a point with respect to a reference point knowing their relative spatial position to a third point. We address this problem in the case of incomplete information where only the angular spatial relationships are known. The use of probabilistic representations allows us to model prior knowledge. We derive exact formulae expressing the conditional probability of the position given the two known angles, in typical cases: uniform or Gaussian random prior distributions within rectangular or circular regions. This result is illustrated with respect to two different simulations: the first is devoted to the localization of a mobile phone using only angular relationships, the second, to geopositioning within a city. This last example uses angular relationships and some additional knowledge about the position.