The geometry of space-time prisms with uncertain anchors

Abstract

Space-time prisms envelop all spatio-temporal locations that moving objects may have visited between two of their known spatio-temporal locations, given a bound on their travel speed. In this context, the known locations are often the result of observations or measurements, and they are called ‘anchor points’. The classic space-time prism, in isotropic two-dimensional space, as well as in transportation networks, assumes that the measurements of these anchor points are exact. Whereas, in many applications, we can assume that time can be measured fairly precisely, this assumption is unrealistic for the spatial components of measured locations (we think of Global Positioning System (GPS) errors, for instance). In this paper, we extend the classical prism from anchor points to circular ‘anchor regions’ that capture the uncertainty or error on their measurement. We define the notion of a space-time prism with uncertain anchor points, called uncertain prism, for short. We study the geometry of uncertain prisms in an arbitrary metric space to make this concept as widely applicable as possible. We also focus on the rims of uncertain space-time prisms, which demarcate the area that a moving object can have visited between two anchor regions (given some local speed limitations).