Moving objects produce trajectories, which are stored in databases by means of finite samples of time-stamped locations. When

also speed limitations in these sample points are known,

beads can be used to model the

uncertainty about the object's location in between sample points.

In this setting, a query of particular interest, that has been

studied in the literature of geographic information systems (GIS),

is the alibi query. This boolean query asks whether two

moving objects can have physically met. This adds up to deciding

whether the necklaces of beads of these objects intersect. This

problem can be reduced to deciding whether two beads intersect.

Since, existing software to solve this problem fails to answer this

question within a reasonable time, we propose an analytical solution

to the alibi query, which can be used to answer the alibi query in

constant time, a matter of milliseconds or less, for two single

beads and in time proportional to the product of their lengths for

necklaces of beads.