| In some circumstances, it is possible to make repeated measurements of the distances (ranges) to a finite set of points (landmarks), which are affixed to a distant rigid body. When the distances to all points on the body are much larger than the distances between points on the body, these distances are shown to approximately satisfy certain invariant equations. These invariant equations relate the observed ranges to the Euclidean geometry of the set of points, and are invariant to the orientation of the body.; When a sequence of observations of the ranges to four or more points is available, then except for the case when the motions of the body are artificially constrained to a lower dimensional set of degenerate conditions, a particular set of invariant equations may be used to uniquely solve for the Euclidean shape of the configuration of points. Explicit, closed form formulas are derived.; For range measurements contaminated with additive noise, naive estimates of the Euclidean invariants are shown to be biased, but given independent estimates of the moments of the noise process, bias corrections for the estimates of the Euclidean invariants can be derived. Also, large sample estimates of the variances of the estimated Euclidean invariants can be derived. |
