Objectives Distances are functions of spatial positions. Precisely revealing the functional relationship which quantitatively embodies the transmission of uncertainty from spatial positions to their distance, a key scientific problem in need of being solved urgently in geomatics, has important theoretical and practical significance.
Methods Aiming at the limitation of presently available solution of the above mentioned problem, under the premise of that the real position corresponding with the observed one of an uncertain point follows the complete spatial random distribution within the error circle, we have derived the probability distribution function of the distance uncertainty and the corresponding density function containing an uncertain point and those between two uncertain points respectively in two-dimensional space. The latter has been employed to explore the transmission law of point uncertainties to distance uncertainties, opening up a new way for studying and solving the problem of distance uncertainties.
Results The results show that for all cases: (1) When the radius of the error circle (corresponding to the point position accuracy) and the observed distance between points change simultaneously, their ratio has a significant positive correlation with the level of distance uncertainties. (2) When the former remains constant, the distance uncertainty has a significant negative correlation with the latter. (3) When the latter remains constant, the distance uncertainty has a significant positive correlation with the former.
Conclusions As far as the distance uncertainty of cases containing an uncertain point and the one of those between two uncertain points are concerned, the latter is obviously greater than the former when the radius of the error circle and the observed distance between points are consistent for both of them. Otherwise they are not comparable.