Simple computation of coordinates and uncertainty of ‘hidden’ points from two coordinated points that are collinear with the hidden point

Abstract

Estimating the measurement uncertainties of coordinates of points that are not accessible by direct geodetic surveying (hidden points) is the topic of this study. The simplicity by which the uncertainties can be found for such hidden points when practicing indirect observations by total station or by Global Navigation Satellite System (GNSS) is demonstrated. A three-dimensional solution is obtained by total station observations using a two-prism hidden-point rod and a two-dimensional solution by GNSS observations on the ground. An intention with the study is that methods and results should apply for general geodetic purposes, like engineering surveying. The derived uncertainty formulae have been verified by Monte Carlo simulations. An important finding is their geometric dependence by the quotient between the two distances involved when two coordinated points are collinear with a third hidden point.