Ambiguities in a problem in planar geodesy (Q2339247)

This paper revisits a problem of planar geodesy: Given unknown target points \(p_1,\ldots,p_t\) and unknown measure points \(q_1,\ldots,q_m\), determine their relative position up to similarity from the angle measures \(\sphericalangle(p_i,q_j,p_k)\). After a suitable algebraic re-formulation, the authors associate an algebraic surface \(S_{\vec{p}}\) to the \(t\)-tuple \(\vec{p} = (p_1,\ldots,p_t)\). This surface is called \textit{profile} and encodes all possible angle measures for given \(\vec{p}\). Similarly, the \textit{co-profile} \(S'_{\vec{q},p_1}\) encodes all angle measures for given \(m\)-tuple \(\vec{q} = (q_1,\ldots,q_m)\) plus one point \(p_1\). In a first step, the authors discuss the problem of determining the profile from measure points or the co-profile from target points. The most interesting cases are \(m = 3\), \(t = 5\) where in general two profiles exist and \(m = t = 4\) where in general two co-profiles exist. Point configurations on cyclic cubics may lead to more than one solution even for a higher number of target and measure points. In a second step, the authors determine target points and measure points (up to similarity) from measurements. For sufficiently many target and measure points, this problem has in general a unique solution. Again, ambiguous configurations are related to cyclic cubics. For \(t = 4\) or \(m = 3\), there are in general two solutions and this ambiguity cannot be resolved by adding more measurements or targets. The authors' approach appeals to results from algebraic geometry and completely avoids tedious calculations. Novel results pertain to exceptional point configurations and also to the elementary geometry of non-cyclic quadrilaterals. The approach via algebraic geometry may well turn out to be even more useful in the corresponding spatial problem.