The positioning configuration optimization is a basic problem in surveying, and the geometric dilution of precision (GDOP) is a key index to handle this problem. Simplex graphs as regular polygons and regular polyhedrons are the well-known configurations with the lowest GDOP. However, it has been proved that there are at most five kinds of regular polyhedrons. We analytically solve the GDOP minimization problem with arbitrary observational freedom to extend the current knowledge.
