Applied Mathematics Seminar

We use Reduce to investigate the mathematics of an idealised form of the Global Positioning System (GPS) in which the speed of propagation of radio signals is constant and discover, by first considering Flatland GPS, that the compact form of the determining equations for the location of the GPS receiver requires the use of Cayley-Menger determinants involving a set of Lorentz invariant quantities. We also find that the idealised GPS problem is isomorphic in structure to a problem first considered in a special case by Descartes, namely, given four spheres with arbitrary centres and radii, to find a sphere which touches the four given spheres. The Lorentz invariant quantities of the GPS problem are found to have a geometrical meaning in terms of tangents common to the four given spheres. Finally we unpack the mathematics that a well-prepared Tripos Candidate of the 1880's would have used to treat this problem.

Note: There are no sophisticated mathematical or physical prerequisites for this talk.