Pasting gauges I: Shortest paths across a hyperplane

We study the largest quasimetric in R ^d no greater than one given gauge on one side of a fixed hyperplane, and another gauge on the other side. We show that such a quasimetric is obtained from shortest paths consisting of at most three linear pieces, each measured by one of the two gauges. For any single-, double- or triple-link path exact optimality conditions are derived and studied in detail, expliciting exactly in which cases they arise as shortest paths. This yields explicit algorithms to construct shortest paths in general and between any two given points of R ^d . Several examples in the plane are fully analysed.