Triangulating on a Sphere

  In Google Earth I selected two positions on the coast of the English Channel in the UK near Dover and another in France for a spherical geometry triangulation check. The positions and the arcs of the spherical triangle connecting them can be seen in the following figure (click to enlarge).

Setting placemarks in Google Earth allows to get the coordinates for the positions chosen to a fraction of a second of arc and the ruler can be used to read the headings of the connecting arcs to a small fraction of a degree. An estimate of the position of the Semaphore Station on the French side was estimated by determining the point of intersection for great circle arcs through the assumed known points on the English side and along the measured headings. A 3D plot of the spherical triangle is shown in the figure below. The arc lengths, α₁ and α₂, were found by doing a search for the values which minimized the distance between the two arcs.

The resulting position for X was within a few seconds of arc of the observed coordinates for the Semaphore station.