Harmonic maps

Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L₀ and relative isometric latitude q=Q–Q₀ gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B–40°,+40°], L–31°,+49°], B₀= ±30°, L₀=9° and (2) B46°,56°], L{4.5°, 7.5°]; 7.5°, 10.5°]; 10.5°,13.5°]; 13.5°,16.5°]}, B₀= 51°, L₀ {6°,9°,12°,15°}.