Harmonic Maps

Abstract

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. Here we construct harmonic functions x(L, B), y(L, B) given as functions of ellipsoidal surface parameters (L, B) of type {Gauss ellipsoidal longitude L, Gauss ellipsoidal latitude B} as well as x(ℓ, q), y(ℓ, q) given as functions of relative isometric longitude ℓ = L−L ₀ and relative isometric latitude q = Q−Q ₀ gauged to a vector-valued boundary condition of special symmetry. {Easting, Northing} or {x(b, ℓ), y(b, ℓ)} and the distortion energy analysis of the new harmonic map are presented as well as case studies for (i) B∈[−40°,+40°], L∈[−31°,+49°], B ₀=±30°, L ₀ =9° and (ii) B∈ [46°, 56°], L∈{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°, 13.5°}; [13.5°, 16.5°]}, B ₀=51°, L ₀ ∈ {6°, 9°, 12°, 15°}.