A platform for research: civil engineering, architecture and urbanism
A platform for research: civil engineering, architecture and urbanism
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. 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(ℓ,q), y(ℓ,q) given as functions of relative isometric longitude ℓ=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,ℓ),y(b,ℓ)} 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) B∈[46°,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°}.
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. 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(ℓ,q), y(ℓ,q) given as functions of relative isometric longitude ℓ=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,ℓ),y(b,ℓ)} 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) B∈[46°,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°}.
Harmonic maps
Grafarend, E.W. (author)
Journal of Geodesy ; 78
2005
Article (Journal)
Electronic Resource
English
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