Optimal Map Projections by Variational Calculus: Harmonic Maps
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartograp
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Optimal Map Projections by Variational Calculus: Harmonic Maps
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartographer. For instance, in Sect. 5-25, we compute the minimumdistortion energy for mapping the “sphere-to-plane”. For maps of type (i) orthographic, (ii) conformal (UPS), (iii) gnomonic, (iv) equiareal (Lambert), (v) equidistant (Postel) and (vi) Lagrange conformal we computed the “best maps” by ordering of type (5.134) and (5.135). Another example is the optimal cylinder projection of the sphere in comparing (i) conformal maps, (ii) equiareal maps and (iii) distance preserving maps with respect to optimal of type Airy and Airy-Kavrajski. Section 10-3, Figs.10.7 and 10.8 lists the optimal. A final example is the optimal design of the Universal Transverse Mercator Projection (UTM), namely the optimal dilatation factor. Section 15-4, Examples 15.3 and 15.4 as well as Fig. 15.5 lists the result. 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 − L0 and relative isometric latitude q = Q − Q0 gauged to a vector-valued boundary condition of special symmetry. {Easting, Northing} or {x(b, l), y(b, l)} of the new harmonic map is given in Tables 22.18 and 22.21. The distortion energy analysis of the new harmonic map is presented as well as case studies for (i) B ∈ [−40◦ , +40◦], L ∈ [−31◦ , +49◦], B0 = ±30◦ , L0 = 9◦ and (ii) 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◦}
22-1 Introduction Harmonic maps are generated as a certain class of optimal map projections: Minimize the distortion energy over a closed and bounded surface or of a part like the Meridian Strip (in general, a two-dimensional Riemann manifold ) to find the Laplace-Beltrami vector-valued partial differential equation as the Euler-Lagrange equations by means of variational calculus. Those harmonic functions x(L, B), y(L, B) given as a function of surface parameters (L, B) specified later generate a harmonic map in the sense of {Δx = 0, Δy = 0} where“Δ”} is the two-dimension LaplaceBeltrami differential operator. Here we aim at generating a harmonic map x(L, B), y(L, B) of the ellipsoid of revolution more specifically the International Reference Ellipsoid as the representative E.W. Grafarend et al., Map Projections, DOI 10.1007/978-3-642-36494-5 22, © Springer-Verlag Berlin Heidelberg 2014
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