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Auxiliary Latitude Formulas: Finding the Coefficients Numerically and Symbolically
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2006 Wolfram Technology Conference
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Champaign IL
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In the theory of map projections based on the ellipsoid model of the earth, several quantities akin to latitude are defined. The version used by geographers and cartographers is “geodetic latitude” (ϕ), but these others (θ, ψ, χ, β, ω) provide the theorist and algorithm developer with the right way to think about the spacing of the parallels for various map projections. They have other purposes as well. They are all zero at the equator and ±90° at the poles.
A plot of the difference of two of these latitudes versus one of them strongly suggests the idea of fitting this relationship to a series of the form a where k is an even number that indexes the summation. This has been done since the 1920s in the work of Oscar Adams, who obviously worked by hand. Now, with the help of Mathematica, many more coefficients a can be computed, both numerically and symbolically, and more accurately. The choices of methods are (1) a series in powers of the ellipsoid’s eccentricity, and (2) a previously problematic numerical integration.
A nifty algorithm for the transverse Mercator map projection requires many a for the formula between conformal latitude and meridional rectifying latitude. This algorithm could not have been developed before the advent of symbolic manipulation of series, like Mathematica provides.
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map projection, ellipsoidal model, Mercator map, conformal latitude, meridional rectifying latitude
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| AdamsO_hist_doc.jpg (227.5 KB) - Supporting image | | Aux_Lat_slides.nb (3.4 MB) - Mathematica Notebook [for Mathematica 5.2] |
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