Geography Reference
In-Depth Information
results
indicate that the derivation efficiency can be significantly improved and formulas
impossible to be obtained by hand can be easily derived with the help of Mathematica,
which renovates the traditional analysis methods and enriches the mathematical theory
basis of cartography to a certain extent.
The main contents and research results presented in this chapter are organized as follows. In
Section II, we discuss the direct transformations from geodetic latitude to three kinds of
auxiliary latitudes often used in cartography, and the direct transformations from these
auxiliary latitudes to geodetic latitude are studied in Section III. In Section IV, the direct
expansions of transformations between meridian arc, isometric latitude, and authalic functions
are derived. In Section V, we discuss the non-iterative expressions of the forward and inverse
Gauss projections by complex numbers. Finally in Section VI, we make a brief summary. It is
assumed that the readers are somewhat conversant with Mathematica and its syntax.
2. The forward expansions of the rectifying, conformal and authalic
latitudes
Cartographers prefer to adopt sphere as a basis of the map projection for convenience since
calculation on the ellipsoid are significantly more complex than on the sphere. Formulas for
the spherical form of a given map projection may be adapted for use with the ellipsoid by
substitution of one of various “auxiliary latitudes” in place of the geodetic latitude. In using
them, the ellipsoidal earth is, in effect, transformed to a sphere under certain restraints such
as conformality or equal area, and the sphere is then projected onto a plane (Snyder, 1987). If
the proper auxiliary latitudes are chosen, the sphere may have either true areas, true
distances in certain directions, or conformality, relative to the ellipsoid. Spherical map
projection formulas may then be used for the ellipsoid solely with the substitution of the
appropriate auxiliary latitudes.
The rectifying, conformal and authalic latitudes are often used as auxiliary ones in
cartography. The direct transformations form geodetic latitude to these auxiliary ones are
expressed as transcendental functions or non-integrable ones. Adams (1921), Yang (1989,
2000) had derived forward expansions of these auxiliary latitudes form geodetic one
through complicated formulation. Due to historical condition limitation, the derivation
processes were done by hand and orders of these expansions could not be very high. Due to
these reasons, the forward expansions for these auxiliary latitudes are reformulated by
means of Mathematica. Readers will see that new expansions are expressed in a power
series of the eccentricity of the reference ellipsoid
e
and extended up to tenth-order terms of
e
. The expansion processes become much easier under the system Mathematica.
2.1. The forward expansion of the rectifying latitude
B
to
B
is
0
The meridian arc from the equator
B
2
2
2
3/ 2
Xa
(1
e
)
(1
e
sin
B
)
dB
(1)
0
