Geography Reference
In-Depth Information
1.
To apply the Lagrange theorem to a trigonometric series;
2.
To write the inverse expansions of the rectifying, conformal and authalic latitude;
3.
To express the coefficients of the inverse expansions as a power series of the
eccentricity.
The detailed derivation processes are given by Li (2010). Confined to the length of the
chapter, they are also omitted. Comparing the results derived by this method with those in
3.1 and 3.2, one will find that they are all consistent with each other even though they are
also formulated in different ways. This fact substantiates the correctness of the derived
formulas, too.
3.4. Accuracies of the inverse expansions
The accuracies of the inverse expansions derived by Yang (1989, 2000) and the author has
been examined choosing the CGCS2000 reference ellipsoid.
The results show that the accuracy of the inverse expansion of rectifying latitude is higher
than 10 -5 ″, while the accuracy of the inverse expansion (32) derived by the author is higher
than 10 -7 ″. The accuracies of the inverse expansion of conformal and authalic latitudes
derived by Yang (1989, 2000) are higher than 10 -4 ″, while the accuracies of the inverse
expansions derived by the author are higher than 10 -8 ″. The accuracies of inverse expansions
derived by the author are improved by 2~4 orders of magnitude compared to those derived
by Yang (1989, 2000).
4. The direct expansions of transformations between meridian arc, isometric
latitude and authalic latitude function
The meridian arc, isometric latitude and authalic latitude function are functions of
rectifying, conformal and authalic latitudes correspondingly. The transformations between
the three variables are indirectly realized by computing the geodetic latitude in the past
literatures such as Yang (1989, 2000), Snyder (1987). The computation processes are tedious
and time-consuming. In order to simplify the computation processes and improve the
computation efficiency, the direct expansions of transformations between meridian arc,
isometric latitude and authalic latitude function are comprehensively derived by means of
Mathematica.
4.1. The direct expansions of transformations between meridian arc and
isometric latitude
4.1.1. The direct expansion of the transformation from meridian arc to isometric latitude
Inserting the known meridian arc X into (23) yields the rectifying latitude . Using the
inverse expansion of the rectifying latitude (32) and the forward expansion of the conformal
latitude (14), one obtains the conformal latitude . Inserting it into (9) yields the isometric
latitude q . The whole formulas are as follows:
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