Geography Reference
In-Depth Information
Table 2 shows that the eighth-orders terms of
e
in coefficients given by Yang(1989, 2000) are
erroneous.
2.4. Accuracies of the forward expansions
In order to validate the exactness and reliability of the forward expansions of rectifying,
conformal and authalic latitudes derived by the author, one has examined their accuracies
choosing the CGCS2000 (China Geodetic Coordinate System 2000) reference ellipsoid with
parameters
f
(Chen, 2008; Yang, 2009), where
f
is the
flattening. The accuracies of the forward expansions derived by Yang (1989, 2000) are also
examined for comparison. The results show that the accuracy of the forward expansion of
rectifying latitude derived by Yang (1989, 2000) is higher than 10
-5
″, while the accuracy of
the forward expansion (5) derived by the author is higher than 10
-7
″. The accuracies of the
forward expansion of conformal and authalic latitudes derived by Yang (1989, 2000) are
higher than 10
-4
″, while the accuracies of the forward expansions derived by the author are
higher than 10
-8
″ . The accuracies of forward expansions derived by the author are improved
by 2~4 orders of magnitude compared to forward expansions derived by Yang (1989, 2000).
a
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3. The inverse expansions of rectifying, conformal and authalic latitudes
The inverse expansions of these auxiliary latitudes are much more difficult to derive than
their forward ones. In this case, the differential equations are usually expressed as implicit
functions of the geodetic latitude. There are neither any analytical solutions nor obvious
expansions. For the inverse cases, to find geodetic latitude from auxiliary ones, one usually
adopts iterative methods based on the forward expansions or an approximate series form.
Yang (1989, 2000) had given the direct expansions of the inverse transformation by means of
Lagrange series method, but their coefficients are expressed as polynomials of coefficients of
the forward expansions, which are not convenient for practical use. Adams (1921) expressed
the coefficients of inverse expansions as a power series of the eccentricity
e
by hand, but
expanded them up to eighth-order terms of
e
at most. Due to these reasons, new inverse
expansions are derived using the power series method by means of Mathematica. Their
coefficients are uniformly expressed as a power series of the eccentricity and extended up to
tenth-order terms of
e
.
3.1. The inverse expansions using the power series method
The processes to derive the inverse expansions using the power series method are as
follows:
1.
To obtain their various order derivatives in terms of the chain rule of implicit
differentation;
2.
To compute the coefficients of their power series expansions;
3.
To integrate these series item by item and yield the final inverse expansions.
