Geography Reference
In-Depth Information
xiy
ae
2
(78)
(1
)
arctanh(sin
K
0
wq l
)
sin
sin 3
sin 5
sin 7
sin 9
1
3
5
7
9
Therefore, the isometric latitude
q
and longitude
l
is known. Inserting
q
into (78) yields
the conformal latitude
arcsin(tanh
q
)
(80)
Then one can compute the geodetic latitude through the inverse expansion of the conformal
latitude (40).
(77) is the solution of the inverse Gauss projection by complex numbers. Its correctness can
be explained as follows:
The two equations in (78) are all elementary complex functions, so the mapping defined by
(78) form
zx y
l
, the
imaginary part disappears and (78) restores to (77). Therefore, (78) meets the second and
third constraints of Gauss projection when
to
wq l
meets the conformal mapping constraint. When
0
l
. Hence, it is clear that (78) is the solution of
0
the inverse Gauss projection indeed.
6. Conclusions
Some typical mathematical problems in map projections are solved by means of computer
algebra system which has powerful function of symbolical operation. The main contents and
research results presented in this chapter are as follows:
1.
Forward expansions of rectifying, conformal and authalic latitudes are derived, and
some mistakes once made in the high orders of traditional forward formulas are pointed
out and corrected. Inverse expansions of rectifying, conformal and authalic latitudes are
derived using power series expansion, Hermite interpolation and Language's theorem
methods respectively. These expansions are expressed in a series of the sines of the
multiple arcs. Their coefficients are expressed in a power series of the first eccentricity of
the reference ellipsoid and extended up to its tenth-order terms. The accuracies of these
expansions are analyzed through numerical examples. The results show that the
accuracies of these expansions derived by means of computer algebra system are
improved by 2~4 orders of magnitude compared to the formulas derived by hand.
2.
Direct expansions of transformations between meridian arc, isometric latitude and
authalic latitude function are derived. Their coefficients are expressed in a power series
of the first eccentricity of the reference ellipsoid, and extended up to its tenth-order
terms. Numerical examples show that the accuracies of these direct expansions are
improved by 2~6 orders of magnitude compared to the traditional indirect formulas.
3.
Gauss projection is discussed in terms of complex numbers theory. The non-iterative
expressions of the forward and inverse Gauss projections by complex numbers are
derived based on the direct expansions of transformations between meridian arc and
isometric latitude, which enriches the theory of conformal projection. In USA, Universal
