Geography Reference
In-Depth Information
w
zx yaj
arcsin(tanh
)
(74)
j
sin 2
j
sin 4
j
sin 6
j
sin 8
j
sin 10
0
2
4
6
8
10
(74) is the solution of the forward Gauss projection by complex numbers. Its correctness can
be explained as follows:
The two equations in (74) are all elementary complex functions. Because elementary
functions in their basic interval are all analytical ones in the complex numbers domain, the
mapping defined by (74) form
wq l
to
zx y
meets the conformal mapping
l
, the imaginary part disappears and (74) restores to (73). Therefore,
(74) meets the second and third constraints of Gauss projection when
0
constraint. When
l
. Hence, it is clear
0
that (74) is the solution of the forward Gauss projection indeed.
5.2. The non-iterative expressions of the inverse Gauss projection by complex
numbers
In principle, the inverse Gauss projection can be iteratively solved in terms of the forward
Gauss projection (74). In order to eliminate the iteration, one more practical approach is
proposed based on the direct expansion of the transformation from meridian arc to
isometric latitude (53).
In order to meet the conformal mapping constraint, the inverse Gauss projection should be
in the following form
1
1
w
q
il
f
()
z
f
(
x
iy
)
(75)
f
1
where
is the inverse function of
f
. According to the second constraint, when
l
,
0
imaginary part disappears and only real part exists, (75) becomes
1
()
qf
x
(76)
Finally, from the third constraint, one knows that
x
in (76) should be the meridian arc
X
, and
(76) is essentially consist with the direct expansion of the transformation from meridian arc to
isometric latitude as (53) shows. Substituting
X
in (53) with
x
gives the explicit form of (76)
x
ae
(77)
2
(1 )
arctanh(sin
K
0
q
)
sin
sin 3
sin 5
sin 7
sin 9
1
3
5
7
9
If one extends the definition of
x
in a real number variable to a complex numbers variable,
or substitutes
x
with
zx y
, the original real number rectifying latitude
will be
automatically extended as a complex numbers variable. We denote the corresponding
complex number latitude as
, and insert it into (77). Rewriting a real variable
q
at the left-
hand of the second equation in (77) as a complex numbers variable
wq l
, one arrives at
