Geography Reference
In-Depth Information
Ⅳ
, the non-iterative expressions of the forward and inverse Gauss projections by complex
numbers are derived.
5.1. The non-iterative expressions of the forward Gauss projection by complex
numbers
Let
w
be complex numbers consisting of isometric latitude
q
and longitude difference
l
before projection,
z
be complex numbers consisting of corresponding coordinates
x
,
y
after projection.
wq l
zx y
(70)
where
i
.
1
In terms of complex functions theory, analytical functions meet conformal mapping
naturally. Therefore, to meet the conformal mapping constraint, the forward Gauss
projection should be in the following form
zx y fw fq l
()
(
)
(71)
where
f
is an arbitrary analytical function in the complex numbers domain. According to
the second constraint, when
l
, imaginary part disappears and only real part exists, (71)
0
becomes
xf q
()
(72)
(72) shows that the central meridian is a straight line after the projection when
l
.
0
Finally, from the third constraint, “scale is true along the central meridian”, one knows that
x
in (72) should be nothing else but the meridian arc
X
, and (72) is essentially consist with
the direct expansion of the transformation from isometric latitude to meridian arc (56).
Substituting
X
in (56) with
x
gives the explicit form of (72)
arcsin(tanh )
sin 2
q
(73)
x
a j
j
j
sin 4
j
sin 6
j
sin 8
j
sin 10
0
2
4
6
8
10
(73) defines the functional relationship between meridian arc and isometric latitude. If one
extends the definition of
q
in a real number variable to a complex numbers variable, or
substitutes
q
with
wq l
, the original real number conformal latitude will be
automatically extended as a complex numbers variable. We denote the corresponding
complex numbers latitude as
, and insert it into (73). Rewriting a real variable
x
at the
left-hand of the second equation in (73) as a complex numbers variable
zx y
, one
arrives at