Geography Reference
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, 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
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