Geography Reference
In-Depth Information
10 k
. The accuracy of the indirect expansion of the transformation from
authalic latitude function to isometric latitude is higher than 10
-2
″, while the accuracy of the
direct expansion (67) is higher than 10
-6
″. The accuracies of the direct expansions derived by
the author are improved by 2~6 orders of magnitude compared to the indirect ones derived
by Yang (1989, 2000).
7
2
higher than
5. The non-iterative expressions of the forward and inverse Gauss
projections by complex numbers
Gauss projection plays a fundamental role in
ellipsoidal geodesy, surveying, map projection
and geographical information system (GIS). It has found its wide application in those areas.
x
N
N
y
O
equator
O
S
S
Figure 1.
Gauss Projection, where
x
and
y
are the vertical and horizontal axes after the projection
respectively,
O
is the origin of the projection coordinates.
As shown in Figure 1, Gauss projection has to meet the following three constraints:
①
conformal mapping;
②
the central meridian mapped as a straight line (usually chosen as a vertical axis of
x
)
after projection;
③
scale being true along the central meridian.
Traditional expressions of the forward and inverse Gauss projections are real functions in a
power series of longitude difference. Though real functions are easy to understand for most
people, they make Gauss projection expressions very tedious. Mathematically speaking,
there is natural relationship between the conformal mapping and analytical complex
functions which automatically meet the differential equation of the conformal mapping, or
the “Cauchy-Riemann Equations”. Complex functions, a powerful mathematical method,
play a very special and key role in the conformal mapping. Bowring (1990) and Klotz (1993)
have discussed Gauss projection by complex numbers. But the expressions they derived
require iterations, which makes the computation process very fussy. In terms of the direct
expansions of transformations between meridian arc and isometric latitude given in section
