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Fig. 6.14 Example of arc-
to-chord correction
checking
6.5.3 Correction of Distance
The Gauss projection is conformal with no angular distortion. However, the pro-
jection distorts distances everywhere except along the central meridian. The dis-
tance correction of the Gauss projection is dependent on the distortion of distance.
Previously, we provided the definitions of the scale and distance distortion. Below
are the derivations of the specific mathematical expressions for the scale so that the
laws and effects of distance distortion and the ways to limit such distortions can be
studied. The formula for the distance distortion will be derived after that.
Formula for Scale Factor
It has been previously stated that the ratio of the arc element at a point on the
projection plane ds to the corresponding arc element on the ellipsoid dS is known as
ds
dS
. For conformal projections, the scale
factor at any point is independent of direction. Hence, any arbitrary directions can
be selected when the formula for the scale factor is derived. In (
6.16
) we provided
the formulae for scale factors along two special directions, where the first expres-
sion is that along the meridian (l
the scale factor at this point, namely m
ﾼ
ﾼ
constant) and the second along the parallel
(q
constant). In conjunction with the formula for the direct solution of the Gauss
projection, it is reasonably convenient to take the partial derivative with respect to l.
Therefore, it is simpler to use the second expression in (
6.16
) for deriving the
formula for the scale factor, namely:
ﾼ
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