Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 299

Fig. 6.12 Approximate derivations of the arc-to-chord correction for the Gauss projection from a
spherical surface (a) to a plane (b)

appear as straight lines A 0 P 0 and B 0 P 0 in the projection. They are both perpendicular
to the x-axis (for point E is projected to an infinite distance). A 0 B 0 P 0 P 0 constitutes a
plane-curved quadrangle, where the side P 0 1 P 0 2 is a curved line.
Let the spherical excess of the spherical quad rangle be ʵ and the angles between

the projected geodesic P 0 P 0 and its chord P 0 1 P 0 2 be

ʴ 1.2 and ʴ 2.1 . For the projection

being conformal, the angle relationship is:

360 ∘ þ ʵ ﾼ 360 ∘ þ ʴ 1:2 þ ʴ 2:1 ð 6 : 54 Þ

Let ʴ 1.2 ﾼ ʴ 2.1 ﾼ ʴ ; then one obtains:

ʴ ﾼ 2 ,

P

where ʵ ﾼ

R 2 and P is the area of the spherical quadrangle. Since the numerical

value of is very small, P can be replaced by the area of the plane quadrangle. We

assume that the plane coordinates of P 0 1 and P 0 2 are (x 1 , y 1 ) and (x 2 , y 2 ); thus:

ʵ

1
2

P ﾼ ð y 1 þ y 2 Þ ð x 2 x 1 Þﾼ y m x 2 ð x 1 Þ ,

and

y m
2R m

ʴ ﾼ ð x 2 x 1 Þ, ð6:55Þ

where R m is the mean radius of curvature at the mid-latitude B m of the two end
points P 1 and P 2 .
The above derivations only result in the absolute values of the arc-to-chord
correction. However, in practical cases, as the position and direction of geodesics
vary, the value of can be either positive or negative. To enable the corrections to

appear in the form of the algebraic sum, make the

ʴ

ʴ obtained be the correction