Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 265

scale factor is independent of direction. This is the fundamental idea in deducing
the general condition for conformal projection. The specific steps are:

1. Based on definition of the scale factor:

ds
dS ,

m ﾼ

where ds is the arc element on the projection plane and dS is the arc element on
the ellipsoid; write down its specific expression.
2. We deduce the general condition for conformal projection according to the fact
that the scale factor m is independent of the azimuth A.

6.2.2 Expression of Scale Factor

In Fig. 6.2 , the left part of the diagram is the ellipsoid surface and the right part is
the projection plane. We establish the Cartesian coordinate system on the projection
plane with o as the origin, the x-axis as the vertical axis, and the y-axis as the
horizontal axis. The coordinate system so defined is a left-hand system, while in
most cases the plane analytical system is displayed as the right-hand system,
because the azimuth defined in geodesy is the angle measured clockwise from the
north direction (which is the same as the angle defined in the left-hand system). In
many topics, the plane coordinates are expressed as a distance in meters to the east,
referred to as the “Easting,” and a distance in meters to the north, referred to as the
“Northing” (FGDC 2001; NIMA 1989, 1990; Maling 1992; DeMers 2005; Galati
2006).
There are two infinitely near points P and P 1 on the ellipsoid surface, which are
projected as P 0 and P 1 0 . Their coordinates are shown in Fig. 6.2 .dS is the geodesic
arc element with an azimuth A and the projected arc element on the plane is ds.We
draw the meridian and parallel through points P and P 1 , respectively, from the
differential triangle PP 1 P 2 according to the differential formulae for the arc lengths
of the meridian and parallel, and the expression of the arc element dS on the
ellipsoid can be written as:

,

r 2 M 2
r 2

dS 2 M 2 dB 2 r 2 dL 2

2 2

ﾼ þ ﾼ ðÞ dB þ ðÞ dL ð 6 : 2 Þ

where the curvature radii M and r are functions of the latitude B. To simplify the
derivation process of the formula, let:

M
r dB,

dq ﾼ ð 6 : 3 Þ

then ( 6.2 ) can be simplified as: