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y-coordinate of 500,000 m and the ordinates are also numbered. The division of
projection zones starts from Greenwich.
The Gauss projection has received much attention and study from geodesists
across the world, among which the investigations done by the Bulgarian geodesist
Vladimir K. Khristov is the most representative. His two works The Gauss-Kr
€
uger
Coordinates on the Ellipsoid (Die Gauss-Kr
uger'schen Koordinaten auf dem
Ellipsoid) published in 1943 and The Gaussian and Geographical Coordinates on
the Krassowski Ellipsoid (Die Gaussschen und geographischen Koordinaten auf
dem Ellipsoid von Krassowski) published in 1955 enriched and developed Gauss
projection in both theory and practice (Hristow 1943, 1955).
€
6.3.2 Conditions for Gauss Projection
In Fig.
6.3a
we imagine an elliptical cylinder wrapped around the Earth ellipsoid
tangential to a meridian on the ellipsoidal surface (which is called the central
meridian or axial meridian). In addition, the central axis of the elliptic cylinder
passes through the center of the ellipsoid. The ellipsoidal elements within a certain
degree of longitude on either side of the central meridian are projected onto the
elliptic cylindrical surface according to the three conditions given below. The
cylindrical surface is then developed along the generating line passing through
the north and south poles of the ellipsoid. The projection plane obtained is known as
the Gauss projection plane, on which the central meridian and equator are projected
as straight lines. The point of intersection of the projected central meridian and
equator is taken as the coordinate origin O. The central meridian is labeled as the x-
axis (north direction) of the projection, while the y-axis (east direction) is the
mapping of the equator. Hence, the Gauss plane rectangular coordinate system is
established; cf. Fig.
6.3b
.
The three conditions for the Gauss projection are:
1. The projection is conformal.
2. The central meridian is projected as a straight line.
3. The length of the central meridian remains unchanged after projection.
The first condition is the general condition for conformal projection, while the
latter two are the particular conditions of the Gauss projection itself. We use
mathematical relations to express the three conditions:
1.
∂
x
q
ﾼ
∂
y
∂
l
,
∂
x
l
ﾼ
∂
y
q
;
∂
∂
∂
2. When l
ﾼ 0, y
ﾼ 0
3. When l
ﾼ
0, x
ﾼ
X (X denotes the length of the meridian arc from the equator)
In terms of solving differential equations, condition 1 can only find the general
solutions of the equation. To find the particular solutions of the differential equa-
tion, one also needs to plug in the initial conditions, i.e. conditions 2 and 3. How to
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