The Gauss-Krüger projection is a type of transverse cylindrical projection that allows the surface of the Earth ellipsoid to be conformally (angle-preserving) transferred to a plane. This projection is scale-free along the central meridian and is primarily used for establishing a plane rectangular coordinate system in geodetic applications.

Historical Development

The foundation of the projection was laid by Carl Friedrich Gauss between 1820 and 1830 during the evaluation of triangulation results in Hanover. Gauss worked on an ellipsoidal transverse Mercator projection with a constant scale along the central meridian but did not publish his work. In 1912, Louis Krüger systematized the projection by expanding Gauss's theory and completing it with detailed formulas in his work titled Konforme Abbildung des Erdellipsoids in der Ebene. Therefore, the projection is known in the literature as the Gauss-Krüger Projection.

Basic Characteristics and Conditions

The projection has three basic conditions:

1. The projection is conformal (angles are preserved).
2. The central meridian is projected as a straight line.
3. The length of the central meridian remains unchanged after projection.

Mathematically, these conditions are expressed as follows:

$\frac{\partial x}{\partial \varphi} = \frac{\partial y}{\partial \lambda}, \quad \frac{\partial x}{\partial \lambda} = -\frac{\partial y}{\partial \varphi}$

Here:

• $x$: north-oriented coordinate (longitude axis),
• $y$: east-oriented coordinate (latitude axis),
• $φ$: latitude,
• $λ$: longitude.

The condition along the central meridian ($λ = 0$) is satisfied:

For the central meridian: $\lambda = 0 \Rightarrow y = 0, \quad x = X$

Here $X$ represents the arc length along the central meridian from the equator:

$X = \int_0^{\varphi} M(\varphi) \, d\varphi$

Here $M(φ)$ is the radius of curvature of the meridian and depends on ellipsoidal parameters.

Zoning and Coordinate System

In this projection, accuracy increases as one approaches the central meridian and decreases as one moves away from it. Therefore, the Earth's surface is divided into narrow projection zones along the longitude direction. A central meridian is determined at the center of each zone, and these zones are typically 3° or 6° wide. Each zone has its coordinate system. By creating overlapping zones in the east-west direction, consistency between maps is ensured.

In the projection plane:

• the x-axis represents the central meridian in the north direction,
• the y-axis is the projection of the equator in the east direction.

To prevent negative values in the y-coordinates within a zone, a false coordinate of 500,000 m is added to the y-coordinate.

$y ′ =y+500000 m$

Areas of Use

The Gauss-Krüger projection is widely used in the national mapping systems of many countries. It is particularly preferred for large-scale maps requiring high precision, in engineering projects, and for reducing geodetic control networks to a plane.