# Gauss-Kruger projection imaging method

#### Introduction

When forming a cartographic image of the terrain in a transverse equiangular cylindrical projection of Gauss-Krueger , problems arise associated with large errors and distortion of the generated image when moving away from the axial meridian. The root of these problems is that the Gauss-Krueger projection consists of sixty “petals” of six-degree zones, between which a distance of 500 km is artificially inserted. This is due to the fact that standard visualization methods do not take into account the narrowing of the zones to the poles, but present them as rectangular. To overcome these problems, there are methods for stitching cards along one axial meridian.

#### GKZone Method

One of these methods is the method of dynamically combining axial meridians, which makes it possible to compensate for the angle and the distance between them. Using this method allows you to display a 2D map in the Gauss-Kruger projection without visible distortion, as well as to avoid redundancy of the original data. The principle of this method is that many fragments of the map sheet obtained during the preparation of data arrays form the so-called petal zones.

To increase the accuracy of the location of objects and reduce visual distortion (negligible), you can set the non-standard size of one zone equal to, for example, 1º or even less. The sizes of the fragments with which the zones are filled are set arbitrarily. The smaller the fragment size, the higher the quality of the visualization.

*Figure 1. An example of dividing the unfolded earth's surface into zones with a step of 6º.*

The combination method is as follows:

- Recalculate the coordinates of the point of interest (relative to the latitude of which gluing is performed) from the geodetic / geocentric coordinate system to the Gaus-Krueger projection.
- Determine the coordinates of two points of contact of adjacent axial meridians.
- Calculate the angle of rotation of one zone relative to another at the current latitude.
- Based on the obtained angle, form a transformation matrix for each neighboring meridians.
- To position a fragment, it is necessary to multiply its coordinates by the corresponding transformation matrix.

After performing the above steps, we get an inextricably stitched card. Figures 2 and 3 show images of stitched zones relative to different northern latitudes.

*Figure 2. Stitched at 20 ° N*

*Figure 3. Stitched at 50 ° N*

Moreover, the latitude coordinate of the fragments of glued areas remains unchanged, only the longitude coordinates change. When zones are merged in this way, during the visualization of the cartographic information of the area, visual deformation is completely absent and the error is minimized. Figure 4 shows that the circle will be depicted without visible distortion if the card is stitched in the manner described.

*Figure 4. Circle image*

#### Conclusion

Of great importance in the visualization of cartographic information is the image quality, but minimizing display errors and the ability to calculate the approximate distance from the resulting image also plays a big role. The described method satisfies both parameters and is successfully used in the development of cartographic systems.

Unfortunately, this method is applicable only for maps of scales up to 1: 1,000,000, since when you zoom in, alignment areas become clearly visible.

#### List of references

- Course of Higher Geodesy, P, S. Zakatov, Moscow "NEDRA" 1976
- GOST R51794-2008

#### From the author

This article is an excerpt from an article for a future conference. I decided not to post details and calculations yet, since they are still being edited. The method name is GKZone, of course, not the most successful, but this is the working name of the method in the source code. Therefore, suggestions for a more understandable method name are welcome.

I hope this article will be useful for people interested in cartography, because the Mercator projection is widely covered in various sources, but the Gauss-Krueger projection is not.

I look forward to constructive criticism of the article and proposals for its improvement.