The method of image formation in the projection of Gauss-Kruger

Introduction

During the formation of a cartographic image of the terrain in a transverse equilateral cylindrical projection of Gauss-Kruger , problems arise associated with large errors and distortion of the formed image with distance from the axial meridian. The root of these problems is that the Gauss-Kruger projection consists of sixty “petals” of six-degree zones, between which an artificial distance of 500 km is introduced. This is due to the fact that standard visualization methods do not take into account the narrowing of zones to the poles, but present them as rectangular. To overcome these problems, there are methods of stitching cards along a single axial meridian.

Method "GKZone"

One of such methods is the method of dynamic alignment of axial meridians, which makes it possible to compensate for the angle and distance between them. Using this method allows you to display a 2D map in the projection of Gauss-Kruger without visible distortion, and also to avoid redundancy of the original data. The principle of this method is that the set of map sheet fragments obtained during the preparation of data arrays, form the so-called zone-petals.
To increase the accuracy of the location of objects and reduce visual distortion (negligible), you can set a non-standard size of one zone equal, for example, 1º or even less. The sizes of the fragments with which the zones are filled are set arbitrarily. The smaller the size of the fragment, the higher the quality of visualization.

Figure 1. Example of dividing the unfolded earth's surface into zones in increments of 6º

The combination method is as follows:

1. Recalculate the coordinates of the point of interest (relative to the width of which gluing is done) from the geodesic / geocentric coordinate system to the Gaus-Kruger projection.
2. Determine the coordinates of two points of contact of adjacent axial meridians.
3. Calculate the angle of rotation of one zone relative to another at the current latitude.
4. Based on the resulting angle to form a transformation matrix for each adjacent meridians.
5. To position a fragment, it is necessary to multiply its coordinates by the corresponding transformation matrix.

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After performing the above actions, we get an inseparable stitched card. Figures 2 and 3 show images of cross-linked zones relative to different northern latitudes.

Figure 2. Stitched at 20 ° N.N.


Figure 3. Stitched at 50 ° N.N.

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


Figure 4. Circle image

Conclusion

Image quality is of great importance in the visualization of cartographic information, 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 with increasing the scale, overlapping areas become clearly visible.

Bibliography

1. The course of higher geodesy, P., S. Zakatov, Moscow “NEDRA” 1976
2. GOST R51794-2008

From the author

This article is an excerpt from an article for a future conference. I decided not to post the details and calculations yet, as they are still being edited. The name of the method - GKZone, is certainly not the best, but this is the working name of the method in the source code. Therefore, suggestions for a clearer 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, and the Gauss-Kruger projection is not.

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