Topographic maps, determining the rectangular coordinates of a point on a topographic map.

Topographic maps are most interesting for tourists, especially large-scale ones. The basis of topographic maps is the geometric concept of the Earth’s surface zone formed by two meridians spaced 6 degrees apart. 

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

The size of such a zone in the form of a “lobule” allows you to create topographic maps with virtually no noticeable distortion. The entire terrestrial ellipsoid is divided into 60 zones. Zones are numbered from west to east, starting from the zero (Greenwich) meridian. The first zone extends from the meridian of 0 degrees to the meridian of 6 degrees. The central (axial) meridian of the first zone is 3 degrees. Topographic maps in the CIS countries are based on the Gauss — Kruger projection. From a mathematical point of view, to create a map projection, the earth’s surface is projected onto a cylinder, the axis lies in the plane of the equator. The lateral surface of the cylinder touches the axial (middle) meridian of the zone.

The zone is projected onto the lateral surface of the cylinder, which then turns into a plane. The axial meridians of each zone are depicted in straight lines and without distortion, retain scale throughout its length. The remaining meridians of the zone and parallels are represented in the projection by curved lines. Distortions of the line lengths increase as you move west or east from the axial meridian and at the borders of the zone become the largest, reaching a value of about 0.1% of the line length measured on the map. For example, if the scale on the axial meridian is 500 meters per 1 cm, then on the edge of the zone it will be 499.5 meters per 1 cm. Due to the insignificance of distortions, it is usually believed that the scale of any topographic map for all its sections is almost constant.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

If the zone number N is known, then the longitude of the central meridian will be N x 6 3 degrees. An ordinary rectangular coordinate system is created on the map of the entire zone with the origin at the point of intersection of the axial meridian with the equator. In cartography, axes are designated differently than generally accepted. In each zone, the axial meridian is taken as the vertical axis (X axis). The horizontal axis of Y is the equator line. Given a coordinate system defined in this way, all X coordinate values ​​in the northern hemisphere will be positive. And the values ​​of the Y coordinates will depend on the location of the selected point with respect to the axial meridian of the zone and, therefore, can be positive or negative.

For convenience, to avoid negative coordinates, we agreed to consider the Y coordinate at the origin not equal to zero, but 500 kilometers. It follows that all points located west of the axial meridian will have a Y coordinate of less than 500 km, and located east of more than 500 km. In the southern hemisphere, an offset of 10,000 km is introduced for the X coordinate for the same purpose. In order to indicate the zone in which the object is located, it is customary to record the zone number at the Y coordinate in the first digits, followed by a six-digit number showing the value of the Y coordinate in meters.

So, if point M is located in the 12th zone and is east of the axial meridian at a distance of 80,300 m, then its Y coordinate is 12,580,300, where the number 12 indicates the zone number, and 500 km is added to the distance of 80,300 meters , the value of the Y coordinate of the axial meridian. If point M is 3,260,700 meters away from the equator, then its X coordinate is 3,260,700.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

On topographic maps, the system of plane rectangular coordinates of a zone is defined as a coordinate kilometer grid. Horizontal lines of the grid are parallel to the equator, and vertical to the axial meridian. These lines are drawn at equal distances from one another and form a set of squares. Each scale has its own grid square sizes, which are shown in the table below..

Sizes of sheets of topographic map and sides of grid squares on topographic maps.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

The table also shows the sizes of the individual most frequently used map sheets. The borders of the map sheets are meridians and parallels. The base sheet of the map is a sheet on a scale of 1: 1 000 000 (millionaire), having an extension in latitude of 4 degrees and longitude of 6 degrees. Cards of a larger scale are formed from the “millionaire” by the corresponding cutting (delineation). In order to be able to easily and quickly find the necessary map sheets, each of them has its own symbol. It should be noted that the direction of the grid lines does not coincide with the north-south and east-west directions, although it is close to them. The greatest deviations are observed at the borders of the zone where they reach 3 degrees.

The deviation of the direction of the true meridian from the vertical line of the coordinate grid is called the approach of the meridians (Sat). The magnitude of the approach of the meridians depends on the location of the point on the map.

Determination of the rectangular coordinates of a point on a topographic map.

As an example, consider determining the coordinates of a point defined on a map of scale 1: 100,000. The figure below shows a section of a map located in the 7th zone with longitudes from 36 to 42 degrees. The grid verticals show the coordinates of the grid lines in kilometers, the first (senior) bits in small numbers, and the last (younger) large ones.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

Moreover, in order not to clutter up the card, small (high) numbers may not be repeated every time, since they are the same everywhere. Horizontally the same, only the first digit 7 is the zone number. Looking at topographic maps, you can see that there are two coordinate grids on it. The first is standard with geographical coordinates indicated only along the edge of the map, and the second is a kilometer grid with a step of 2 cm (2 km). Applying a ruler to the nearest grid lines, we determine the displacement (in mm) inside the square and translate them into the distance according to the scale.

By digitizing the grid lines, we determine their coordinates. Summing up the found values, we determine the coordinates of the point: X = 409,080 m, Y = 6,200,450 m (zone number is not included). It is more convenient to make measurements with a special scale having vertical and horizontal axes calibrated in accordance with the map scale. To do this, it is necessary to impose a scale on the map so that the crosshairs of the axes coincide with the object on the map, and the axes are directed parallel to the map grid. Then the necessary offsets are read from both scales at the points of intersection with the map grid.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

Making a scale for determining the rectangular coordinates of a point on a topographic map.

Similar scales, either separately or with a compass, are issued in the USA, but for us they are useless, since our cards are issued on other scales. But such a scale can be done independently. To do this, it must be printed on a transparent film on the printer and glued to the compass tablet. The proposed option is made for the compass of the Azimuth series, this is a liquid compass with a rectangular tablet, in the middle of which there is a large magnifying glass. The scale is glued with tape on the back of the tablet strictly under the magnifying glass. It must be glued carefully around the entire perimeter so that there is no penetration of the water. It is preferable to use a wide transparent adhesive tape covering the entire surface, in this case it is better to print the scale in mirror image.

Topographic maps, determining the rectangular coordinates of a point on a topographic map.

What can be done with the coordinates of a given point defined in this way?

If a map is recorded in your GPS navigator, you can enter the received coordinates into the device, mark a point and then hit the road. If it is necessary to mark a point on a paper map according to the coordinates defined by the navigator, then this will be a task opposite to that already considered. It is solved in a similar way, only in the reverse order. The highest digit (thousands of meters) is the square, and the remainder is the offset within the square.

Topographic maps in the universal transverse projection of the Mercator (UTM Universal Transverse Mercator).

This projection features topographic maps of the United States. The concept and dimensions of the zone in the UTM projection are the same as in the Gauss-Krueger projection. However, there are differences. In the UTM projection, when creating a map, the lateral surface of the cylinder intersects the surface of the zone at two points that are 180,000 meters away from the axial meridian. As a result, the scale on the axial meridian is different from unity and amounts to 0.9996, at the points of intersection of the zone with the cylinder, the scale is 1. However, for practical use this is not very important, since measurements are made using the coordinate grid. The dimensions of the squares of the grid can be given in inches, and the distance in miles of a land or statute mile is 1609 meters.

Based on materials from the book All About GPS Navigators.
Naiman V.S., Samoilov A.E., Ilyin N.R., Sheinis A.I..

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