# Polar coordinates, the relationship between the directional angle, true and magnetic azimuths, the determination of polar coordinates on the map.

The polar coordinates became popular with the advent of radar and radio direction finding, when it became necessary to determine the position of individual points on the topographic map and on the ground using the angle relative to any direction and the distance to them from some selected point, which is called the pole.

## Polar coordinates, the relationship between the directional angle, true and magnetic azimuths, how to determine the polar coordinates on the map.

If we take instead of two mutually perpendicular axes x and y in a system of plane rectangular coordinates, only one x axis and the starting point on it are 0 (pole) and from it we determine the angle a (alpha), which is called the angle of position, and also the distance D (from the pole to the point), then these two quantities are called “polar coordinates”.

The polar coordinates have the x axis, called the polar axis, and the angle of the individual point. It can have three designations and, accordingly, three names:

• Directional angle a.
• True azimuth A
• Magnetic Azimuth Am.

Such a large number of position angles and their different names are explained by what we will take as the polar axis when we use the polar coordinates. From which direction will we measure the position angle.

If we take the direction of the vertical line of the coordinate grid as the polar axis, then this angle will be called the directional angle and denoted by a. If for the polar axis we take the direction of the true meridian (and it is on the map), then this angle will be called the true azimuth and denoted by A.

And finally, if we take the magnetic meridian (the direction of the magnetic needle of the compass) as the polar axis, then this angle is called the magnetic azimuth and will be denoted by Am. In all these cases, the position angle varies from 0 to 360 degrees. And always measured by the hour hand.

## Correlation between directional angle, true and magnetic azimuths.

If we establish the ratio of the polar axes to each other, then the ratio between the directional angle a, the true and magnetic azimuths A and Am will be determined. The vertical lines of a rectangular grid of coordinates make up a certain angle with the meridians. That is, the sides of the map frame. The reason for this is that all meridians converge at the poles, and the vertical lines of the grid remain parallel to their axial meridian of the zone.

## Meridian Rapprochement.

The angle composed by the true meridian at a given point and the vertical line of the grid passing through the same point is called the convergence of the meridians and is designated by the Greek letter gamma. Meridian approach is eastern (with a + sign), when the grid has a slope to the right relative to the map frame. And the western one (with the – sign), when the grid has a slope to the left.

If the angle of convergence of the meridians reaches 1 degree or more, it must be taken into account when passing the drift angle to the true azimuth. Its value at the edges of the zone reaches 3 degrees.

## Magnetic declination.

The true meridian, in turn, does not coincide with the magnetic one, which shows the compass. This angle between them is called magnetic declination and is denoted by the Greek letter delta. Magnetic declination is considered east (with a + sign) if the north end of the magnetic needle of the compass deviates east of the true meridian. And western (with a – sign) when dodging west.

The difficulty, taking into account the magnetic declination at the transition from the directional angle to the magnetic azimuth, lies in the fact that, due to the magnetic properties of the Earth, it is not the same at different points on the earth’s surface. Moreover, in the same place it also does not remain constant, and changes from year to year.

## Magnetic needle deflection angle or directional correction.

Thus, it can be seen from the foregoing that the vertical lines of the coordinate grid and the magnetic meridians form an angle between themselves, representing the sum of the approach of the meridians and the magnetic declination. This angle is called the deflection angle of the magnetic needle. Or a directional amendment, and is denoted by a capital letter – P.

The correction of the direction P is counted from the northern direction of the vertical line of the coordinate grid and is considered positive (with a + sign) if the northern end of the magnetic arrow deviates east of this line. And negative (with a – sign) with a western deflection of the magnetic needle. Data on the magnitude of the correction of direction (P) and its constituent quantities: the approach of the meridians, the magnetic declination, are placed in the form of a diagram under the lower frame of the sheet with explanations.

These data are necessary in order to quickly switch from the directional angles measured on the map to the corresponding magnetic azimuths on the ground. For this scheme, the relationship between the position angle and the correction will look like this:

## a = Am + P

If we know the directional angle and we need to determine the magnetic azimuth from it, then the formula will take the form:

## Am = a P

All this is true only with eastern magnetic declination (+) and western convergence of meridians (-). For other schemes, the direction correction may not be equal to the sum of these angles, but to the difference. Or, moreover, she herself can become negative. Then, when passing from the directional angle to the magnetic azimuth, in the first formula it must be subtracted. And in the second formula, on the contrary, add.

This circumstance forces everyone working with the map to carefully study the layout of the vertical grid line, the true and magnetic meridians, and the correction data placed on each topographic map.

Errors made in determining the directional correction (P), and even more so in its sign when determining map data for movement in azimuths over the terrain, are dangerous because when they are 5 degrees and when moving a distance of up to 1 km, deviation at the end of the path can make up about 100 meters. If it is in an open area, then a landmark can still be found. But in a closed area (in the forest) it is almost impossible to find it.

Based on the book “Map and Compass My Friends”.
Klimenko A.I..