It is very well known to every high school student that any point on the plane is characterized by its coordinates (x, y). They coincide with the coordinates of the vector 0A that comes from the point 0, which is the origin of coordinates. Let us suppose that A and B are arbitrary points on the plane with the coordinates (x1, y1) and (x2, y2), respectively. In this case, the vector AB has, obviously, the coordinates (x1 - x2, y2 - y1). It is known that the square of the length of the vector is equal to the sum of the squares of its coordinates. Therefore, the distance d between points A and B, or, equivalently, the length of vector AB, is defined by the following condition: d2 = (x1 - x2)2 + (y2- y1)2. This formula allows a student to find the distance between any two points on the plane provided he or she knows the coordinates of these points. Every time when we talk about the coordinates of a point on the plane, we refer to a well-defined coordinate system x0y. Generally speaking, a coordinate system on the plane can be chosen differently in order to calculate the distance between two points with a higher degree of precision. Thus, instead of the coordinate system x0y we can consider the coordinate system x'0y', which is obtained by turning around the axes of the old starting point 0 counterclockwise by an angle α.

If a point on the plane within the coordinate system x0 has the coordinates (x, y), then it will have different coordinates (x', y') in the new system of coordinates x'0y'. As an example, we can consider the point M located on the axis 0x' and spaced by a distance equal to 1 from the point 0. Obviously, this point has the coordinates (cos α, sin α) in this coordinate system x0y and the coordinates (1,0) in the coordinate system x'0y'. The coordinates of any two points A and B on the plane depend on how the plane coordinate system has been set. However, in this case, the distance between two points does not depend on what method of setting the coordinate system has been chosen.

Let us suppose that A is a point in a predetermined coordinate system. The orthogonal projection of point A on the plane Oxy is denoted as A', and the length of the vector OA - as r. The angle of inclination of the vector to the plane Oxy can be denoted as ψ, and we assume that it changes from -90o to + 90o. If the point A is located in the upper half of our coordinate system, then the angle ψ is considered positive, and if it is located the lower half – the angle is negative. The angle between the vector and the axis Ox is denoted as φ. Usually, the triplet (r, ψ, φ) is called the spherical coordinates of the point A in the space. Cartesian coordinates (x, y, z) of the point in the space are expressed through its spherical coordinates in accordance with the following formulas: 1) x = r cos ψ cos φ; 2) y = r cos ψ sin φ; 3) z = r sin ψ.

Spherical coordinates comprise several exclusive concepts relevant to measuring the distance between two points, namely:

- Parallel. The points on the sphere that have the same angle ψ, forms a circumference, which is known as a parallel.
- Meridian. The points on the sphere that have the same angle φ, form a semicircle, which is known as a meridian.
- Orthodrome. The arc of a great circle that joins two points of the sphere is the shortest way to the area between the two points. This path is called orthodrome, and a Greek translation of this means ‘a straight run’
- Loxodrome. The curve forming equal angles with different meridians is called a rhumb line or loxodrome, and a Greek translation of the latter means ‘an oblique run’

If you happen to be in need of knowing how the distance between two points is estimated by military intelligence here is a comprehensive guide that will help you get a good grasp of the topic. Primarily, the data required for azimuthal movement (magnetic azimuth directions between the turning points along the route and the distance between them), is determined by using a large-scale map. Preparing the data for azimuthal movement includes route selection as well as the study of a map and landmarks on particular sites of the route; also, a specialist needs to determine magnetic azimuth directions and the distances between selected reference points, transfer obtained data on the map or draw up a scheme of the movement.

Studying an area, one must assess its passability, masking and protective properties as well as define impassable and impenetrable obstacles and alternative routes. Making a tracing of a route depends on the nature of the terrain, the presence of landmarks on it and special conditions of the forthcoming movement. The chief aim is to measure exactly the distance between two points and choose a route that allows you to move quickly and secretly in order to reach the destination (an end point or facility). A route should be selected in such a way that it has a minimum number of turns. The turning points on a route must be scheduled and connected with landmarks that can be easily identified in the area (for instance, tower constructions, intersections of roads, bridges, overpasses, geodetic marks). A specialist circles selected landmarks and connects them by straight lines. The route lines that do not intersect the vertical grid line might be promptly continued to the intersection which is the nearest to them so that it would be easier later to measure directional angles. The latter are measured using a protractor or milrule, which provide a precise angle measurement with an error of ± 1-2 °. Later on, the measured azimuth of a direction is converted into magnetic bearings. The distance between two points on the route is measured with a caliper and a ruler with millimeter divisions. If a route is planned for a hilly (mountainous) terrain, then the distances measured on the map must be corrected for relief distortions.

A movement scheme can be created according to the following sequence. A cartographer transfers the starting point, the landmarks related to turning points and the end point of a route to a blank sheet of paper. All the landmarks (or reference points) on the scheme should be placed on a new scheme similarly to their position on a copied map. Moreover, all the landmarks must be depicted on a scheme using conventional signs analogous to those on the map. Then the landmarks are to be numbered and connected by straight lines. Each line should be undersigned with the initial details of the movement in the form of a fraction, where the numerator is the magnetic azimuth and the denominator is the distance between two points in meters and the time of the movement in minutes. If the azimuth movement will be performed on foot, the distance in meters then is converted into pairs of steps and fixed on the scheme in parentheses. After this the arrow ‘north – south’ is added to the scheme as well as all possible intermediate or auxiliary additional routes.

To make it easier to sustain the direction of a movement, specialists use auxiliary and supporting landmarks in addition to temporary ones. These landmarks usually are visible heavenly bodies: the Sun, the Moon, and bright stars. If you must use them check the azimuth direction after each 15 minutes of the movement, since heavenly bodies (except for the North Star) are constantly moving across the sky. If you move in their direction without controlling your movement for a long time, it is highly probable that you will lose the correct route soon enough. In order to keep moving in the chosen directions, you also can use linear landmarks or traces of vehicles. How precise the result of a movement will be dependent on the nature of the terrain, visibility conditions, and, undoubtedly, on errors in determining the bearings and measuring the distance between two points. Usually, the deviation from the pivot point. which you need to reach amounts to no more than 1/10 of the distance traveled, i.e. 100 m per kilometer of the distance traveled. Therefore, if a predetermined distance were covered and the intended landmark did not become visible, it would be necessary to search within a circle, which radius is equal to 1/10 of the distance traveled from the previous turning point. In some cases, such as winter skiing, the distance between two points is measured approximately taking into account time and speed. To avoid loss of orientation due to an inaccurate distance measurement, an observer should choose landmarks that are easily visible from a distance. Also, in some cases, a landmark can hardly be seen behind an obstacle or is difficult to identify when approaching it. To verify the correctness of a route to the reference point, it is recommended to install a makeshift milestone or make notches on a tree. When moving to the next point the magnetic azimuth direction to the previous point A should be determined; this reverse azimuth normally differs from the azimuth of a predetermined direction of a movement in this section of the route by 180. Only after measuring the distance between two points establishing the connection with the point A according to reverse azimuth and ensuring that the direction to the point A coincides exactly with the selected route it is possible for an observer to continue the movement.

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