It is known that the Earth is spherical, i.e., it does not have the shape of a perfect sphere. Its shape is irregular, and like any rotating body, it is slightly flattened at the poles. In addition, due to the uneven distribution of masses of Earth’s substance and tectonic deformations, the Earth has extensive convexities and concavities. For this reason, the Earth’s surface is replaced with some regular surface, which is called the surface of relativity.
In the most precise approximation, such a surface is the surface of a geoid (a figure bounded by the level surface of the ocean). It is practically impossible to determine its shape accurately. Therefore, in the theory and practice of cartography, the Earth ellipsoid or a sphere of a certain radius (when creating small-scale maps (when we can neglect polar compression) are taken as the surface of relativity.
The terrestrial ellipsoid is an ellipsoid of rotation with low compression, the dimensions of which are chosen in such a way that for a given territory it is the least deviated from the geoid. It is assumed that the equatorial plane and the center of the ellipsoid of rotation coincide with the equatorial plane and the Earth’s center of mass. Such an Earth ellipsoid is otherwise called a reference ellipsoid. By the Decree of the Council of Ministers of April 7, 1946, Krasovsky’s reference ellipsoid was adopted in our country as such a reference ellipsoid. It has the following parameters:
- a = 6,378,245 km – semi-major axis;
- b = 6 356 863 km – semi-major axis;
c = 1 : 298.3 – polar compression.
The ellipsoid of rotation is formed by rotating the ellipse PNE1PSE2 around the polar axis PNPS (Fig. 1). They are obtained by crossing the PNPS axis of the surface of the ellipsoid.
Sections of the surface of the ellipsoid of rotation by planes parallel to the equator plane form circles – parallels. Sections of the surface of the ellipsoid of rotation by planes passing through the axis of rotation form ellipses – meridians.
Let O’K’ be the normal to the surface of ellipsoid at point K. The planes passing through the normal are called normal planes. Sections of these planes with the surface of the ellipsoid yield normal sections, or verticals. Then a meridian is a normal section whose plane passes through the polar axis. The normal section perpendicular to the plane of the meridian PNE1PSE2 gives the section of the 1st vertical.
The radii of curvature of these sections are defined by the following formulas:
- radius of curvature of the meridian;
- radius of curvature of the 1st vertical;
where – 1st eccentricity;
a and b are the major and minor half-axes of the ellipsoid of rotation.
The radius of parallel (r) is calculated through the radius of curvature of the first vertical
Coordinate system on the surface of an ellipsoid and a sphere
The position of a point on the surface of an ellipsoid can be defined in one or another coordinate system. The basic coordinate system is geographic with ?, ?
Geographical latitude (?) is the angle between the plane of the equator and the normal OM (plumb line) of the current point M. Latitude varies from 0 to 90°.
Geographical longitude (?) is the dihedral angle between the planes of the prime meridian and the meridian of the current point M. Longitude varies from 0 to 180° west and east of the prime meridian. In cartographic calculations western longitudes are taken with a “minus” sign, eastern – with a “plus” sign.
In addition to the considered coordinate system, there is a number of others used in mathematical cartography:
- rectangular spheroidal;
- spherical polar, etc.
Coordinate lines should be understood as geometric places of points, for which one of the coordinates is constant. For example, parallel is the geometric place of points of equal latitude (? = const), and meridian is the geometric place of points of equal longitude (? = const).
When the Earth is taken as a sphere, the geographic coordinates are the spherical coordinates ?, ? with the pole of the coordinate system coinciding with the geographic pole.