The earth is an oblate spheroid, length of equatorial axis being 12756.75 km and polar axis being 12713.80 km. Since the difference between these two axes and irregularities on the earth surface are very small (Note. Height of Mount Everest is 8.79 km) compared to these two axes, the earth may be treated as a sphere, Figure 11.1 shows a circular plane passing through a point A on the earth surface. The gravitational force is always directed towards the centre of the earth.
Hence, the plumb-line shown in Fig. 11.1 is a vertical line. Line perpendicular to vertical line (tangential to earth surface) is known as horizontal line. In surveying all measurement at any point are in the direction of these two lines.
Obviously, the vertical and horizontal lines at another point B are not parallel to the respective lines at A. It should be noted that all lines lying on the earth’s surface are curved lines and all triangles are spherical triangles as shown in Fig. 11.2. Hence, surveying involves spherical trigonometry.
If the area to be surveyed is small, the curvature of the earth may be neglected and all plumb lines treated as the same vertical. Hence, the lines normal to plumb line at any point in the area are treated as the same horizontal. All triangles in the area may be treated as plane triangles.
The survey in which earth curvature is neglected is called Plane Surveying and the survey in which earth’s curvature is considered is known as Geodetic Surveying.
No definite value can be assigned to the area up to which a survey may be treated as plane, since the degree of accuracy required forms the controlling factor. However, the following points should be noted:
(i) The length of an arc of 1.2 km on earth’s mean surface is only 1 mm more than the straight line connecting those two points.
(ii) The sum of the interior angles of a geometrical figure laid on the surface of the earth differs from that of the corresponding figure only to the extent of one second for about 200 square kilometres of area.
Hence, in most of engineering projects plane surveying is used. The geodetic surveying is used to determine the precise positions of control stations on the surface of the earth to which plane survey details are connected in works of larger magnitude like preparing maps of countries. Thus, in surveying there are two primary divisions viz. Geodetic Surveying and Plane Surveying.