in mathematics, the partition of a surface into triangles, in general, curvilinear triangles. For example, if a tetrahedron or octahedron is inscribed in a sphere and the surface of the tetrahedron or octahedron is projected onto the surface of the sphere along radii, the sphere (that is, the surface of the sphere) will be separated into four or eight spherical triangles, respectively; the spherical triangles constitute a triangulation.
A generalization of the concept of the triangulation of a surface is the concept of multidimensional triangulation, that is, the n-dimensional triangulation of an n-dimensional polytope. This generalization is the same as the concept of a simplicial complex. A topological space is said to be triangulable if it is homeo-morphic to some polytope. In any topological mapping of a given polytope onto a given triangulable set, every triangulation of the polytope induces a (curvilinear) triangulation of the set. Thus, triangulable sets may also be called curvilinear polytopes.
a method of laying out a network of geodetic control points; a network laid out by this method.
Triangulation consists in the construction of chains or networks of adjacent triangles and the determination of the positions of the vertices of the triangles in a chosen system of coordinates. In each triangle, all three angles are measured, and the length of one of the sides is computed on the basis of the solution of the preceding triangle; in the first triangle one of the sides is measured directly. If the side of a triangle is obtained from direct measurement, it is called a base. In the past, a short line, called a base line, was directly measured instead of a base; the side of the triangle was obtained from the base line by trigonometric calculations using a special network of triangles. This side of the triangle is usually called the known side, and the network of triangles used to calculate the known side is called the base net. To control and improve the accuracy of triangulation chains or networks, the number of base lines or bases measured is greater than the minimum number necessary.
The method of triangulation is commonly regarded as having been invented and first applied by W. Snell between 1615 and 1617: he laid out a chain of triangles in the Netherlands to measure an arc of meridian. The first attempts to use the method for topographic surveys in prerevolutionary Russia were made at the end of the 18th century and the beginning of the 19th century. By the early 20th century the triangulation method had come into general use.
Triangulation has great scientific and practical value. It is used, for example, to determine the figure and size of the earth by means of angle and arc measurements, to study the horizontal movements of the earth’s crust, to carry out topographic surveys on different scales and for different purposes, and to perform various types of geodetic work associated with the surveying for, and the planning and construction of, major structures and with the planning and construction of cities.
A triangulation is constructed on the basis of the principle of going from the whole to the part, that is, from large triangles to smaller ones. For this reason, triangulations are subdivided into classes, or orders, that differ in the precision of measurement and in the construction procedure. In countries with small areas, triangulations of the highest class are constructed in the form of continuous networks of triangles. In countries with large areas, such as the USSR, Canada, the People’s Republic of China, and the USA, triangulations are constructed according to certain schemes and rules. The most efficient system of schemes and rules for the construction of triangulations is that used in the USSR.
State triangulation in the USSR is divided into four classes (see Figure 1). A class 1 triangulation is constructed in the form of chains of triangles with sides 20–25 km long; the chains are laid out approximately along meridians of longitude and parallels of latitude and form polygons with perimeters of 800–1,000 km. The angles of the triangles are measured by highly accurate theodolites with an error of not more than 0.7”. At the places where class 1 triangulation chains intersect, base lines are measured by means of wires; the error of base-line measurement does not exceed 1/1,000,000 of the base-line length, and the known sides of base nets are measured with an error of approximately 1/300,000. After highly accurate electrooptical range finders were invented, bases were measured directly with an error of less than 1/400,000.
The spaces within the class 1 polygons are filled in with continuous networks of class 2 triangles having sides of about 10–20 km; the angles in the class 2 triangles are measured with the same level of precision as in the class 1 triangulation. The base in the class 2 network within the class 1 polygon is measured with the precision indicated above. At the ends of each base in class 1 and class 2 triangulations, the latitude and longitude are determined astronomically with an error of not more than ± 0.4”, and the azimuth is determined in the same way with an error of about ± 0.05”. In addition, latitude and longitude are determined astronomically at intermediate points about 100 km apart in class 1 triangulation chains, but much more closely along certain specially selected chains.
Figure 1
The points of class 3 and 4 triangulations are determined on the basis of the chains and networks of class 1 and 2 triangulations; the spacing of such points depends on the scale of the topographic survey. For example, for a survey scale of 1:5,000, there should be one triangulation point for every 20–30 sq km. The error of angle measurement in class 3 and 4 triangulations does not exceed 1.5” and 2.0”, respectively.
In Soviet practice traversing may be substituted for triangulation. In this case, it is required that the same accuracy in determining the positions of points on the earth’s surface be achieved when either method is used to construct a geodetic control network.
The locations of the vertices of triangulation triangles are marked by wooden or metal towers 6–55 m high, depending on the terrain. To ensure that the triangulation points are preserved for a long period, their locations are fixed in the ground by metal pipes or concrete blocks with metal markers. The markers fix the positions of points for which coordinates are given in appropriate catalogs.
To determine the coordinates of triangulation points, the real earth is replaced by some reference ellipsoid, to the surface of which the results of angle and base measurements are referred.
The Krasovskii reference ellipsoid has been adopted in the USSR. The construction of triangulations and the use of a reference ellipsoid yield a single system of coordinates for a country’s entire area. Such a coordinate system makes it possible to carry out topographic and geodetic tasks in different parts of the country either simultaneously or independently. In addition, a single coordinate system ensures that the results can be combined into a whole and that a uniform national topographic map of the country can be produced on a standard scale.
REFERENCES
Krasovskii, F. N., and V. V. Danilov. Rukovodstvo po vysshei geodezii, 2nd ed., part 1, fase. 1–2. Moscow, 1938–39.
Instruktsiia o postroenii gosudarstvennoi geodezicheskoi seti SSSR, 2nd ed. Moscow, 1966.A. A. IZOTOV
triangulation - a trigonometric method of determining the position of a fixed point from the angles to it from two fixed points a known distance apart; useful in navigation
triangulation - a method of surveying; the area is divided into triangles and the length of one side and its angles with the other two are measured, then the lengths of the other sides can be calculated