Geodetic Lines

Geodetic Lines

 

lines on a surface whose arcs are the shortest distance between the lines’ ends. On a plane surface, geodetic lines are straight; on a cylinder they are wavy; and on a sphere they are great circles. Not every arc of a geodetic line is the shortest path between its ends. For example, on a sphere, the arc of a great circle—a greater semicircle—will not be the shortest path between its ends. A property of geodetic lines is that their principal normals are normals to the surface.

Geodetic lines first appeared in the works of Johann Bernoulli and L. Euler. Since the definition of geodetic lines is linked only with measurements on a surface, the lines belong to objects of so-called interior geometry of surfaces. An understanding of geodetic lines can also be applied in the geometry of elliptical space. The Soviet mathematicians A. D. Aleksandrov and A. V. Pogorelov have done research on the analogues of geodetic lines on convex surfaces.

The concept of geodetic lines is used extensively in theoretical and practical questions of geodesy. Points on the earth’s surface are projected on the surface of a terrestrial ellipsoid and are connected by geodetic lines. There are several special methods for transferring the distances and angles on the earth’s surface to corresponding arcs of geodetic lines and angles between them onto the surface of the terrestrial ellipsoid.

REFERENCES

Liusternik, L. A. Geodezieheskie linii, 2nd ed. Moscow-Leningrad, 1940.
Aleksandrov, A. D. Vnutrenniaia geometriia vypuklykh poverkhnostei. Moscow-Leningrad, 1948.
Pogorelov, A. V. Lektsii po differentsial’noi geometrii, 4th ed. Kharkov, 1967.
Kell’, N. G. Vysshaia geodeziia i geodezicheskie raboty, part 1. Leningrad, 1932.
Krasovskii, F. N. Rukovodstvo po vysshei geodezii, part 2. Moscow, 1942.

E. G. POZNIAK

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