If longitudinal is constant and not a circle of latitude, then is a geodetic line.
proof
Let be a parameterization of the area , where we o. B. d. A. as arc length of the generating curve may take. With this we add the coefficients to the 1st fundamental form
, , .
Be o. B. d. A. parameterized according to the arc length . In order to be able to apply Liouville's theorem , we explicitly calculate the geodetic curvatures of the lines (parallels) and lines (meridians):
Hence the geodesic curvature of the curve to
(1)
Differentiating the function yields:
With follows from (1)
and with it the claim.
Use in national surveying
In national surveying , the problem arises of calculating a geodetic line for a given starting point and direction, the so-called first geodetic main task .
The azimuth is the angle of intersection of the line with the north direction. With this follows from Clairaut's theorem the constancy of
along the geodetic. If the reduced width is introduced according to the formula , then the constancy of follows
This value is called Clairaut's constant of the geodetic line.
literature
Michael Spivak : A Comprehensive Introduction to Differential Geometry. Volume 3. 3rd edition. Publish or Perish Press, Houston TX 1999, ISBN 0-914098-72-1 , pp. 214-216.