Clairaut's relation

The set of Clairaut (named after Alexis -Claude Clairaut ) is a statement of classical differential geometry.

Statement

Be a surface of revolution and a regular curve. Denote the radius of the parallel of by as well as the angle of intersection of the curve with this parallel. Then apply:

• If a geodesic, then the function is constant along.
• Is constant along and not parallel, as is a geodesic.

Evidence

Be a parameterization of the surface, we can assume as arc length of the generating curve wlog. Thus, we calculate the coefficients of the first fundamental form to

Be wlog parameterized by arc length. To apply the theorem of Liouville, we explicitly calculate the geodesic curvatures of the lines ( parallels ) and lines ( meridians):

Hence the geodesic curvature of the curve to

Differentiating the function returns:

With the following ( 1)

And thus the assertion.

Application in the national survey

In the national survey, there is the problem of computing, for a given starting point and direction of a geodesic line, the so-called first geodesic principal task.

And are the semi-axes of the reference ellipsoid and the square of the (first) numerical eccentricity. Is the radius of the parallel of latitude with the ellipsoidal

As an azimuth is defined as the angle of intersection of the line with the direction of North. Thus, from the set of Clairaut follows the constancy of

Along the geodesic. Performs to the reduced width according to the formula, it follows that the constancy of

This value is called the Clairaut constant of the geodesic.