Clairaut's theorem (differential geometry)

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

statement

Is a surface of revolution and with a regular curve on . It denotes the radius of the circle of latitude through and the angle of intersection of the curve with this circle of latitude. Then apply: ${\displaystyle S}$ ${\displaystyle c:]a,b[\to S}$ ${\displaystyle s\mapsto c(s)}$ ${\displaystyle S}$ ${\displaystyle r(s)}$ ${\displaystyle c(s)}$ ${\displaystyle \alpha (s)}$

• If there is a geodetic line , the function is constant along its length . ${\displaystyle c}$ ${\displaystyle r\cos \alpha \,}$ ${\displaystyle c}$
• If longitudinal is constant and not a circle of latitude, then is a geodetic line. ${\displaystyle r\cos \alpha \,}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle c}$

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 ${\displaystyle (u,v)\rightarrow (r(v)\cos u,r(v)\sin u,h(v))}$ ${\displaystyle S}$ ${\displaystyle v}$ ${\displaystyle (r(v),h(v))}$

${\displaystyle E(u,v)=(r(v))^{2}}$, , . ${\displaystyle F(u,v)=0}$ ${\displaystyle G(u,v)=1}$

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): ${\displaystyle s\rightarrow {\vec {c}}(s)}$ ${\displaystyle s}$ ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle \kappa _{g,u}=-{\frac {1}{r}}{\frac {dr}{dv}},\qquad \kappa _{g,v}=0.}$

Hence the geodesic curvature of the curve to ${\displaystyle {\vec {c}}}$

${\displaystyle \kappa _{g}=-{\frac {1}{r}}{\frac {dr}{dv}}\cos \alpha +{\frac {d\alpha }{ds}}.}$ (1)

Differentiating the function yields: ${\displaystyle C(s)=r({\vec {c}}(s))\cos \alpha (s)}$

${\displaystyle {\frac {dC}{ds}}={\frac {dr}{dv}}{\frac {dv}{ds}}\cos \alpha -r\sin \alpha \cdot {\frac {d\alpha }{ds}}.}$

With follows from (1) ${\displaystyle {\frac {dv}{ds}}={\frac {\sin \alpha }{\sqrt {G}}}}$

${\displaystyle {\frac {dC}{ds}}=-r\sin \alpha \cdot \kappa _{g}}$

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 .

Let and be the semi-axes of the reference ellipsoid and the square of the (first) numerical eccentricity . The radius of the circle of latitude with the ellipsoidal latitude is ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle e^{2}=(a^{2}-b^{2})/a^{2}}$ ${\displaystyle \phi }$

${\displaystyle P(\phi )=N(\phi )\cos \phi ={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}}\cos \phi .}$

The azimuth is the angle of intersection of the line with the north direction. With this follows from Clairaut's theorem the constancy of

${\displaystyle {\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}}\cos \phi \sin A}$

along the geodetic. If the reduced width is introduced according to the formula , then the constancy of follows ${\displaystyle \beta }$ ${\displaystyle \tan \beta ={\sqrt {1-e^{2}}}\tan \phi }$

${\displaystyle \cos \beta \sin A.}$

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.