Geodesic curvature

In differential geometry, the geodesic curvature vector is a property of curves in a metric space which reflects the deviance of the curve from following the shortest arc length distance along each infinitesimal segment of its length.

The vector is defined as follows: at a point "P" on a curve "C", the geodesic curvature vector k_g is the curvature vector k of the projection of the curve "C" onto the tangent plane at "P".

The scalar magnitude of the geodesic curvature vector is simply called the geodesic curvature $k\_g$. A curve for which the geodesic curvature is everywhere vanishing is called a geodesic.

ome theorems involving geodesic curvature

*At a point $p$ on a curve $C$, the geodesic curvature vector $kappa\_g$ is the projection of the curvature vector $kappa$ of $C$ at $p$ onto the tangent plane at $p$.

*The relation to the regular curvature of the curve is given by: $kappa^2\; =\; kappa\_g^2\; +\; kappa\_N^2$, where $kappa$ is the regular curvature and $kappa\_N$ is the normal curvature.

*The Gauss-Bonnet theorem.

ee also

* Curvature
* Darboux frame

References

*citation | last = do Carmo|first =Manfredo P. | title=Differential Geometry of Curves and Surfaces | publisher=Prentice-Hall | year=1976 | id = ISBN 0132125897
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External links

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