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.

Some theorems involving geodesic curvature

• At a point p on a curve C, the geodesic curvature vector κ_g is the projection of the curvature vector κ of C at p onto the tangent plane at p.

• The relation to the regular curvature of the curve is given by: , where κ is the regular curvature and κ_N is the normal curvature.

• The Gauss–Bonnet theorem.

See also

• Curvature
• Darboux frame

References

• do Carmo, Manfredo P. (1976), Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0132125897 
• Guggenheimer, Heinrich (1977), "Surfaces", Differential Geometry, Dover, ISBN 0-486-63433-7 .
• Slobodyan, Yu.S. (2001), "Geodesic curvature", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .

External links

• Weisstein, Eric W., "Geodesic curvature" from MathWorld.