I Meaning of the sign of the geodesic curvature

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My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.
 

lavinia

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My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.
how do you define geodesic curvature?
 
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Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.
 

lavinia

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Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.
OK.

Since the surface is oriented the unit tangent to the curve determines an oriented orthogonal direction by 90 degree rotation. This direction may be opposite to the projection of the curve's curvature vector onto the tangent plane. If the projection points opposite to the oriented normal then the geodesic curvature is negative.
 
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