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Leo Mar
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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.
Leo Mar said:My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.
Leo Mar said:Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.
The sign of the geodesic curvature represents the direction of the curvature of a curve on a surface at a given point. It is positive if the curve is curving in the same direction as the surface, and negative if it is curving in the opposite direction.
The sign of the geodesic curvature is calculated using the Frenet-Serret formulas, which take into account the curvature and torsion of the curve as well as the shape of the surface at the given point.
A positive sign of the geodesic curvature indicates that the curve is convex at the given point, meaning that it is curving outward from the surface.
A negative sign of the geodesic curvature indicates that the curve is concave at the given point, meaning that it is curving inward toward the surface.
The sign of the geodesic curvature is used in various applications in mathematics, physics, and engineering, particularly in the study of curved surfaces and differential geometry. It is also used in navigation, such as in determining the shortest path between two points on a curved surface.