Meaning of the sign of the geodesic curvature

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Discussion Overview

The discussion revolves around the meaning of the sign of the geodesic curvature for coordinate patches and surfaces. Participants explore definitions and implications of geodesic curvature in the context of differential geometry.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants inquire about the definition of geodesic curvature and its significance for both coordinate patches and surfaces.
  • One participant describes the geodesic curvature of a unit speed curve as the component of the curve's second derivative in the direction of a specific vector derived from the normal and tangent vectors.
  • Another participant elaborates that the orientation of the surface and the unit tangent to the curve can determine the sign of the geodesic curvature, noting that if the projection of the curve's curvature vector onto the tangent plane points opposite to the oriented normal, the geodesic curvature is negative.

Areas of Agreement / Disagreement

Participants express similar inquiries regarding the definition and implications of geodesic curvature, but there is no consensus on a singular interpretation or conclusion regarding its meaning.

Contextual Notes

The discussion includes assumptions about the orientation of surfaces and the definitions of curvature, which may not be universally agreed upon or fully resolved.

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.
 
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Leo Mar said:
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?
 
Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.
 
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.

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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