geodesic torsion

geodesic torsion

[¦jē·ə¦des·ik ′tȯr·shən]
(mathematics)
For a given point on a surface and a given direction, the torsion of the geodesic on the surface through the point and in the given direction.
For a given curve on a surface at a given point, the torsion of the geodesic through the point in the same direction as the given curve.
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where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 1/[T.sub.g], 1/[R.sub.n] and 1/[R.sub.g] are geodesic torsion, normal curvature and geodesic curvature, respectively.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1, 2) and 1/[R.sub.n] are geodesic torsion, normal curvatures of the parameter curves and the normal curve, respectively.
where 1/[T.sub.g] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1,2) are geodesic torsion of normal curve and normal curvature of parameter curves, respectively.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] {i = 1,2) and 1/[T.sub.g] are geodesic torsions, normal curvatures of the parameter curves and the normal torsion of normal curve, respectively.
In these formulae at (4) and (5) , the functions [[kappa].sub.n], [[kappa].sub.g] and [[tau].sub.g] are called the geodesic curvature, the normal curvature and the geodesic torsion, respectively.
where [k.sub.n](s) = <[??]"(s), n(s)), [k.sub.g](s) = <[??]"(s), g(s)> and [[tau].sub.g](s) = -<n'(s), g(s)> are called the normal curvature, the geodesic curvature and the geodesic torsion of [??], respectively.
The relations for the normal curvature, the geodesic curvature and the geodesic torsion are given by