second curvature

second curvature

[′sek·ənd ¦kər·və·chər]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Here geodesic circle means a curve in M whose first curvature is constant and whose second curvature is identically zero.
Then its second curvature [k.sub.2](s) can be equal to zero or different from zero.
Every null rectifying curve in [E.sup.4.sub.1] with the second curvature [k.sub.2](s) = 0 is an osculating curve of the second kind.
For an arbitrary curve [phi] with first and second curvature, [kappa] and [tau] in the space [E.sup.3.sub.1], the following Frenet formulae are given in [4]:
In general, a geodesic circle (a curve whose first curvature is constant and second curvature is identically zero) does not transform into a geodesic circle by the conformal transformation
There are only two curvatures in this case, the second curvature is determined only up to a constant factor.
For a space-like curve [phi] with first and second curvature, K and T in the space [E.sup.3.sub.1], the following Frenet formulae are given in [4].
For an arbitrary curve [phi] with first and second curvature, [kappa] and [tau] in the space [E.sup.3], the following Frenet formulae are given in [3], [5]
Here, first and second curvature are defined by [kappa] = [kappa](s) = [absolute value of T'(s)] and [tau](s) = <N, B'>.
Taking the norm of both sides, we get second curvature and second binormal as
For the unit speed curve f with the first and second curvatures, [kappa] and [tau] in the space [R.sup.3], the following Frenet-Serret formulae are given by