Here geodesic circle means a curve in M whose first curvature is constant and whose second curvature
is identically zero.
Then a is congruent to an osculating curve of the second kind if and only if the second curvature
For an arbitrary curve [phi] with first and second curvature
, [kappa] and [tau] in the space [E.
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.
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.