On the explicit characterization of spherical curves in n-dimensional Euclidean space

It is known that a curve in 3-dimensional Euclidean space is spherical if and only if 1/k1 k2 + [1/k2 (1/k1)] = 0 (k1 ≠ 0, k2 ≠ 0) (1) where k1 and k2 are its first curvature function and second curvature function, respectively. In 1971, integral form of (1) was given [2] as 1/k1 = A cos ( ∫ k2(s) ds) + B sin (k2 (s) ds) (2) In the present work, a) it is given another method for (2); b) it is shown that the differential equation characterizing a spherical curve in n-dimensional Euclidean space n ≥ 3 can be solved explicitly to express nth curvature function of the curve in terms of its curvatures and its other curvature functions; c) it is shown that integral form of the generalization of (1) gives us (2) as a spherical case for n = 3.