As is well known from Fenchel’s theorem, the total absolute curvature of any closed space curve in Euclidean 3-space is always not less than \(2\pi \) . By the fundamental theorem of space curves, a curve is determined by its curvature and torsion functions, and in the case of a closed curve, these functions must be periodic. However, it remains an open problem to decide which periodic curvature and torsion functions actually give rise to closed space curves; this is the so-called closed curve problem. In this paper, we propose a constructive method to obtain closed spherical curves by controlling only the values of the first integral of a periodic function, which then determines their periodic curvature and torsion. We further provide a solution to the closed curve problem in the special case of rotationally symmetric curves. In addition, our approach extends naturally to closed curves with singularities, including closed planar Legendre curves, closed spherical fronts and closed framed curves in Euclidean 3-space. For these cases, we show precise conditions that characterize when (2,3)-cusps occur on such curves.
