Differential Geometry and Topology of Curves 2024, 40: Closed Curves

A closed curve on the unit sphere that is not contained to any hemisphere is a spherical indicatrix of some space curve. A planar curve γ with position vector r(s). If its curvature k(s) is continuous and periodic with a period of T, then

for closed γ. Curvature and torsion of a space curve can define a function with

iff γ is closed and its length is L. For polygonal lines formed by segments {P} the notion of curvature and torsion can be approximated by two independent angles φ, θ. The planes spanned by the basis of vectors running along the curve segments determines a unique unit normal vector. One of the independent identifying angles is equal to the angle between said normal vectors. If the curve is closed, then the last segment is determined by the other previous segments, so a closed polygonal line with n segments is determined by a space motion with up to 3n - 6 parameters. A matrix characterizing the curve can be expressed exclusively in terms of the angles, so the closedness of polygonal lines can be expressed through matrix equations constructed by those quantities.

For a fixed positive length l, there is a class of C²-regular closed curves on the unit sphere. A continuous function F(x, y) is either periodic in y with a period of l, or