Differential Geometry and Topology of Curves 2024, 41: Circles and Periodic Closed Curves

A closed rectifiable plane curve with length L and the area F of a plane domain bounded by it span the "isoperimetric" inequality: L² ≥ 4πF. The equality holds, if the curve is a circle, because it has the smallest lengths among closed rectifiable plane curves bounding a plane domain of the same area. A curve of infinite length is bounded iff the position vector r(T) with function period T is orthogonal to an eigenvector of the space rotation transforming from the natural frame at 0 to that at T, corresponding to the eigenvalue l. l can be viewed as the axis of rotation, and a vector a spanning l is stationary. If the natural frames at 0 and T are different, then a is unique. For boundedness, (r(T), a) = 0, meaning the curve is bounded, iff it's closed.

For total curvature k* and torsion κ* as defined below for a non-planar curve satisfy either one:

If κ is non-negative, the curve is unbounded.

Delaunay's Problem posits two points, through which the curve with constant curvature k = 1 with either the smallest or greatest length is placed. It's not always solvable, but from attempts to solve it arise several smaller theorems. If the distance between the points is less than 2, with α < π, β = 2π where α is the arclength of the unit circle connecting the points. Any other curve with k = 1 connecting them has a length that is either less than α, or larger than β. A plane curve with endpoints P, T with PT forming a closed convex curve, twisting the curve increases the length of PT.

A plane closed Jordan curve decomposes the plane it exists in into two connected components, and is their mutual boundary. For disjoint polygonal lines h, g where either h is closed, or the g is between two vertical lines passing through the end points of h, then all points on g have a constant value N(Q, h), where N is a special function that maps to 0 if the number of points where a ray collinear with a vertical ray through the origin instead passing through Q, intersects g an even amount of times, and to 1 everywhere else. Closed Jordan curves (the image under a homeomorphism) decompose planes into at least two connected components, while simple arcs (the image of an interval under a homeomorphism) don't decompose the plane at all. When a closed curve decomposes the plane into connected components, then it is the boundary of each component.