Differential Geometry and Topology of Curves 2024, 39: Curves in Hyperplanes

Space-filling curves should already be a familiar notion from topology. By a Poincare map g with some naive composition, since g preserves arclengths, the composition will have an exact multiple of g's arc length. Every such composition has an inverse map, which in itself can be included in a composition. The intersection of the composition and its space argument is non-empty, so the points of an n-times applied map g decompose into infinitesimally short, equal arcs and the resulting set of points is dense everywhere on the arc length. This allows curves to fill space.

A moving straight line l parallel to a vector along a space curve γ forms a cylinder, with the curve as its "directrix". l is the "generator" of the cylinder. If a plane-curve on the (x, y)-plane and l is parallel to the z-axis. The regular curve lying on the cylinder, with a position vector of r(t) = r₀(t) + z(t) where z(t) is the z-axis coordinate. The space curve is the projection of the regular curve into the (x, y)-plane.

When the curvature of the regular curve is non-vanishing, the torsion is bounded by a positive constant and the total length is infinite. If the curvature of the space curve is bounded from above, then its total length is infinite. Assume the regular curve is of infinite total length and a positive curvature k > 0, then its torsion is bounded from below by a positive constant and a closed strongly convex projection, and the regular curve is unbounded. If it's projected into a locally convex curve with positive, but bounded from above torsion, then it's unbonded as well.

Let Γ be a closed regular curve in Euclidean space, then Fenchel's inequality states that

where the inequality holds iff Γ is a convex plane curve. A positive curvature defines a spherical indicatrix of tangents by unit tangent vectors of Γ at some points P, which when gathered into a translation that maps P into the Cartesion origin. The end point of the translated vector is on the unit sphere. If Γ is closed and non-planar, any plane that exists on Γ, points of Γ at maximal/minimal distance from the plane have parallel tangent vectors. The curve intersects any great circle at least in two points and is therefore not fully inside any hemisphere. A closed curve on the unit circle with the length L < 2π, then there is a point Q on the unit circle exists that has an inner distance between itself and any point on the curve of at most L/4.