Differential Geometry and Topology of Curves 2024, 38: Osculating Spheres & Planar Curves

Four points on a space curve that aren't on a plane, there is a unique sphere passing through those points. If the limit sphere exists, it's the osculating sphere of the curve at the limit point. The center and radius of the osculating sphere are the limits of centrs and radii of spheres that have all other limits at the convergence point. The position vector of the curve r(s) with the natural parameter s sees a fixed value a at the convergence point and each of the other fixed points corresponds to values s = a + h where each point has its own offset h. The center of the sphere is determined by taking the convergence points and one of the fixed points and place a plane orthogonally through the center of the segment. The center is a point of intersection of each plane determined this way. The center of the osculating sphere is situated in the normal plane to the curve at the convergence point. The curvature and torsion factor into an expression that characterizes spherical curves

Besides curves include ellipses, hyperbolas and parabolas other planar curves include the Bernoulli lemniscate, which is a curve of the forth degree. It relies on two fixed points F, G with a curve formed by points M so that the lengths of MF, MG are constant. F, G are the foci of the lemniscate, defined by

A curve that for every point M sees MF₁...MFₙ = ρ = constant, is a lemniscate with n foci.

An Archimedean spiral is a curve represented by ρ = c₀φ + c₁. It consists of an infinite number of twists. Charted out, it comes out as a line on the positive x-axis, with a regular spirals propagating from 0. Opposed to it, there is a logarithmic spiral described by

The curve intersects all rays whose origin is 0 at a constant angle. With an offset ρ₀, this spiral winds into a circle at the center around the pole with a radius ρ₀. A fraction of two of such expression results in a spiral winding into two curves.