Differential Geometry and Topology of Curves 2024, 36: Algebraic Properties of Curves

Take a point P on a curve γ containing the points a, b on different sides with respect to P, and with a plane passing through the three of them. When a, b tend toward P, the resulting plane is called the "osculating plane" on γ at P. If it exists, then it's unique. If γ is C²-regular represented as r = r(t). If vectors r', r'' are not collinear, then the osculating plane is the one spanned between said vectors. If the vectors r', r'' are colinear, the limit position of the plane is not determined and the corresponding point of γ is called the straightening point of γ. The osculating plane is independent from the parametrization of γ. On a planar curve, it coincides with the plane containing the curve. [r'(t₀), r''(t₀)] is orthogonal to the osculating plane. The length of a curve is determined by the integral analogous to the length of polygonal curves. Assuming, it's smooth (and thus rectifiable, meaning that the length of all polygonal lines within it is bounded from above),

The length is independent of parametrization. The curvature k of γ exists and is calculated by its curvature vector, which has a length equal to the curvature. Its torsion κ derives from the the derivative of the angle on the curve with respect to some distance on the curve.

For plane curves, the curvature is defined analogously, from which derives the osculating circle.

Given a plane curve γ characterized by a an equation φ(x, y) = 0, a point M on γ is a singularity of γ if

If D > 0, then M is an isolated point, if D < 0, M is a point of self-intersection, and if D = 0, M is either an isolated point, a cusp, or an osculate point. In cusps, the tangent ray does not decompose into different branches, while this is the case in osculate points.

A continuous map of an interval whose image is a square is Peano's Curve, constructed by decomposing an interval into four equal intervals indexed left to right, doing the same to a square into four equal squares so that each smaller square shares a side with a consecutive square. Doing so recursively will give a curve with correspondence between intervals and squares. For every point on the interval there is a point of the large square with ordering preserved. Each point on the interval belongs to an infinite sequence of enclosed intervals, with the corresponding squares forming an infinite sequence of enclosed squares. As the lengths of the squares converge to 0, there is a unique point P in the square, belonging to all squares of the sequence and hence a continuous map from the interval to the sequence of squares. A plane curve is an "envelope" of a family of plane curves, if it tangents at each point to a curve of the family passing through P.