Differential Geometry and Topology of Curves 2024, 37: Frenet Formulas

A space curve γ given by its position vector r as a vector-valued function with arc length s on which unit tangent vector τ, principal normal ν and binormal β depend. The three vector quantities are mutually orthogonal, forming a basis of Euclidean space called the "natural frame". The derivatives of these vectors are decomposed into linear combinations to it. The coefficients of the decomposition follow from the principal normal. The decompositions of the vectors are the "Frenet Formulas", used to investigate the behaviour of γ at a neighbourhood U of a point P on γ by way of Taylor expansion. The definitions of the Frenet Formulas conveniently replace some of the more difficult expressions following from the differentiation.

For continuous functions f(s), g(s) with s in [0, l] so that f is positive everywhere. Assume a point p and three orthogonal unit vectors are fixed in Euclidean space. The unique C²-regular curve γ passes through p and its arc length is counted from p. The unit vectors form the natural frame at p, and f is the curvature and g is the torsion of γ. Any curve then is determined uniquely through curvature and torsion up to a motion of Euclidean space. They are the "natural equations" of γ. It can be reduced to one Ricatti equation for a complex vector. Number sequences {s}, {φ}, {ϕ} satisfying sᵢ > 0, 0 < φᵢ < π, 0 ≤ ϕᵢ ≤ 2π. With an arbitrary point p, an oriented plane e passing through p and an arbitrary parallel vector a, there is a unique polygonal line g with {s} the lengths of segments {a} forming g. {φ} are the curvature angles and {ϕ} are the torsion angles of g.