fundamental theorem of space curves

Informal summary.

The curvatureMathworldPlanetmathPlanetmath and torsionMathworldPlanetmath of a space curveMathworldPlanetmath are invariant with respect to Euclidean motions. Conversely, a given space curve is determined up to a Euclidean motion, by its curvature and torsion, expressed as functions of the arclength.


Let 𝜸:I be a regular, parameterized space curve, without points of inflection. Let κ(t),τ(t) be the corresponding curvature and torsion functions. Let T:33 be a EuclideanPlanetmathPlanetmath isometry. The curvature and torsion of the transformed curve T(𝜸(t)) are given by κ(t) and τ(t), respectively.

Conversely, let κ,τ:I be continuous functionsMathworldPlanetmathPlanetmath, defined on an interval I, and suppose that κ(t) never vanishes. Then, there exists an arclength parameterization 𝜸:I of a regular, oriented space curve, without points of inflection, such that κ(t) and τ(t) are the corresponding curvature and torsion functions. If 𝜸^:I is another such space curve, then there exists a Euclidean isometry T:33 such that 𝜸^(t)=T(𝜸(t)).

Title fundamental theorem of space curvesMathworldPlanetmath
Canonical name FundamentalTheoremOfSpaceCurves
Date of creation 2013-03-22 13:23:28
Last modified on 2013-03-22 13:23:28
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Theorem
Classification msc 53A04
Related topic SpaceCurve