Intrinsic Equations of a Curve

(also natural equations of a curve), equations that express the curvature k and torsion σ of a curve as functions of the arc length of the curve: k = k(s) and σ = σ(s). The name “intrinsic equations” is chosen because the functions k(s) and σ(s) are independent of the position of the curve in space (independent of the choice of the coordinate system) and these functions depend only on the shape of the curve. Two curves that can be continuously differentiated three times and that have the same intrinsic equations can differ only in their positions in space. In other words, the shape of the curve is uniquely defined by the intrinsic equations of the curve. If we are given two continuous functions k(s) and σ(s), with the first being positive, then there always exists a curve for which the given functions are, respectively, the curvature and the torsion of the curve. (SeeDIFFERENTIAL GEOMETRY.)