intrinsic equation

A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the following problem: given two functions of one parameter, find the space curve for which the functions are the curvature and torsion. Euler gave an integral solution for plane curves (which always have torsion τ = 0). Call the angle between the tangent line to the curve and the x-axis ϕ the tangential angle, then ϕ = integral κ(s) d s, where κ is the curvature.

Cesàro equation | natural parametric equations | Whewell equation