Cartesian coordinates of an intrinsically defined curve

Summarizing some old research on the dynamics of a pointall body along its own trajectory, this paper established the differential relationships between the principal curvatures of a 3D curve, that is the normal curvature and the torsional curvature, and its Cartesian coordinates. The differential system thus derived is actually a dynamical system of a representative point of the curve moving along it. This dynamic system is analyzed to see the possibilities of finding analytical solutions in finite terms, using Frobenius' integrability theorem for the general case and usual integration methods for the particular case consisting of the constant ratio between the two curvatures.