Principal Curve

A curve $\boldsymbol{\alpha}$ on a Regular Surface $M$ is a principal curve Iff the velocity $\boldsymbol{\alpha}'$ always points in a Principal Direction, i.e.,

$$S(\boldsymbol{\alpha}')=\kappa_i \boldsymbol{\alpha}',$$

where $S$ is the Shape Operator and $\kappa_i$ is a Principal Curvature. If a Surface of Revolution generated by a plane curve is a Regular Surface, then the Meridians and Parallels are principal curves.

References

Gray, A. ``Principal Curves'' and ``The Differential Equation for the Principal Curves.'' §18.1 and 21.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 410-413, 1993.