## Principal Curve

A curve on a Regular Surface is a principal curve Iff the velocity always points in a Principal Direction, i.e.,

where is the Shape Operator and 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.