Principal Vector

A tangent vector ${\bf v}_{\bf p}=v_1{\bf x}_u+v_2{\bf x}_v$ is a principal vector Iff

$$\mathop{\rm det}\left[{\matrix{{v_2}^2 & -v_1v_2 & {v_1}^2\cr E & F & G\cr e & f & g\cr}}\right]=0,$$

where $e$, $f$, and $g$ are coefficients of the first Fundamental Form and $E$, $F$, $G$ of the second Fundamental Form.

See also Fundamental Forms, Principal Curve

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 410, 1993.