Noncylindrical Ruled Surface

A Ruled Surface parameterization ${\bf x}(u,v)={\bf b}(u)+v{\bf g}(u)$ is called noncylindrical if ${\bf g}\times{\bf g}'$ is nowhere 0. A noncylindrical ruled surface always has a parameterization of the form

$${\bf x}(u,v) = \boldsymbol{\sigma}(u)+v\boldsymbol{\delta}(u),$$

where $\vert\boldsymbol{\delta}\vert=1$ and $\boldsymbol{\sigma}'\cdot \boldsymbol{\delta}'=0$, where ${\boldsymbol{\sigma}}$ is called the Striction Curve of x and $\boldsymbol{\delta}$ the Director Curve.

See also Distribution Parameter, Ruled Surface, Striction Curve

References

Gray, A. ``Noncylindrical Ruled Surfaces.'' §17.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 345-349, 1993.