Striction Curve

A Noncylindrical Ruled Surface always has a parameterization of the form

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

(1)

where $\vert\boldsymbol{\delta}\vert=1$ , $\boldsymbol{\sigma}'\cdot \boldsymbol{\delta}'=0$ , and ${\boldsymbol{\sigma}}$ is called the striction curve of x. Furthermore, the striction curve does not depend on the choice of the base curve. The striction and Director Curves of the Helicoid

$\begin{displaymath} {\bf x}(u,v)=\left[{\matrix{0\cr 0\cr bu\cr}}\right]+av\left[{\matrix{\cos u\cr \sin u\cr 0}}\right] \end{displaymath}$

(2)

are

$\displaystyle \boldsymbol{\sigma}(u)$	$\textstyle =$	$\displaystyle \left[\begin{array}{c}0\\ 0\\ bu\end{array}\right]$	(3)
$\displaystyle \boldsymbol{\delta}(u)$	$\textstyle =$	$\displaystyle \left[\begin{array}{c}a\cos u\\ a\sin u\\ 0\end{array}\right].$	(4)

For the Hyperbolic Paraboloid

$\begin{displaymath} {\bf x}(u,v)=\left[{\matrix{u\cr 0\cr 0\cr}}\right]+v\left[{\matrix{0\cr 1\cr u\cr}}\right], \end{displaymath}$

(5)

the striction and Director Curves are

$\displaystyle \boldsymbol{\sigma}(u)$	$\textstyle =$	$\displaystyle \left[\begin{array}{c}u\\ 0\\ 0\end{array}\right]$	(6)
$\displaystyle \boldsymbol{\delta}(u)$	$\textstyle =$	$\displaystyle \left[\begin{array}{c}0\\ 1\\ u\end{array}\right].$	(7)

References

Gray, A. ``Noncylindrical Ruled Surfaces'' and ``Examples of Striction Curves of Noncylindrical Ruled Surfaces.'' §17.3 and 17.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 345-350, 1993.