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Striction Curve

A Noncylindrical Ruled Surface always has a parameterization of the form

{\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
{\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)

$\displaystyle \boldsymbol{\sigma}(u)$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}0\\  0\\  bu\end{array}\right]$ (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
{\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)

See also Director Curve, Distribution Parameter, Noncylindrical Ruled Surface, Ruled Surface


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.

© 1996-9 Eric W. Weisstein