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Striction Curve

A noncylindrical ruled surface always has a parameterization of the form

x(u,v)=sigma(u)+vdelta(u),
(1)

where |delta|=1, sigma^'·delta^'=0, and 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

x(u,v)=[0; 0; bu]+av[cosu; sinu; 0]
(2)

are

sigma(u)=[0; 0; bu]
(3)
delta(u)=[acosu; asinu; 0].
(4)

For the hyperbolic paraboloid

x(u,v)=[u; 0; 0]+v[0; 1; u],
(5)

the striction and director curves are

sigma(u)=[u; 0; 0]
(6)
delta(u)=[0; 1; u].
(7)

See also

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

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References

Gray, A. "Noncylindrical Ruled Surfaces" and "Examples of Striction Curves of Noncylindrical Ruled Surfaces." §19.3 and 19.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 445-449, 1997.

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Striction Curve

Cite this as:

Weisstein, Eric W. "Striction Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StrictionCurve.html

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