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Right Conoid

A Ruled Surface is called a right conoid if it can be generated by moving a straight Line intersecting a fixed straight Line such that the Lines are always Perpendicular (Kreyszig 1991, p. 87). Taking the Perpendicular plane as the $xy$-plane and the line to be the x-Axis gives the parametric equations

$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle v\cos \vartheta(u)$  
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle v\sin \vartheta(u)$  
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle h(u)$  

(Gray 1993). Taking $h(u)=2u$ and $\vartheta(u)=u$ gives the Helicoid.

See also Helicoid, Plücker's Conoid, Wallis's Conical Edge


Dixon, R. Mathographics. New York: Dover, p. 20, 1991.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 351-352, 1993.

Kreyszig, E. Differential Geometry. New York: Dover, 1991.

© 1996-9 Eric W. Weisstein