View source for Conoid

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A [[Catalan surface|Catalan surface]] all straight line generators of which intersect a fixed straight line, called an axis of the conoid. For example, a hyperbolic paraboloid is a conoid with two axes.

The position vector of a conoid is given by

$$r=\{u\cos v+\alpha f(v),u\sin v+\beta f(v),\gamma f(v)\},$$

where $\{\alpha,\beta,\gamma\}$ is a unit vector having the same direction as an axis of the conoid and $f(v)$ is some function. For a right conoid $\alpha=\beta=0$, $\gamma=1$, and then its axis is a line of striction. A right conoid with $f(v)=av$ is a [[Helicoid|helicoid]].



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Géométrie différentielle: variétés, courbes et surfaces" , Presses Univ. France  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR></table>