# Conoid

From Encyclopedia of Mathematics

A 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.

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#### References

[a1] | M. Berger, B. Gostiaux, "Géométrie différentielle: variétés, courbes et surfaces" , Presses Univ. France (1987) |

[a2] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) |

**How to Cite This Entry:**

Conoid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Conoid&oldid=32670

This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article