Conoid

Conoid
The hyperbolic paraboloid z=xy is a conoid with x-axis as its axis.
The hyperbolic paraboloid z=xy is a conoid with y-axis as its axis.

In geometry, a conoid is a Catalan surface all of whose rulings intersect a fixed line, called the axis of the conoid. If all its rulings are perpendicular to its axis, then the conoid is called a right conoid.[1]

For example, the hyperbolic paraboloid z = xy is a conoid (in fact, a right conoid) with x-axis and y-axis as its two axes.

A conoid can be represented by parametric equations

$x=v\cos u+lf(u), y=v\sin u+mf(u), z=nf(u) \,$

where {mn} is a vector parallel to the axis of the conoid and ƒ(u) is some function.

If  = m = 0 and n = 1, then the conoid is a right conoid.

References

• A. Gray, E. Abbena, S. Salamon,Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, FL:CRC Press, 2006. [2] (ISBN:9781584884484)
• Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE [3] (ISBN 978-0-8176-4074-3)