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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]

Hyperbolic paraboloid as a conoid.gif

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 {ℓ, m, n} is a vector parallel to the axis of the conoid and ƒ(u) is some function.

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

Ruled surface
Right conoid
Catalan surface

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)