Right conoid

A right conoid as a ruled surface.

In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the axis of the right conoid.

Using a Cartesian coordinate system in three-dimensional space, if we take the z-axis to be the axis of a right conoid, then the right conoid can be represented by the parametric equations:

x=v\cos u,y=v\sin u,z=h(u)

where h(u) is some function for representing the height of the moving line.

Examples

Generation of a typical right conoid

A typical example of right conoids is given by the parametric equations

x=v\cos u,y=v\sin u,z=2\sin u

The image on the right shows how the coplanar lines generate the right conoid.

Other right conoids include:

Helicoid: $x=v\cos u,y=v\sin u,z=cu.$
Whitney umbrella: $x=vu,y=v,z=u^{2}.$
Wallis’s conical edge: $x=v\cos u,y=v\sin u,z=c{\sqrt {a^{2}-b^{2}\cos ^{2}u}}.$
Plücker’s conoid: $x=v\cos u,y=v\sin u,z=c\sin nu.$
hyperbolic paraboloid: $x=v,y=u,z=uv$ (with x-axis and y-axis as its axes).

Hazewinkel, Michiel, ed. (2001) [1994], "Conoid", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Right Conoid from MathWorld.
Plücker's conoid from MathWorld

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