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 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: $x=v\cos u, y=v \sin u, z=c\sqrt{a^2-b^2\cos^2u}.\,$
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).

See also

External links

Hazewinkel, Michiel, ed. (2001), "Conoid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Right Conoid from MathWorld.
Plücker's conoid from MathWorld