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:

${\displaystyle x=v\cos u,y=v\sin u,z=h(u)\,}$

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

${\displaystyle x=v\cos u,y=v\sin u,z=2\sin u\,}$

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

Other right conoids include:

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

See also

• Conoid
• Helicoid
• Whitney umbrella
• Ruled surface

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