Right conoid

Ruled surface made of lines orthogonal to an axis
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 {\displaystyle x=v\cos u}
y = v sin ⁡ u {\displaystyle y=v\sin u}
z = h ( u ) {\displaystyle 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 {\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: x = v cos ⁡ u , y = v sin ⁡ u , z = c u . {\displaystyle x=v\cos u,y=v\sin u,z=cu.}
  • Whitney umbrella: x = v u , y = v , z = u 2 . {\displaystyle x=vu,y=v,z=u^{2}.}
  • Wallis's conical edge: x = v cos ⁡ u , y = v sin ⁡ u , z = c a 2 − b 2 cos 2 ⁡ u . {\displaystyle 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 ⁡ n u . {\displaystyle x=v\cos u,y=v\sin u,z=c\sin nu.}
  • hyperbolic paraboloid: x = v , y = u , z = u v {\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

  • "Conoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Right Conoid from MathWorld.
  • Plücker's conoid from MathWorld


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