Right conoid

 ( Surfaces ) 

In geometry, a right conoid is a ruled surface generated by a family of that all intersect 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 to be the axis of a right conoid, then the right conoid can be represented by the parametric equations:

$x=v\backslash cos\; u$

$y=v\backslash sin\; u$

$z=h(u)$

where is some function for representing the height of the moving line.

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

$x=v\backslash cos\; u,\; y=v\backslash sin\; u,\; z=2\backslash sin\; u$

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

Other right conoids include:

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

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See also

• Conoid
• Helicoid
• Whitney umbrella
• Ruled surface

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External links

• Right Conoid from MathWorld.
• Plücker's conoid from MathWorld

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