Right conoid

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:

${\displaystyle x=v\cos u}${\displaystyle x=v\cos u}

${\displaystyle y=v\sin u}${\displaystyle y=v\sin u}

${\displaystyle z=h(u)}${\displaystyle z=h(u)}

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

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

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

• Conoid
• Helicoid
• Whitney umbrella
• Ruled surface

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

• Surfaces
• Geometric shapes
• Geometry stubs

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