Plücker's conoid

{{Short description|Right conoid ruled surface}}

Image:Plucker's conoid (n=2).jpgImage:Plucker's conoid (n=3).jpgImage:Plucker's conoid (n=4).jpg

{{No footnotes|date=September 2022}}

In geometry, Plücker's conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a conical wedge or cylindroid; however, the latter name is ambiguous, as "cylindroid" may also refer to an elliptic cylinder.

Plücker's conoid is the surface defined by the function of two variables:

: $z=\backslash frac\{2xy\}\{x^2+y^2\}.$

This function has an essential singularity at the origin.

By using cylindrical coordinates in space, we can write the above function into parametric equations

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

Thus Plücker's conoid is a right conoid, which can be obtained by rotating a horizontal line about the {{nowrap|{{mvar|z}}-axis}} with the oscillatory motion (with period 2π) along the segment {{math|[–1, 1]}} of the axis (Figure 4).

A generalization of Plücker's conoid is given by the parametric equations

: $x=v\; \backslash cos\; u,\backslash quad\; y=v\; \backslash sin\; u,\backslash quad\; z=\; \backslash sin\; nu.$

where {{mvar|n}} denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the {{nowrap|{{mvar|z}}-axis}} is {{math|{{sfrac|2π|n}}}}. (Figure 5 for {{math|1=n = 3}})

File:Plucker conoid (n=2).gif

File:Plucker conoid (n=3).gif

File:Plucker conoid (n=2).gif|Animation of Plucker's conoid with {{math|1=n = 2}}

File:Plucker's conoid (n=2).jpg|Plucker's conoid with {{math|1=n = 2}}

File:Plucker's conoid (n=3).jpg|Plucker's conoid with {{math|1=n = 3}}

File:Plucker's conoid (n=2).gif|Animation of Plucker's conoid with {{math|1=n = 2}}

File:Plucker's conoid (n=3).gif|Animation of Plucker's conoid with {{math|1=n = 3}}

File:Plucker's conoid (n=4).jpg|Plucker's conoid with {{math|1=n = 4}}