Plücker's conoid

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.

Figure 1. Plücker's conoid with n = 2.

Figure 2. Plücker's conoid with n = 3.

Figure 3. Plücker's conoid with n = 4.

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

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

This function has an essential singularity at the origin.

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

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

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

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

${\displaystyle x=v\cos u,\quad y=v\sin u,\quad z=\sin nu.}$

where n denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the z-axis is 2π/n. (Figure 5 for n = 3)

Figure 4. Plücker's conoid with n = 2.

Figure 5. Plücker's conoid with n = 3

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  Animation of Plucker's conoid with n = 2
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  Plucker's conoid with n = 2
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  Plucker's conoid with n = 3
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  Animation of Plucker's conoid with n = 2
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  Animation of Plucker's conoid with n = 3
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  Plucker's conoid with n = 4