Concept# Plücker's conoid

Summary

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=\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\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
: x=v \cos u,\quad y=v \sin u,\quad z= \sin nu.
where n denotes

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