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.

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 $n = 2$ 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 the number of folds in the surface. The difference is that the period of the oscillatory motion along the $z$-axis is $n = 3$. (Figure 5 for $n = 4$)