Plücker's conoid

In geometry, Plücker's conoid is a ruled surface named after the German mathematician Julius Plücker.

Plücker's conoid is the surface defined by the function of two variables: This function has an essential singularity at the origin.

By using cylindrical coordinates in space, we can write the above function into parametric equations 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 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 1. Plücker's conoid with n = 2 .
Figure 2. Plücker's conoid with n = 3 .
Figure 3. Plücker's conoid with n = 4 .
Figure 4. Plücker's conoid with n = 2 .
Figure 5. Plücker's conoid with n = 3