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: 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 2"π") 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 2"π"/"n". (Figure 5 for "n" = 3)