title: Plücker's conoid
text: # 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 [–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 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)
  
  * Animation of Plucker's conoid with n = 2
  * Plucker's conoid with n = 2
  * Plucker's conoid with n = 3
  * Animation of Plucker's conoid with n = 2
  * Animation of Plucker's conoid with n = 3
  * Plucker's conoid with n = 4
  