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.

Figure 1. Plücker's conoid with n = 2.

Figure 2. Plücker's conoid with n = 3.

Figure 3. Plücker's conoid with n = 4.

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)

Figure 4. Plücker's conoid with n = 2.

Figure 5. Plücker's conoid with 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

See also

• Ruled surface
• Right conoid
• Wallis's conical edge

References

• A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, Florida:CRC Press, 2006. [1] (ISBN 978-1-58488-448-4)
• Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE [2] (ISBN 978-0-8176-4074-3)

External links

• Weisstein, Eric W. "Plücker's Conoid". MathWorld.