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:

: $z=\frac.$

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 with the oscillatory motion (with period 2''π'') along the segment 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 denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the is . (Figure 5 for )

File:Plucker conoid (n=2).gif, Animation of Plucker's conoid with
File:Plucker's conoid (n=2).jpg, Plucker's conoid with
File:Plucker's conoid (n=3).jpg, Plucker's conoid with
File:Plucker's conoid (n=2).gif, Animation of Plucker's conoid with
File:Plucker's conoid (n=3).gif, Animation of Plucker's conoid with
File:Plucker's conoid (n=4).jpg, Plucker's conoid with

See also

*Ruled surface
*Right conoid

References

* A. Gray, E. Abbena, S. Salamon, ''Modern differential geometry of curves and surfaces with Mathematica'', 3rd ed. Boca Raton, Florida:CRC Press, 2006.

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* Vladimir Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE'

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External links

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Surfaces
Geometric shapes

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