Plücker's conoid

Figure 1. Plücker's conoid with Template:Math.

Figure 2. Plücker's conoid with Template:Math.

Figure 3. Plücker's conoid with Template:Math.

Template:No footnotes 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

$x=v\cos u,y=v\sin u,z=\sin 2u.$

Thus Plücker's conoid is a right conoid, which can be obtained by rotating a horizontal line about the Template:Nowrap with the oscillatory motion (with period 2π) along the segment Template:Math of the axis (Figure 4).

A generalization of Plücker's conoid is given by the parametric equations

$x=v\cos u,y=v\sin u,z=\sin nu.$

where Template:Mvar denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the Template:Nowrap is Template:Math. (Figure 5 for Template:Math)

Figure 4. Plücker's conoid with Template:Math.

Figure 5. Plücker's conoid with Template:Math

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  Animation of Plucker's conoid with Template:Math
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  Plucker's conoid with Template:Math
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  Plucker's conoid with Template:Math
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  Animation of Plucker's conoid with Template:Math
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  Animation of Plucker's conoid with Template:Math
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  Plucker's conoid with Template:Math

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] (Template:Isbn)
• Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE [2] (Template:Isbn)

External links

• Template:MathWorld

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