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[[Image:Plucker's conoid (n=2).jpg|right|thumb|240px|Figure 1. Plücker’s conoid with ''n''=2.]][[Image:Plucker's conoid (n=3).jpg|right|thumb|240px|Figure 2. Plücker’s conoid with ''n'' = 3.]][[Image:Plucker's conoid (n=4).jpg|right|thumb|240px|Figure 3. Plücker’s conoid with ''n'' = 4.]]
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In [[geometry]], the '''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]].
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The Plücker’s conoid is defined by the function of two variables:
 
: <math>z=\frac{2xy}{x^2+y^2}.</math>
 
By using [[cylindrical coordinates]] in space, we can write the above function into parametric equations
 
: <math> x=v\cos u,\quad y=v\sin u,\quad z=\sin 2u. </math>
 
Thus the 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 [&minus;1,&nbsp;1] of the axis (Figure 4).  
 
A generalization of the Plücker’s conoid is given by the parametric equations
 
: <math> x=v \cos u,\quad y=v \sin u,\quad z= \sin nu. </math>
 
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&nbsp;2''π''/''n''. (Figure&nbsp;5 for ''n''&nbsp;=&nbsp;3)
 
[[File:Plucker conoid (n=2).gif|thumb|480px|Figure 4. Plücker’s conoid with ''n''&nbsp;=&nbsp;2.]]
[[File:Plucker conoid (n=3).gif|thumb|720px|Figure 5. Plücker’s conoid with ''n''&nbsp;=&nbsp;3]]
 
==See also==
*[[Ruled surface]]
*[[Right conoid]]
 
==External links==
*[http://mathworld.wolfram.com/PlueckersConoid.html Plücker’s conoid ] from MathWorld
 
==References==
* A. Gray, E. Abbena, S. Salamon,''Modern differential geometry of curves and surfaces with Mathematica'', 3rd ed. Boca Raton, FL:CRC Press, 2006. [http://www.crcpress.com/ecommerce_product/product_detail.jsf?catno=C4487&isbn=0000000000000] (ISBN 9781584884484)
 
* Vladimir Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE'' [http://books.google.com/books?id=K31Nzi_xhoQC&pg=PA277&dq=conoid+maple&lr=&ei=B9hvSs_qKYzSkASR8c3XDg] (ISBN 978-0-8176-4074-3)
 
{{DEFAULTSORT:Plucker's Conoid}}
[[Category:Surfaces]]
[[Category:Geometric shapes]]
 
 
{{geometry-stub}}

Latest revision as of 00:28, 2 October 2014

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