https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=75.165.90.51formulasearchengine - User contributions [en]2026-05-24T01:53:52ZUser contributionsMediaWiki 1.43.0-wmf.28https://en.formulasearchengine.com/index.php?title=Beltrami_vector_field&diff=24505Beltrami vector field2013-08-26T00:45:48Z<p>75.165.90.51: The sign of the right-hand side for the curl of the curl of F was wrong. The next two equations also needed correction re sign.</p>
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<div>[[Image:Catalan surface.gif|right|thumb|600px| A Catalan surface.]]<br />
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{{About|the ruled surfaces|the minimal surface|Catalan's minimal surface}}<br />
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In [[geometry]], a '''Catalan surface''', named after the Belgian [[mathematician]] [[Eugène Charles Catalan]], is a [[ruled surface]] all of whose rulings are parallel to a fixed [[Plane (geometry)|plane]]. The [[vector equation]] of a Catalan surface is given by <br />
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:'''''r''''' = '''''s'''''(''u'') + ''v'' '''''L'''''(''u''),<br />
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where '''''r''''' = '''''s'''''(''u'') is the space curve and '''''L'''''(''u'') is the [[unit vector]] of the ruling at ''u'' = ''u''. All the vectors '''''L'''''(''u'') are parallel to the same plane, called the ''[[Directrix (rational normal scroll)|directrix]] plane'' of the surface. This can be characterized by the condition: the [[Scalar triple product|mixed product]] ['''''L'''''(''u''), '''''L' '''''(''u''), '''''L" '''''(''u'')] = 0.[http://books.google.com/books?id=K31Nzi_xhoQC&pg=PA279&dq=catalan+surface&ei=7GFySs2CN5qIlQTCsvSDAQ#v=onepage&q=catalan%20surface&f=false]<br />
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The [[parametric equations]] of the Catalan surface are [http://www.mathcurve.com/surfaces/catalan/catalan.shtml]<br />
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<math> x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) \,</math><br />
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If all the rulings of a Catalan surface intersect a fixed [[Line (geometry)|line]], then the surface is called a [[conoid]].<br />
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Catalan proved that the [[helicoid]] and the [[Plane (geometry)|plane]] were the only [[Ruled surface|ruled]] [[minimal surfaces]].<br />
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==See also==<br />
*[[Ruled surface]]<br />
*[[Conoid]]<br />
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==References==<br />
* 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)<br />
* {{springer|title=Catalan surface|id=p/c020710}}<br />
* V. 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)<br />
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[[Category:Surfaces]]<br />
[[Category:Geometric shapes]]<br />
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{{geometry-stub}}</div>75.165.90.51