Catalan surface

Quality 0.50 · 1 view · Updated 2 months ago

Mathematical concept

title: "Catalan surface" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["surfaces", "geometric-shapes"] description: "Mathematical concept" topic_path: "general/surfaces" source: "https://en.wikipedia.org/wiki/Catalan_surface" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept ::

::callout[type=note] the ruled surfaces ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b4/Catalan_surface.gif" caption="A Catalan surface."] ::

In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose generators are parallel to a fixed plane.

The vector equation of a Catalan surface is given by

:r = s(u) + v L(u),

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 plane of the surface. This can be characterized by the condition: the mixed product [L(u), ***L' ***(u), ***L" ***(u)] = 0.https://books.google.com/books?id=K31Nzi_xhoQC&dq=catalan+surface&pg=PA279

x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) ,

If all the generators of a Catalan surface intersect a fixed line, then the surface is called a conoid.

Catalan proved that the helicoid and the plane were the only ruled minimal surfaces.

A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, Florida:CRC Press, 2006. https://www.crcpress.com/product/isbn/9781584884484 ()
V. Y. Rovenskii, Geometry of curves and surfaces with MAPLE https://books.google.com/books?id=K31Nzi_xhoQC&dq=conoid+maple&pg=PA277 ()

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::