geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a Catalan surface, named after the Belgian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Eugène Charles Catalan Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic ...

, is a

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directri ...

all of whose rulings are parallel to a fixed plane.

Equations

The

vector equation In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...

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 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...

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'')=

The parametric equations of the Catalan surface ar

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

Special cases

If all the rulings of a Catalan surface intersect a fixed Line (geometry), line, then the surface is called a conoid. Catalan proved that the

helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similar ...

and the plane were the only

ruled ''Ruled'' is the fifth full-length LP by The Giraffes. Drums, bass and principal guitar tracks recorded at The Bunker in Brooklyn, NY. Vocals and additional guitars recorded at Strangeweather in Brooklyn, NY. Mixed at Studio G in Brooklyn, NY ...

minimal surfaces.

References

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

() * * V. Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE'

({{ISBN, 978-0-8176-4074-3) Surfaces Geometric shapes

Equations

Special cases

See also

References