geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 periodi ...

, is a

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

all of whose generators are parallel to a fixed

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

Equations

The

vector equation Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics ...

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 (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...

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 In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. In the case of a single parameter, parametric equations are commonly used to ...

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 generators of a Catalan surface intersect a fixed Line (geometry), line, then the surface is called a

conoid In geometry a conoid () is a ruled surface, whose rulings (lines) fulfill the additional conditions: :(1) All rulings are parallel to a plane, the '' directrix plane''. :(2) All rulings intersect a fixed line, the ''axis''. The conoid is a rig ...

. Catalan proved that the

helicoid The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

and the

were the only ruled

minimal surfaces In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

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