Catalan surface
Encyclopedia
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. The vector equation of a Catalan surface is given by
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.http://books.google.com/books?id=K31Nzi_xhoQC&pg=PA279&dq=catalan+surface&ei=7GFySs2CN5qIlQTCsvSDAQ#v=onepage&q=catalan%20surface&f=false
The parametric equations of the Catalan surface are http://www.mathcurve.com/surfaces/catalan/catalan.shtml
If all the rulings 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.Helicoid
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a Catalan surface, named after the Belgian mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Eugène Charles Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...
, is a ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
all of whose rulings 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.http://books.google.com/books?id=K31Nzi_xhoQC&pg=PA279&dq=catalan+surface&ei=7GFySs2CN5qIlQTCsvSDAQ#v=onepage&q=catalan%20surface&f=false
The parametric equations of the Catalan surface are http://www.mathcurve.com/surfaces/catalan/catalan.shtml
If all the rulings of a Catalan surface intersect a fixed line
Line
- Science and technology :* Line , a circuit or loop.** A power line for electric power transmission** line power or lines power, domestic mains electricity.** telephone line** RF transmission line...
, then the surface is called a conoid
Conoid
In geometry, a conoid is a Catalan surface all of whose rulings intersect a fixed line, called the axis of the conoid. If all its rulings are perpendicular to its axis, then the conoid is called a right conoid....
.
Catalan proved that the helicoid
Helicoid
The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
and the plane were the only ruled
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
minimal surfaces.Helicoid
Helicoid
The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...