Catalan surface

Catalan surface: Example of a helical surface
Guide curve is a helix, the directional plane is the xy plane

A Catalan surface is described in the Belgian mathematician in the geometry Eugène Charles Catalan designated ruled surface whose generatrices (straight lines) are all at a fixed level, the target level are parallel.

A Catalan surface with the straightening plane can be represented by a parametric representation ${\displaystyle \mathbf {x} \cdot \mathbf {n} _{0}=0}$

• ${\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)\ ,}$ With

${\displaystyle \mathbf {r} (u)\cdot \mathbf {n} _{0}=0}$

describe. Every surface curve with a fixed parameter is a generator , describes the guide curve and the vectors are all parallel to the straightening plane. ${\displaystyle \mathbf {x} (u_{0},v)}$ ${\displaystyle u=u_{0}}$ ${\displaystyle \mathbf {c} (u)}$ ${\displaystyle \mathbf {r} (u)}$

If it is sufficiently differentiable, the planarity condition can also be passed through ${\displaystyle \mathbf {r} (u)}$

${\displaystyle \det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0}$

express.

Examples:

(1) Level: ${\displaystyle \mathbf {x} (u,v)=\mathbf {p} +u\mathbf {a} +v\mathbf {r} }$

The guide curve is a straight line.

(2) cylinder : ${\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,0)+v(0,0,1)}$

The guide curve is a circle. Any plane parallel to the z-axis can be used as a guide plane.

(3) helical surface : ${\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,u)+v(\cos u,\sin u,0)}$

The guide curve is a helix (screw line) and the straightening plane is parallel to the xy plane.

This helical surface can also be created with a straight line (z-axis) as a guide curve:

${\displaystyle \mathbf {x} (u,v)=(0,0,u)+v(\cos u,\sin u,0)}$

(4) A conoid is a Catalan surface in which the straight lines intersect on a fixed straight line, the axis .

Catalan proved that the plane and the helical surface are the only ruled surfaces among the minimal surfaces .

Comment:

1. Ruled surfaces but not Catalan surfaces are e.g. E.g .: cone , single-shell hyperboloid .
2. One should not confuse a Catalan surface with a Catalan minimal surface !

See also

• Ruled surface
• Conoid

Individual evidence

1. ^ W. Kühnel: Differentialgeometrie. Vieweg-Verlag, Stuttgart 2003, ISBN 3-528-17289-4 , p. 78.

literature

• A. Gray, E. Abbena, S. Salamon: Modern differential geometry of curves and surfaces with Mathematica. 3. Edition. CRC Press, Boca Raton, FL 2006, ISBN 1-58488-448-7 .
• VY Rovenskii: Geometry of curves and surfaces with MAPLE. 2000, ISBN 0-8176-4074-6 . (Excerpts online)