DARBOUX SURFACE

 Gaston Darboux (1842-1917): French mathematician.

 Cartesian parametrization: where A(v) is an orthogonal matrix.

A Darboux surface is a surface that is the union of "equal" curves (i.e. the images of one another by isometries of space), called its generatrices.
Kinematically, it is therefore a surface generated by the movement of a curve.

When the generatrix is in:
- translation (A = I3), we get the translation surfaces.
- rotation (A is a rotation matrix with constant axis with direction vector and angle v, a = b = c = 0), we get the surfaces of revolution.
- a helicoidal motion (A is a rotation matrix with a constant axis with direction vector and angle v ), we get the helicoids.

When the generatrix is a line, we get the ruled surfaces, and when it is a circle, the circled surfaces.

When the generatrices are planar and their orthogonal trajectories are parallel, we get the Monge surfaces.

Example of Darboux surface that is not of any of the previous types:
Example of surface that is not a Darboux surface: ellipsoid that is not of revolution??