Dupin's theorem

In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:

• The intersection curve of any pair of surfaces of different pencils of a threefold orthogonal system is a curvature line.

A threefold orthogonal system of surfaces consists of three pencils of surfaces such that any pair of surfaces out of different pencils intersect orthogonally.

The most simple example of a threefold orthogonal system consists of the coordinate planes and their parallels. But this example is of no interest, because a plane has no curvature lines.

A simple example with at least one pencil of curved surfaces: 1) all right circular cylinders with the z-axis as axis, 2) all planes, which contain the z-axis, 3) all horizontal planes (see diagram).

A curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature). The set of curvature lines of a right circular cylinder consists of the set of circles (maximal curvature) and the lines (minimal curvature). A plane has no curvature lines, because any normal curvature is zero. Hence, only the curvature lines of the cylinder are of interest: A horizontal plane intersects a cylinder at a circle and a vertical plane has lines with the cylinder in common.

The idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories. Special examples are systems of confocal conic sections.