Dupin's theorem
In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:[1] 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).
Special examples are systems of confocal conic sections.
Dupin's theorem is a tool for determining the curvature lines of a surface by intersection with suitable surfaces (see examples), without time-consuming calculation of derivatives and principal curvatures.
The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique.
3. pencil: Planes through the cone's axis (purple).
The purple planes intersect at the lines of cone C (green).
The points of the space can be described by the spherical coordinates
3. pencil: Planes through the axis of cone C (purple):
3. pencil: Planes containing the axis of the given torus (purple).
The blue cones intersect the torus at horizontal circles (red).
The purple planes intersect at vertical circles (green).
Usually a surface of revolution is determined by a generating plane curve (meridian)
2. pencil: Cones with apices on the axis of revolution with generators orthogonal to the given surface (blue).
3. pencil: Planes containing the axis of revolution (purple).
The cones intersect the surface of revolution at circles (red).
They are a prominent example of a non trivial orthogonal system of surfaces.
(Curvature lines of rotational quadrics are always conic sections !)
The curvature lines are sections with one (blue) and two (purple) sheeted hyperboloids.
The curvature lines are intersections with ellipsoids (blue) and hyperboloids of two sheets (purple).
A Dupin cyclide and its parallels are determined by a pair of focal conic sections.
The diagram shows a ring cyclide together with its focal conic sections (ellipse: dark red, hyperbola: dark blue).
2. pencil: right circular cones through the ellipse (their apexes are on the hyperbola) 3. pencil: right circular cones through the hyperbola (their apexes are on the ellipse) The special feature of a cyclide is the property: Any point of consideration is contained in exactly one surface of any pencil of the orthogonal system.
Hence any point can be represented by: For the example (cylinder) in the lead the new coordinates are the radius
can be considered as the cylinder coordinates of the point of consideration.
The condition "the surfaces intersect orthogonally" at point
This is true, if Hence Deriving these equations for the variable, which is not contained in the equation, one gets Solving this linear system for the three appearing scalar products yields: From (1) and (2): The three vectors
and hence are linear dependent (are contained in a common plane), which can be expressed by: From equation (1) one gets
Curvature lines: green, red
