They are an object of great significance in the fascinating discipline of Lie sphere geometry, and the notion of Dupin cyclides can be extended to higher dimensions by 'Dupin submanifolds', with their classification of particular interest.

Dupin cyclides and similar objects are still being studied at the forefront of mathematical research. For example, they accurately describe the behaviour of gravitational and electric potential. Solutions to Laplace's equation give rise to harmonic functions, which are important in many areas of science. More concretely, a standard torus can be parameterised by: $$ egin^3$, using cyclides to determine coordinate systems. A standard torus means one swept out by a circle rotating about an axis, rather than the topological sense of any surface with one hole. As in other contexts, cylinders and cones are taken to be infinite, with cones consisting of two components meeting at their sharp ends. These three surfaces are also Dupin cyclides themselves. See to read some (advanced) material on the applications of Dupin cyclides in geometric modelling.Ī Dupin cyclide is constructed by 'inverting' (we shall make this notion more concrete later) a cylinder, cone, or standard torus in a sphere - see here for helpful images. An elegant sculpture by Florian Claar has some cyclide-like structure, as does this pavilion design. The spheres in question all have their centres lying along a ( space) curve.Ĭyclides and similar structures can be of interest to artists and architects. Recall that an envelope of a family of spheres is a surface which is tangent to every sphere at some point. Also, a Dupin cyclide is a 'channel surface', meaning it is an envelope of a certain family of spheres. In a sense, this generalises the notion of a ruled surface (made up of straight lines) to surfaces made up of circlines. A Dupin cyclide is characterised by the property that all of its lines of curvature are (arcs/segments of) either circles or lines. Mathematicians sometimes refer to this generalisation of a circle or line as a ' circline'. They received much attention including works by Arthur Cayley and James Clerk Maxwell . The points of the space can be described by the spherical coordinates ( r, φ, θ ) is true, too.Dupin cyclides were discovered by French mathematician Charles Dupin in 1882. The purple planes intersect at the lines of cone C (green). The blue cones intersect the given cone C at a circle (red). These three pencils of surfaces are an orthogonal system of surfaces. pencil: Planes through the cone's axis (purple). pencil: Cones with apexes on the axis of the given cone such that the lines are orthogonal to the lines of the given cone (blue).ģ. pencil: Shifting the given cone C with apex S along its axis generates a pencil of cones (green).Ģ. Orthogonal system (purple, green, blue) of surfaces for the cone (green), curvature lines: green, redġ. Given: A right circular cone, green in the diagram. The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique.Įxamples Right circular cone Special examples are systems of confocal conic sections.ĭupin'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 idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories. 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. A plane has no curvature lines, because any normal curvature is zero. The set of curvature lines of a right circular cylinder consists of the set of circles (maximal curvature) and the lines (minimal curvature). But this example is of no interest, because a plane has no curvature lines.Ī 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).Ī curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature). The most simple example of a threefold orthogonal system consists of the coordinate planes and their parallels.

The intersection curve of any pair of surfaces of different pencils of a threefold orthogonal system is a curvature line.Ī threefold orthogonal system of surfaces consists of three pencils of surfaces such that any pair of surfaces out of different pencils intersect orthogonally.

In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement: Two planes (purple, blue) as members of a threefold orthogonal system intersect a cylinder at curvature lines (blue circle, purble line)