There are several equivalent definitions of Dupin cyclides. Young.ĭupin cyclides are used in computer-aided design because cyclide patches have rational representations and are suitable for blending canal surfaces (cylinder, cones, tori, and others). This property means that Dupin cyclides are natural objects in Lie sphere geometry.ĭupin cyclides are often simply known as cyclides, but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions.ĭupin cyclides were investigated not only by Dupin, but also by A. 2 Arakelyan A G 2016 Infinite sequences of mutually tangent circles (Geometry and Graphics 4 Salkov N A 2015 Properties of Dupin cyclides. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one-parameter family of spheres) in two different ways. They were discovered by (and named after) Charles Dupin in his 1803 dissertation under Gaspard Monge. In particular, these latter are themselves examples of Dupin cyclides. The points of the space can be described by the spherical coordinates ( r, φ, θ ) is true, too.In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. The purple planes intersect at the lines of cone C (green). The blue cones intersect the given cone C at a circle (red). Over the past few years, interest in Dupin cyclides has re-emerged: they can be used both in mechanical engineering and in construction, for covering the spans of civil buildings and in temple. 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. CommonInscribed Sphere This work was partially supported by the National Science Foundation undergrant CCR-9696084 (formerly CCR-9410707),DUE-9653244 and DUE-9752244,and by a Michigan Research Excellence Fund 19981999. Orthogonal system (purple, green, blue) of surfaces for the cone (green), curvature lines: green, redġ. Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. Given: A right circular cone, green in the diagram. These surfaces have a variety of interesting properties and are aesthetic from a geometric and algebraic viewpoint. The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique.Įxamples Right circular cone Dupin cyclides are algebraic surfaces of order three and four whose lines of curvature are circles. 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. Dupin cyclides are canal surfaces in two ways 10, 16: A Dupin cyclide can be defined as the envelope of a smooth one-parameter family F1 of spheres touching. 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: The misc3d package provides a great implementation of the marching cubes algorithm, allowing to plot implicit surfaces. Two planes (purple, blue) as members of a threefold orthogonal system intersect a cylinder at curvature lines (blue circle, purble line)
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