Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form QM(u) = x^2 + y^2 + z^2 - c^2 t^2where (x, y, z) are the spacial components of the vector u and t is the time component of u and c is the constant of the speed of light. In this Minkowski-Lorentz space, a Dupin cyclide is the union of two conics on the unit pseudo-hypersphere, called the space of spheres, and a singular point of a Dupin cyclide is represented by an isotropic vector. Then, we model Dupin cyclides using rational quadratic Bézier curves with mass points. The subdivisions of a surface i.e. a Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space ε3, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are circles which look like ellipses.

Optical Microscopy Observations and Construction of Dupin Cyclides at the Isotropic/Smectic A Phase Transition

In this work, we are interested in the nucleation of bâtonnets at the Isotropic/Smectic A phase transition of 10CB liquid crystal. Very often, these bâtonnets are decorated with a large number of focal conics. We present here an example of a bâtonnet obtained by optical crossed polarized microscopy in a frequently observed particular area of the sample. This bâtonnet presents bulges and one of them consists of a tessellation of ellipses. These ellipses are two by two tangent, one to each other, and their confocal hyperbolas merge at the apex of the bâtonnet. We propose a numerical simulation with Python software to reproduce this tiling of ellipses as well as the shape of the smectic layers taking the well-known shape of Dupin cyclides within this particular bâtonnet area.

Möbius automorphisms of surfaces with many circles

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Möbius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides.

Application of Dupin Cyclide’s Properties to Inventions

Dupin cyclide belongs to channel surfaces. These surfaces are the single known ones whose focal surfaces, i.e. surfaces consisting of point sets of centers of curvatures, have been degenerated into two confocal second order curves. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various cyclides’ properties and demonstrated application of these surfaces in various industries, mostly in construction. Based on Dupin cyclides’ properties have been developed several inventions relating to drawing devices and having the opportunity to apply in various geometric constructions with the use of computer technologies. It is possible because the Dupin cyclides’ geometric properties suppose not only to create devices recognized as inventions, but also provide an opportunity to apply these properties to write programs for drawing v arious kinds of curves on a display screen: the second order curves, their equidistant in the direction of normals or tangents, as well as to perform other constructions. It should be said that in inventions belonging to technical areas, which include the drawing devices, the geometric component is always decisive. This position with the express aim of technical inventions can justify any copyright certificate of the USSR, any patent of Russia and foreign countries. Unfortunately, currently in schools geometry is not studied as a component of pupil’s intellectual horizons, that broadens his area of interests and teaches to analytical understanding the world, but as an inevitable, almost unnecessary appendage in preparation for the Unified State Examination.

Method of Rotation of Geometrical Objects Around the Curvilinear Axis

Rotation is the motion of geometric objects along a circle. This is one of geometric techniques used to form lines and surfaces. In this paper has been considered the rotation of objects in a three-dimensional space around a straight axis. It is known that a straight line can be considered as a particular case of a circle with a radius equal to infinity. Such circle’s center is at infinite distance from the considered straight line segment. Then in the general case, the rotation axis is a closed curve, for example, a circle with a radius of finite magnitude. Rotation of a point around a straight axis now splits into two trajectories. One of them is a circle with a radius, the second is a straight line crossing with the axis, and the center of this trajectory is at an infinite distance from the point. The method of point rotation about an axis of finite radius was considered. Note that a circle is a special case of an ellipse. When the actual focus of the circle is stratified into two, the line itself loses its curvature constancy, and is called an ellipse. The point, rotating around the elliptical axis, is stratified into four ones, forming four circles (trajectories). Axis foci appearing in turn in the role of the main one determine two trajectories by each with a trivial and nontrivial center of rotation. We have considered the variant for arrangement of the generating circle so that its center coincided with one of the elliptic axis’s foci. The obtained surfaces are a pair of co-axial Dupin cyclides, since they have identical properties. Changing the circle generatrix radius, other things being equal, we get different types of closed cyclides.

Representations of Dupin Cyclides

We know very little about such an interesting surface as Dupin cyclide. It belongs to channel surfaces, its special cases are tor, conical and cylindrical surfaces of rotation. It is known that Dupin cyclides are the only surfaces whose focal surfaces, that are surfaces consisting of sets of curvatures centers points, have been degenerated in second-order curves. Two sets give two confocal conics. That is why any study of Dupin cyclides is of great interest both scientific and applied. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various properties of cyclides, and demonstrated application of these surfaces in various industries, mostly in construction. Based on the cyclides’ properties in 1980s have been developed numerous inventions relating to devices for drawing and having the opportunity to be applied in various geometric constructions with the use of computer technologies. In the present paper have been considered various options for representation of Dupin cyclides on a different basis – from the traditional way using the three given spheres unto the second-order curves. In such a case, if it is possible to represent four cyclides by three spheres, and when cyclide is represented by the second-order curve (konic) and the sphere their number is reduced to two, then in representation of cyclide by the conic and one of two cyclide’s axes a single Dupin cyclide is obtained. The conic itself without any additional parameters represents the single-parameter set of cyclides. Representations of Dupin cyclides by ellipse, hyperbola and parabola have been considered. The work has been sufficiently illustrated.

Generalised cyclidic nets for shape modelling in architecture

The aim of this article is to introduce a bottom-up methodology for the modelling of free-form shapes in architecture that meet fabrication constraints. To this day, two frameworks are commonly used for surface modelling in architecture: non-uniform rational basis spline modelling and mesh-based approaches. The authors propose an alternative framework called generalised cyclidic nets that automatically yield optimal geometrical properties for the envelope and the structural layout, like the covering with planar quadrilaterals or hexagons. This framework uses a base circular mesh and Dupin cyclides, which are natural objects of the geometry of circles in space, also known as Möbius geometry. This article illustrates how complex curved shapes can be generated from generalised cyclidic nets. It addresses the extension of cyclidic nets to arbitrary topologies with the implementation of a ‘hole-filling’ strategy and also demonstrates that this framework gives a simple method to generate corrugated shells.