Dupin Cyclides Research Articles

Dupin Cyclides as a Subspace of Darboux Cyclides

Dupin Cyclides as a Subspace of Darboux Cyclides

Classification of Dupin cyclidic cubes by their singularities

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

Java Script Programs for Calculation of Dihedral Angles with Manifold Equations

Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δ<sub>Xn</sub>[ppm] and vicinal coupling constant <sup>3</sup><I>J</I><sub>HnHn+1</sub>[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances l<sub>CnCn+1</sub>[A<sup>0</sup>] between two atoms of carbon are under different polar equations (<i>i.e.</i> limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance l<sub>CnCn+1</sub>[A<sup>0</sup>], with application on conformational and configurational analysis.

Dupin Cyclides Passing through a Fixed Circle

Dupin cyclides are classical algebraic surfaces of low degree. Recently, they have gained popularity in computer-aided geometric design (CAGD) and architecture owing to the fact that they contain many circles. We derive algebraic conditions that fully characterize the Dupin cyclides passing through a fixed circle. The results are applied to the basic problem in CAGD of the blending of Dupin cyclides along circles.

Geometry of focal conics in sessile cholesteric droplets.

Focal conic domains are defects characteristic of layered liquid crystal phases. Their association can build flowers where petals are the ellipses of the Dupin cyclides involved in these defects. We report here the observation of focal conic flowers in cholesteric droplets sessile on a glass surface and surrounded by glycerol. The observation of the droplets in different directions helps to solve the three dimensional architecture of the flower. The effects of the droplet size and of the pitch value are also reported.

Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology

Although electrons (fermions)and photons (bosons) produce the same interference patterns in the two-slit experiments, known in optics for photons since the 17th Century, the description of these patterns for electrons and photons thus far was markedly different. Photons are spin one, relativistic and massless particles while electrons are spin half massive particles producing the same interference patterns irrespective to their speed. Experiments with other massive particles demonstrate the same kind of interference patterns. In spite of these differences, in the early 1930s of the 20th Century, the isomorphism between the source-free Maxwell and Dirac equations was established. In this work, we were permitted replace the Born probabilistic interpretation of quantum mechanics with the optical. In 1925, Rainich combined source-free Maxwell equations with Einstein’s equations for gravity. His results were rediscovered in the late 1950s by Misner and Wheeler, who introduced the word "geometrodynamics” as a description of the unified field theory of gravity and electromagnetism. An absence of sources remained a problem in this unified theory until Ranada’s work of the late 1980s. However, his results required the existence of null electromagnetic fields. These were absent in Rainich–Misner–Wheeler’s geometrodynamics. They were added to it in the 1960s by Geroch. Ranada’s solutions of source-free Maxwell’s equations came out as knots and links. In this work, we establish that, due to their topology, these knots/links acquire masses and charges. They live on the Dupin cyclides—the invariants of Lie sphere geometry. Symmetries of Minkowski space-time also belong to this geometry. Using these symmetries, Varlamov recently demonstrated group-theoretically that the experimentally known mass spectrum for all mesons and baryons is obtainable with one formula, containing electron mass as an input. In this work, using some facts from polymer physics and differential geometry, a new proof of the knotty nature of the electron is established. The obtained result perfectly blends with the description of a rotating and charged black hole.

Relationship between tetrahedral and dihedral on hypersphere coordinates

The wave character of the NMR data (chemical shift ΔδCn[ppm] – vicinal coupling constant 3JHH[Hz]) enable calculation the tandem dihedral-vicinal or tetrahedral-internal angles with 3-sphere approach. Golden ratio, invers of Fibonacci number, and manifold inversion torus to Dupin cyclide under 3-sphere units ensure the calculation of the tetrahedral angles of five membered ring in opposite with dihedral angles sin versus tan functions. The transformation from space/time to space under electric and magnetic rule gives dodecahedron and icosahedron values for calculation tetrahedral angles from polyhedron and hypersphere equations. Tetrahedral angles of five membered ring can be calculated from carbon chemical shift ΔδCn[ppm] with Java Script program and hypersphere equations.

Hypersphere and Antiviral Activity of Three Alkyl Chain Iminocyclitols with D and L Ribitol Stereochemistry

<I>N</I>-Alkyl-C₁-dialkyl chains iminocyclitols with D or L-ribitol stereochemistry are synthesized with high diastereoselectivity after Grignard reagents addition to <I>N</I>-quaternary pyrrolines salts, and tested for antiviral activity in bovine viral diarrhea virus (BVDV), surrogate for hepatitis C virus (HCV). Dihedral angles are calculated from carbon chemical shift (δ_Cn[ppm]) with 3-sphere method without building units. 3-Sphere, a hypersphere in 4D, under Hopf fibration and Lie algebra mathematics theories enable calculation of the dihedral angles from the NMR data (vicinal coupling constant ³<I>J</I>_HnHn+1[Hz], chemical shift δ_Cn[ppm]). Instead of 3D manifold equations on seven sets unit or six sets units are proposed equations between 4D – 2D, in function of the curvature. The relationship between the antiviral activity and the iminocyclitol structure reveals that monoalkyl chain, <I>N</I>-n-C₁-dodecyl β-L-ribitol trifloroacetate salt 30 (IC₅₀ 1.5 uM) has higher antiviral activity in tangential space, relative to three alkyl chain, <I>N</I>-Methyl-C₁-butil, nonyl-L-ribitol. HCl 26 (IC₅₀ < 2 uM) with torus and Dupin cyclide coordinate, both with coordinates in 2D. Three alkyl chain isopropylidene protected pyrrolidine 25 has in 4D with all equations for calculation of the dihedral angles, and in protected pyrroline 19b double bond moves the coordinates in 2D.

Об одном способе формирования коник

In the articles of the Geometry and Graphics magazine devoted to the properties of the Dupin cyclide, the construction of a conic – ellipse, hyperbola and parabola – using the properties of the cyclide was considered. At the same time, the center of the transformation was located on a straight line connecting the centers of the two base circles, and its location on such a straight line was negotiated separately and was located as the center of homology. To construct a parabola, it was necessary to take a straight line instead of the second circle, and the center of the transformation – the center of homology – had to be located at the intersection point of a straight line passing through the center of the first circle perpendicular to the second circle-a straight line with the first circle. Two different parabolas were obtained as a result of the transformation. In this paper, it is proved that if we take the center of correspondence that does not belong to a circle, we get other second–order curves - ellipses and hyperbolas. The construction of an ellipse is geometrically proved. To do this, the center of correspondence must lie on a straight line connecting the centers of the circles, but outside the actual circle. Several examples are considered. If the center of correspondence is inside the circle, we will have a hyperbola. Thus, having initially given only one con-figuration from a straight line and a circle, it is possible to obtain all conics: ellipses, parabolas, and hyperbolas, passing into one another. The proposed scheme for constructing conics can be used for computer drawing of all conics, which is more convenient than with the available options sewn into today's graphical drawing systems.

Topological Classification and Determination of Non-Degenerate Intersections of Two Dupin Cyclides

The surface-surface intersection is a fundamental task in CAD/CAM. We present the classification and a full enumeration of the topology of all non-degenerate intersections of two Dupin cyclides. Inversion geometry is first used to transform the intersection of two Dupin cyclides to that of a cyclide and a torus. Then the topology of the intersection curve is reduced to the arrangements of the main circle of the torus and another pair of circles, which can be characterized by a simple algebraic sequence. Based on the classification and enumeration results, an efficient determination algorithm for the topology of two Dupin cyclides is also provided. • A classification of the non-degenerate intersections of two Dupin cyclides is given by a simple algebraic sequence. • A full enumeration of all non-degenerate intersections of two Dupin cyclides based on a simple algebraic sequence is provided. • An efficient and robust algorithm of determining the inter- section topology of two Dupin cyclides is provided.

Mass Points, Spaces of Spheres, ”hyperbolae” and Dandelin and Quételet Theorems

The use of mass points eases the definition of a branch of a hyperbola in the Euclidean plane based on a Rational Quadratic Bézier Curve. In the space of spheres, any circular cone, circular cylinder, torus, pencil of spheres or Dupin cyclide is represented by a rational quadratic Bézier curve that is conic arc seen as circle arc. The limit points of the Poncelet pencil or the singular points of the Dupin cyclide can be determined using the asymptotes of this circle. This article shows that the use of mass points simplifies the modelling of these pencils or these Dupin cyclides in the space of spheres. The determination of the Dandelin spheres ends this work as an application.

Discrete cyclic systems and circle congruences

We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.

Subdivisions of Horned or Spindle Dupin Cyclides Using Bézier Curves with Mass Points

This paper shows the same algorithm is used for subdivisions of Dupin cyclides with singular points and quadratic Bézier curves passing through infinity. The mass points are usefull for any quadratic Bézier representation of a parabola or an hyperbola arc. The mass points are mixing weighted points and pure vectors. Any Dupin cyclide is considered in the Minkowski-Lorentz space. In that space, the Dupin cyclide is defined by the union of two conics laying on the unit pseudo-hypersphere, called the space of spheres. The subdivision of any Dupin cyclide, is equivalent to subdivide two Bézier curves of degree 2 with mass points, independently. The use of these two curves eases the subdivision of a Dupin cyclide patch or triangle.

Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

The paper deals in the Computer-Aided Design or Computer-Aided Manufacturing domain with the Dupin cyclides as well as the Bézier curves. It shows that the same algorithms can be used either for subdivisions of ring Dupin cyclides or Bézier curves. The Bézier curves are described with mass points here. The Dupin cyclides are considered in the Minkowski-Lorentz space. This makes a Dupin cyclide as the union of two conics on the unit pseudo-hypersphere, called the space of spheres. And the conics are quadratic Bézier curves modelled by mass points. The subdivision of any Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space, 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 bounded 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.

Математическое описание частного случая квазивращения фокуса эллипса вокруг эллиптической оси

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Математическое описание частного случая квазивращения фокуса эллипса вокруг эллиптической оси

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Setting of the Dupin cyclide by three straight lines and sphere

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 architecture; the properties of the Dupin cyclide can be widely used in computer graphics. The article considers a method for setting of Dupin cyclides using three lines and a sphere, which greatly the possibilities for the construction of these surfaces, especially when it is necessary to cut compartments of a certain size from them and to dock them.

Möbius automorphisms of surfaces with many circles

AbstractWe 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 the Dupin cyclide in temple architecture

In recent years, the construction of churches and temples has been rapidly developed, so the need for design and support for such constructions arose. A new department called “Temple architecture” even appeared at MArchI (Moscow Architectural Institute). The article considers a method for geometric construction of the surface of such objects as the dome of temples and churches from the parts of Dupin cyclides. As a result, two variants of domes made from the compartments of the Dupin cyclide were proposed. Based on the theoretical research, it becomes clear that there can be more design options, and if we use a special case of cyclides for construction - compartments of rotating cones - then many times more.