G1-Blend between a Differentiable Superquadric of Revolution and a Plane or a Sphere Using Dupin Cyclides

2008 ◽  
Author(s):  
Lionel Garnier ◽  
Sebti Foufou ◽  
Yohan Fougerolle
Keyword(s):  
Dupin Cyclides

Freeze fracture imaging and computer simulation of liposome structure: Membrane geometry and defects

1983 ◽  
Vol 41 ◽  
pp. 646-647
Author(s):  
Joseph A. Zasadzinski
Keyword(s):  
Liquid Crystalline ◽  
Freeze Fracture ◽  
Continuum Theory ◽  
Closed Surfaces ◽  
Straight Line ◽  
Low Weight ◽  
Concentric Spheres ◽  
Membrane Geometry ◽  
Dupin Cyclides ◽  
Limiting Case

At low weight fractions, many surfactant and biological amphiphiles form dispersions of lamellar liquid crystalline liposomes in water. Amphiphile molecules tend to align themselves in parallel bilayers which are free to bend. Bilayers must form closed surfaces to separate hydrophobic and hydrophilic domains completely. Continuum theory of liquid crystals requires that the constant spacing of bilayer surfaces be maintained except at singularities of no more than line extent. Maxwell demonstrated that only two types of closed surfaces can satisfy this constraint: concentric spheres and Dupin cyclides. Dupin cyclides (Figure 1) are parallel closed surfaces which have a conjugate ellipse (r1) and hyperbola (r2) as singularities in the bilayer spacing. Any straight line drawn from a point on the ellipse to a point on the hyperbola is normal to every surface it intersects (broken lines in Figure 1). A simple example, and limiting case, is a family of concentric tori (Figure 1b).To distinguish between the allowable arrangements, freeze fracture TEM micrographs of representative biological (L-α phosphotidylcholine: L-α PC) and surfactant (sodium heptylnonyl benzenesulfonate: SHBS)liposomes are compared to mathematically derived sections of Dupin cyclides and concentric spheres.

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

Materials ◽  
2021 ◽  
Vol 14 (16) ◽  
pp. 4539
Author(s):  
Mikhael Halaby Macary ◽  
Gauthier Damême ◽  
Antoine Gibek ◽  
Valentin Dubuffet ◽  
Benoît Dupuy ◽  
...  
Keyword(s):  
Phase Transition ◽  
Numerical Simulation ◽  
Liquid Crystal ◽  
Optical Microscopy ◽  
Smectic A ◽  
Polarized Microscopy ◽  
Smectic A Phase ◽  
Dupin Cyclides

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.

On shell membranes of Enneper type: generalized Dupin cyclides

2009 ◽  
Vol 42 (40) ◽  
pp. 404016 ◽  
Author(s):  
W K Schief ◽  
A Szereszewski ◽  
C Rogers
Keyword(s):  
Dupin Cyclides

Three views of Dupin cyclides and blending of cones

2002 ◽  
Vol 40 (1-2) ◽  
pp. 39-47
Author(s):  
E. Méndez ◽  
A. Müller ◽  
M. Paluszny
Keyword(s):  
Dupin Cyclides

scholarly journals Symmetries of canal surfaces and Dupin cyclides

2018 ◽  
Vol 59 ◽  
pp. 68-85
Author(s):  
Juan Gerardo Alcázar ◽  
Heidi E. I. Dahl ◽  
Georg Muntingh
Keyword(s):  
Canal Surfaces ◽  
Dupin Cyclides

Representation of Dupin cyclides using quaternions

2015 ◽  
Vol 82 ◽  
pp. 110-122 ◽  
Author(s):  
Severinas Zube ◽  
Rimvydas Krasauskas
Keyword(s):  
Dupin Cyclides

Foliations of 𝕊3 by Dupin cyclides

2013 ◽  
pp. 67-102
Author(s):  
Rémi Langevin ◽  
Jean-Claude Sifre
Keyword(s):  
Dupin Cyclides

Construction of 3D Triangles on Dupin Cyclides

2011 ◽  
Vol 1 (2) ◽  
pp. 42-57 ◽  
Author(s):  
Bertrand Belbis ◽  
Lionel Garnier ◽  
Sebti Foufou
Keyword(s):  
Algebraic Surfaces ◽  
Parametric Equation ◽  
French Mathematician ◽  
Lines Of Curvature ◽  
Implicit Equations ◽  
Cad Cam ◽  
Dupin Cyclide ◽  
Dupin Cyclides ◽  
Definition Of ◽  
The 19Th Century

This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

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