Parabolic cyclides

Focal Conic Defects Dupin and Parabolic Cyclides... 

The description of parabolic cyclide surfaces is based upon two confocal parabolas in mutually perpendicular planes, with the vertex of one parabola passing through the focus of the other. These parabolas represent line defects in smectic liquid crystals. Parts of some typical parabolic cyclide surfaces are pictured in... 

Figure 6.6 The two types of parabolic cyclide surface only parts of these surfaces are depicted. The left plot shows the tunnel type and the middle plot shows the bridge type. The bridge can be omitted in the context of smectic liquid crystals, leaving that part of the parabolic cyclide surface of relevance shown in the right plot. These plots were obtained using the parametrisation in equations (6.114), (6.115) and (6.116) with —4 and, respectively, /x = 1 for the tunnel type and p = —1...

It is now clear that fixing a value for fj, and varying 0 and t will generate a complete parabolic cyclide surface. For some values of only parts of the surface will be relevant for liquid crystals, as pointed out by Rosenblatt et al [234], but we shall not go into such details here briefly, bridges appear as shown in the middle part of Fig. 6.6 which are thought to have no physical relevance to smectic liquid crystals. (The above parametrisation is based on that of Forsyth [90], but note that the surface equation and parametrisation introduced in [90] contain some misprints a full derivation is given in Reference [260].)... 

Similar to the Dupin cyclides discussed above, a transformation from Cartesians to the local (/a, 6y t) coordinate system can be made. This new coordinate system is orthogonal with unit basis vectors t, where the / -direction is parallel to the local layer normal of the parabolic cyclide surface. It can be shown [260], using the surface equation (6.113), that V/x = Ji coincides precisely with the unit layer normal, and therefore a = fulfils the requirements a a = 1 and V x a = 0. By setting the vectors... 

Figure 6.7 Parts of equally spaced parabolic cyclide surfaces are shown in the upper illustration. These were obtained from the parametrisation introduced at equations (6.114) to (6.117) with the mutually orthogonal parabolic defects (6.111) and (6.112) indicated by the two bold curves. The lower illustration shows a cross-section of the upper figure in the plane z = 0 the parabolic defect is evident and the corresponding smectic layers are arranged as shown with the other parabola being perpendicular to the page and passing through the point (0,0,0). By symmetry, there is a similar cross-section in the plane y — 0.

I.W. Stewart, On the parabolic cyclide focal-conic defect in smectic hquid crystals, Liq. Cryst, 15, 859-869 (1993). 

Parabolic focal conics are a special case of generic focal conic defects, which are composed of layers curved to form toroidal surfaces called Dupin cyclides (see Fig. 10-31). Each such structure contains a pair of disclination lines—one an ellipse and the other... 

Figure 30. Dupin s cyclides and their deformations, (a) A toroidal Dupin cyclide and its circular lines of curvature y and /, the planes of which intersect along the axes A (joining O and Q) and A, respectively. Two tangential planes along two symmetrical circles T intersect along A. (b) A parabolic Dupin s cyclide, with its two focal parabolae P and P, its circular curvature lines y and /, and its axes A and A, (c) Three parallel, but separated, parabolic Dupin s cyclides. The lines represent sections by planes x=0,y = 0 and = 0 at the top of the figure, two conical points are present, as in (b), but the spindle-shaped sheet has been removed (Rosenblatt et al.

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