Focal Conic Defects Dupin and Parabolic Cyclides

4 Focal Conic Defects Dupin and Parabolic Cyclides 

The focal conics consisting of an ellipse and hyperbola which enable the construction of a Dupin cyclide may be described in Cartesian coordinates by 

A transformation from the Cartesian coordinate system to the local (r, u, v) frame can then be made. It can be shown that this transformed coordinate system is orthogonal with unit basis vectors r, u, v where the r-direction is parallel to the local layer normal of the Dupin cyclide. Moreover, it can be shown [210] that Vr = r is the unit layer normal to the cyclide surface, so that a = f automatically fulfils the necessary requirements a a = 1 and V x a = 0, stated at equations (6.3)i and (6.4). It consequently follows that for a fixed value of r, the inner part of a Dupin cyclide surface is obtained by varying the values of u and v in the above expressions for the Xy y and z coordinates of the surface. An alternative parametrisation for the full Dupin cyclide surfaces is available in the book by Forsyth [90, p.326], but care needs to be exercised in the context of parallel layers for liquid crystals because complicated restrictions on the range of the parameters must be introduced. 

It can be shown that for the six term elastic energy for the SmCM phase given by equation (6.46), the vectors 

The bottom part of Fig. 6.5 depicts a cross-section taken in the plane y = 0 showing the hyperbola, and displays how the smectic layers match up near the hyperbola the ellipse is perpendicular to the page and passes through the two points indicated by dots at (—5,0,0) and (5,0,0). 

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