Dupin cyclides cross-sections

Fig. 8.30 Focal-conic defect structure in SmA A pair of cones with a common elliptical base and a hyperbola connecting cone apices (a) cross-section of the upper cone by plane ABC with gaps between lines (Dupin cyclides) indicating the smectic layers (b) filling the space of the sample by cones of different size (c)...

Figure 9.5. A vertical cross-section through a set of Dupin cyclides.

Smectic A phases in which the layers are not uniformly parallel to the glass slides confining the sample (i.e. not in a planar orientation) are characterized by fan-like textures (Fig. 5.10b), made up of focal conics (Fig. 5.12). A focal conic is an intersection in the plane of a geometric object called a Dupin cyclide (Fig. 5.13), which results from lamellae forming a concentric roll (like a Swiss roll) being bent into an object based on an elliptical torus of non-uniform cross-section. The straight line that would define the rotation axis of the torus is distorted into a hyperbola in the Dupin cyclide. 

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