Torsion geodesic

For a surface characterised by ki=-K2, the Gaussian curvature is simply related to the normal curvature and geodesic torsion ... 

In this case, the magnitude of the geodesic torsion at a point on a straight line lying in the surface is equal to the magnitude of the principal curvatures of the surface at that point. 

If the (curved) edges lie along the principal directions on the surface the geodesic torsion of the network vanishes (xg=0). In this case, the density is ... 

Eq. (2.10) reveals that the density is minimised - aOowing the widest pores -when the geodesic torsion of the framework vanishes. This occurs when the net edges are parallel to the principal directions. If all O atoms lie inside rings... 

Two characteristics determine the shape of molecular aggregates. The first is the shape of the constituent molecules, which sets the curvature of the aggregate. The second is coupled to the chirality of the molecules, which also determines the curvature of the aggregate, via the geodesic torsion. The bulk of this chapter is devoted to an exploration of the effect of molecular shape on aggregation geometry. An account of the theory of self-assembly of chiral molecules is briefly discussed at the end of this chapter. 

This (local) double twist configuration clearly involves a hyperbolic deformation of the imaginary layers. In contrast to the hyperbolic layers found in bicontinuous bilayer lyotropic mesophases, the molecules within these chiral thermotropic mesophases are oriented parallel to the layers, to achieve nonzero average twist. The magnitude of this twist is deternuned by the direction along which the molecules lie (relative to the principal directions on the surface), and a function of the local curvatures of the layers (K1-K2), cf. eq. 1.4. Just as the molecular shape of (achiral) surfactant molecules determines the membrane curvatures, the chirality of these molecules induces a preferred curvature-orientation relation, via the geodesic torsion of the layer. 

The torsion of a curve describes its pitch a helix exhibits both constant curvature and torsion. Its curvature is measured by its projection in the tangent plane to the curve - which is a circle for a helix - while its torsion describes the degree of non-planarity of the curve. Thus a curve on a surface (even a geodesic), generally displays both curvature and torsion. 

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