Geodesic curvature

The positivity of the right-hand side of Inequality (8.4) implies the positivity of the left-hand side. Thus, in view of Equation (8.1), the curvature 2 of the geodesic -gon is positive. The resulting inequality 2(r + q) — qr > 0 contradicts the assumption made. Hence, a finite non-extensible polycycle cannot have parabolic or hyperbolic parameters (r, q) these parameters are elliptic. ... 

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

Figure 1.7 Decomposition of a curve in a surface (left) into orthogonal geodesic and normal curvatures (right).

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. 

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. 

Figure 1.8 Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (flj) and the geodesic curvature along the arcs AB, BC, CD and DA.

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. 

To determine how much a curve is curving near a point on the surface the normal curvature components suffice, since the geodesic curvature rather concerns a property of curves in a metric space which reflects the deviance of... 

In this section, we introduce the computational method in the form used for the surfaces exhibited in Section IV, i.e., where the prescribed mean curvature of the computed surface is everywhere constant and the boundary conditions are determined by two dual periodic graphs. We also give generalizations of the method for the computation for a surface of prescribed—not necessarily constant—mean curvature, with prescribed contact angle against surface. Generalization to the computation of space curves of prescribed curvature or geodesic curvature is available (Anderson 1986). 

Kenmotsu, K. 1978, Generalized Weierstrass Formula for Surfaces of Prescribed Mean Curvature, in Minimal Submanifolds and Geodesics, Morio Obata Kaigai Publications, Tokyo, 73-76. 

Anosov, D. V. Geodesic Flows on Closed Manifolds of Negative Curvature. Trudy Matemat. Inst. Steklova, v. 90. Nauka, Moscow (1967). 

A curve in a curved surface that is as straight as possible. Geodesic curvature... 

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