Surface constant curvature

Gauss has shown [14] that Equation 9.6a is true for any surface with constant curvature H for which the reversible fluctuations do not break the relation... 

Given a map M, its circle-packing representation (see [Moh97]) is a set of disks on a Riemann surface E of constant curvature, one disk D(v, rv) for each vertex v of M, such that the following conditions are fulfilled ... 

The group T (l, m, n) can be realized as a group of isometries of a simply connected surface X of constant curvature, where ... 

We now consider another geometric viewpoint on (r, )-polycycles. In the above consideration, the curvature was uniform and the triangles were viewed as embedded into a surface of constant curvature. Consider now the curvature to be constant, equal to zero, in the triangle itself, and to be concentrated on the vertices, where r-gons meet. 

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. 

The infinitely long cylinder with no motion of the interface or of the fluid within the cylinder is, of course, a possible equilibrium configuration, in the sense that it is a surface of constant curvature so that the stationary constant-radius fluid satisfies all of the conditions of the problem, including the Navier-Stokes and continuity equations (trivially), as well as all of the interface boundary conditions including especially the normal-stress balance, which simply requires that the pressure inside the cylinder exceed that outside by the factor 1//a. The question for linear stability theory is whether this stationary configuration is stable to infinitesimal perturbations of the velocity, the pressure, or the shape of the cylinder. 

Another simple example is a sphere of radius R. The principal curvatures at any point K = l/i and the Gaussian curvature, K1K2 = 1/ , which is a positive constant. The only other quadric of constant curvature is the projective plane. The curvature of any dome-shaped surface, although... 

We assume that the curvatures are slowly varying in the plane of the membrane (or are constant as they are for cylindrical and spherical curvature) so that the pressure depends only on < . The surface of curvature is measured... 

Although the pressure is constant, the droplets are not spherical. In order to produce a constant curvature surface (Laplace s condition) joining up with the thin cylinder, the droplet must assume a wavelike shape. Indeed, at the join with the fibre, there must be a negative radius of curvature of the order of b to compensate for the positive curvature imposed there by the fibre. This point of inflection, and the wavelike shape, make direct measurement of the contact angle e rather difficult. The best way to determine Oe is to deposit droplets of known volume, measure their geometrical dimensions (1) and then use the tables of 6e values established by Carroll (at Unilever). 

We may calculate the details of the drop s profile z x). The surface has a constant curvature throughout given by... 

Upon choosing the surface of tension as the dividing surface for an isotropic interface (C = 0) with constant curvature, the Laplace equation (Eq. (27)) reduces to the familiar form AP = 2Hy because then both the bending moment B as well as the torsion moment will vanish as we will show in the following. 

In his pioneering article from 1956, Buff [7] assumed constant curvature and, furthermore, put the shearing tension C = 0. Additionally, following Gibbs, he assumed 0 to be equal to zero even for an arbitrary dividing surface, resulting in the approximate Laplace equation ... 

Starting with transition state theory and assuming simple quadratic potential surfaces with equal force constants (curvature) for the initial and final states, Marcus derived an expression for the rate of thermal electron transfer (Marcus, 1956) ... 

In order to facilitate the calculation of capillary forces, several approximations on the meniscus shape have been proposed. They are mainly applied for experimental conditions where the radius of curvature of the meniscus interface is much smaller than the radius of curvature of the solid surfaces. This is relevant for the surface force apparatus where the surface has centimetric radius, while the meniscus is typically tens of hundreds of nanometers. The most used approximation is the toroidal approximation assuming the liquid interface has a circular profile. Obviously, such a meniscus does not exhibit a constant curvature. Nevertheless, this approximation gave good results, in particular for small contact angles, and is therefore widespread (see Ref. 15 for its application in various geometries and section 9.3.1.1 for an example of its application in atomic force microscopy [AFM]). In the case of capillary condensation between a plane and a sphere with a large radius of curvature R, in contact, the tension term of the capillary force is negligible and the Laplace term leads to the simple formula F = AnRy cos 9 A parabolic profile is also sometimes used to eliminate some numerical difficulties inherent in circle approximation. 

Figure 13.1 Three basic deformations for a membrane element considered as a mathematical surface, (A) Initially, the undeformed state of the membrane element is assumed to be flat. (B) A shear deformation occurs when tangential forces parallel to the side of the membrane are applied at constant area. (C) Tangential forces perpendicular to the membrane sides increase or decrease the membrane area at constant curvature. The two last sketches represent bending deformations occurring when the forces applied to the membrane are out of the plane, in the cases of (D) positive or (E) negative Gaussian curvature.

Let us say some words on the memoir of Mile, which appeared in 1838. The author expounds a theory of capillary phenomena, in which he also makes use of the tension but, as he considers it, that tension would obey laws now inadmissible. He starts from the principle that, in a homogeneous liquid, the molecules seek to be arranged in a regular and identical way everywhere however he shows that this identity is possible only when the surface of the liquid is plane, and he concludes that, in consequence of the abnormal arrangement of the molecules, curved surfaces constantly try to become plane that effort, which results from the tensions, is all the more energetic as the curvatures are pronounced, and this same effort determines, on the mass, a pressure in the case of convex surfaces, and a traction in the case of concave surfaces finally that from this are bom capillary phenomena. According to Mile, one sees, tension would exist only in curved surfaces, and it would vary with the curvature. 

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