Euler characteristic surface areas

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

Figure 27. Seven possible cases of the polygonal surface representation in a single pyramid. The Euler characteristic is calculated as a sum of the number of faces and the number of vertices minus the number of edges of the polygons. The black and white circles represent points with higher and lower values relative to the threshold one. The gray area is the schematic representation of the surface inside a pyramid [225].

The parallel surface method (PSM) has been invented to measure the average interface curvature (and the Euler characteristic) from the 3D data images [222]. First, a parallel surface to the interface is formed by translating the original interface along its normal by an equal distance everywhere on the surface (see Fig. 33). The change of the surface area at the infinitely small parallel shift of the surface is... 

Figure 37. The maximum of the mean curvature distribution scaled with the interface density increases very rapidly (up 2.5 times) within a short time interval, x , after the noise term has been switched off in the simulation. The Euler characteristic and the average domain size, / o, remain constant, and the surfaces area decreases by 3%. This illustrates that the curvature distributions are very sensitive to the thermal undulations of the interface. The times are x = 0.0, 0.032, 0.085, 0.225, 0.896, 2.05 from bottom to top at /// ] 0.

The general necessity for twelve pentagons, with a (nearly) arbitrary number of hexagons follows from Euler s theorem a network of trivalent nodes comprising P pentagons and H hexagons contains n = Euler characteristic n — b + a must be 2, which implies P = 12, with no explicit constraint on H. (In fact, polyhedra exist for any... 

Many important questions and conjectures remain unresolved. It is not known whether these solutions are the only embedded //-surfaces for the five dual pairs of skeletal graphs studied, for example. An important issue is whether or not there exists a bound on the mean curvature attainable in such families for all of the branches studied here, and for the family of unduloids with a fixed repeat distance (Anderson 1986), the dimensionless mean curvature H = HX is always less than n, where X is the sphere diameter in the sphere-pack limit. It is possible that there exists an upper bound on H lower than n that depends on the coordination number, or the Euler characteristic. For the P, D, I, WP, F, and RD branches, the islands over which K > 0 coalesce wih neighboring R regions at a critical mean curvature that is the same (to within an error in H of about 0.15) as the value H corresponding to the local minimum in surface area. We have given what we suspect to be the analytical value for the area of the F-RD minimal surfaces, and for the first nonzero coefficient in both the area and volume expansions about // = 0 in the P family. 

The integral curvature of a surface is linked to the Euler-Poincare characteristic of that surface (x) by eq. (1.12). This allows the average geometry of orientable surfaces to be related to the number of holes or handles, characterised by the surface genus, g, and the area of the surface. A, by the relation ... 

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