Surface elliptic/hyperbolic

When k = fc2 at every point, the surface is called minimal, which implies nonpositive K. The surface is elliptic if K > 0, hyperbolic if K < 0, and parabolic if K = 0. 

Note 3 Neighbouring domains form a family with a common apex where the hyperbolae of these domains join each other. This common point is located at the surface that is opposite to the surface containing the ellipses (see Fig. 26). Each family is bounded by a polygon formed by hyperbolic and elliptical axes these are parts of focal-conic domains that provide a smooth variation of smectic layers between the domains of different families. These domains are the tetrahedra in Fig. 26. 

In the vicinity of elliptic points, the surface can be fitted to an ellipsoid, whose radii of curvature are equal to those at that point (Fig. 1.5a). The surface lies entirely to one side of its tangent plane, it is "synclastic" and both curvatures have the same sign. About a parabolic point, the surface resembles a cylinder, of radius equal to the inverse of the single nonzero principal curvature (Fig. 1.5b). H3rperbolic ("anticlastic") points can be fitted to a saddle, whic is concave in some directions, flat in others, and convex in others (Fig. 1.5c). At hyperbolic points, the surface lies both above and below its tangent plane. 

Thus, surfaces that are free of holes or handles like the sphere have positive and Xi are elliptic. Surfaces with a single handle or a hole (a donutshaped torus, or - plus or minus a single point - a cylinder or a plane) are on average parabolic, or two-dimensionally Euclidean (= p=0). All surfaces with more than one hole (or handle), are hyperbolic (negative , x)-... 

These local calculations predict the formation of planar, hyperbolic, parabolic (cylindrical) and elliptic (globular) interfaces as the volume fraction of the iMger block increases from 50%. The compositional range of existence of the various interfacial geometries depends to a limited extent on the effective surface tension acting at the interface. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

All plane sections of surfaces of the second order are either circular, parabolic, hyperbolic, or elliptical, and are comprised under the generic word conicoids, of which spheroids, paraboloids, hyperboloids and ellipsoids are special cases. 

However, a two-dimensional (2D) interface separating three-dimensional (3D) domains has two independent curvatures, which can be either concave or convex. The product of those curvatures determines the intrinsic geometry both convex (or concave) leads to an elliptic cap , one vanishing curvature gives a planar, cylindrical or conical parabolic sheet, and opposite curvatures to a saddle-shaped hyperbolic surface (Figure 16.2). 

Figure 16.2. Different types of interfacial curvatures (surface patches) (a) elliptic (b) parabolic (c) hyperbolic...

The second order tensor in 2D images was also used by Noble [31], who pointed out that the local image surface can be classified according to the Hessian matrix determinant as a planar point (zero determinant), parabolic point (zero determinant), hyperbolic point (negative determinant) and elliptic point (positive determinant). Points of interest are those which contain strong intensity variation such as the hyperbolic and elliptic points. 

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