When the interfacial energy is a nonconvex function of orientation, the anisotropic curvature flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in threedimensions that incorporates surface diffusion and bulk-s...

The dynamic behaviour of spiral waves rotating on surfaces of curved reaction–diffusion systems depends strongly on the curvature of the surface. It was shown in an earlier experiment that a spiral in a Belousov–Zhabotinsky system on a paraboloid drifts to the point of highest Gaussian curvature, as predicted numerically and analytically. Beyond this, theoretical work predicts an increase of th...

A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...

1 Review of the geodesic curvature of the curves in the surface We first review the geodesic curvature of the curves in the surface introduced in section 4.1. Let M be a surface and let α be a curve in the surface M, parametrized by its arc-length (.i.e. α ′ = 1). By the " trick of the unit-vector " , α ′′ is perpendicular to α ′ , so (4.2.1) α ′′ = κ n n + κ g n × α ′ .

We use the generalized Weierstrass representation to analyze the asymptotic behavior of a constant mean curvature surface that locally arises from an ODE with a regular singularity. We show that if system is a perturbation of that of a Delaunay surface, then the corresponding constant mean curvature surface has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is em...

A variational formulation of an image analysis problem has the nice feature that it is often easier to predict the e ect of minimizing a certain energy functional than to interpret the corresponding EulerLagrange equations. For example, the equations of motion for an active contour usually contains a mean curvature term, which we know will regularizes the contour because mean curvature is the r...

Surface representation as a boundary-valued problem of partial differential equations (PDE) is an important topic of computer graphics and computer aided design. In existing references, various free-from surfaces were created with a fourth order PDE which is only able to meet the tangential conditions at the surface boundaries. The need for a sixth order PDE in surface modelling arises in two s...

We read with great interest the article by Kanellopoulos and Asimellis1 and the subsequent correspondence with Reinstein et al.2 with the debate regarding the law that drives epithelial remodeling and the healing process. Kanellopoulos and Asimellis explained their findings of epithelial thickening after myopic LASIK with their hypothesis that epithelial behavior is related to the biomechanics ...

Although, most of the abnormal structures of human brain do not alter the shape of outer envelope of brain (surface), some abnormalities can deform the surface extensively. However, this may be a major problem in a surface-based registration technique, since two nearly identical surfaces are required for surface fitting process. A type of verification known as the circularity check for th...

The purpose of this paper is to offer demonstrations of 'illusory volumes' in the spirit of the illusory flat surfaces described by Kanizsa. These demonstrations of illusory volumes exploit a new cue to the recovery of surface curvature from ambiguous images: conformation. In assuming conformation, the visual system assumes that the surface of a volume conforms to the curvature of its neighbori...