Alternative Method for Segmenting Digitized Data in Reverse Engineering

The study aims to propose an alternative method of segmentation of digitized data. It begins by layering the item's surface. It analyzes the nodes inside a point cloud to detect any consistent shape change. If one detected, then it holds it and looks for the next one which may describe a possible shape change. The points between those two are classified and marked as part of the shape change curvature line. The method remembers the marked points and holds them as central nodes which will later form reference regions. It uses normal vectors behavior methods to detect shape changes along X and Y axis. As any other direction would not be detectable by the bi-dimensional approach it then introduces a morphological parameter capable on its own to fully describe the curvature variation of a given item's surface by means of Gauss's curvature. To evaluate curvature variation, the method proposes that the central node's curvature should be compared with every found limit points. Because of the noise present in any points cloud it establishes a threshold value beyond which points may describe accurately any shape change. This procedure takes place for all analyzed reference regions and collects only those who have a greater value than the threshold one. This considerations may be extrapolated to other types of geometries as well, as it is the case with cylinders or cones.

Constructing isothermal curvature line coordinates on surfaces which admit them

Isothermic parameterizations are synonyms of isothermal curvature line parameterizations, for surfaces immersed in Euclidean spaces. We provide a method of constructing isothermic coordinate charts on surfaces which admit them, starting from an arbitrary chart. One of the primary applications of this work consists of numerical algorithms for surface visualization.

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

Conformally flat hypersurfaces in Euclidean 4-space

We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.