ABOUT SMOOTH INTERPOLATION OF A TRIANGULATED SURFACE

A method and algorithm for rebuilding a surface triangulation in three-dimensional space defined by an STL file is proposed. An initial surface in 3D space (STL file) is represented as a polyhedron composed of triangular faces. The method is based on the analytical representation of the surface as a piecewise polynomial function. This function is built on a polyhedral surface composed of triangles and satisfies the following requirements: 1) within one face, the function is an algebraic polynomial of the third degree; 2) the function is continuous on the entire surface and preserves the continuity of the first partial derivatives; 3) the surface determined by the function passes through the vertices of the initial triangulated surface. The restructuring of computational meshes is required in cases of distortion of the shape of cells when solving problems of mathematical physics using mesh methods (finite-difference, FEM, etc.). Cell distortion can be due to various reasons. These can be large distortions of moving Lagrangian meshes in the calculations in the current configuration, with instability of the hourglass type, with distortion of the faces of the interface between interacting gaseous, liquid and elastoplastic bodies. The rebuilding of the mesh reduces to solving the problem of constructing a smooth surface passing through the nodes of an existing triangulated surface or part of it. Later the nodes of the new mesh are placed on the constructed smooth surface with existing requirements for the size and shape of the cells. The construction of a smooth piecewise polynomial surface is based on the ideas of spline approximation and reduces to the building of a cubic polynomial on each triangular face, taking into account the smooth conjugation of polynomial pieces of the surface constructed on adjacent faces. The proposed method for rebuilding surface triangulation can be useful for calculating the motion of deformable bodies when solving problems of the dynamics of continuous media on immovable Euler grids.

Green Function and Self-adjoint Laplacians on Polyhedral Surfaces

Using Roelcke’s formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface$X$and compute the$S$-matrix of$X$at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the$S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

Identification of Primary Shape Descriptors on 3D Scanned Particles

The number of global mechanical equilibria as a shape descriptor (among others, for sedimentary particles) is at the forefront of current geophysical research. Although the technology is already available to provide scanned, 3D images of the particles (appearing as fine spatial discretization of smooth surfaces), nevertheless, the automated identification and measurement of global equilibria on such 3D images has not been solved so far. The main difficulty lies in the algorithmic distinction between local equilibria (associated with the small un-evenness of the pebble’s surface) and global equilibria, associated with the overall shape. The former are easily measured, however, only the latter provide meaningful physical information. Here we provide and illustrate an algorithm to detect global equilibrium points on a finely discretized, polyhedral surface provided by 3D scan of sedimentary particles.

FITTING A POINT CLOUD TO A 3D POLYHEDRAL SURFACE

The ability to measure parameters of large-scale objects in a contactless fashion has a tremendous potential in a number of industrial applications. However, this problem is usually associated with an ambiguous task to compare two data sets specified in two different co-ordinate systems. This paper deals with the study of fitting a set of unorganized points to a polyhedral surface. The developed approach uses Principal Component Analysis (PCA) and Stretched grid method (SGM) to substitute a non-linear problem solution with several linear steps. The squared distance (SD) is a general criterion to control the process of convergence of a set of points to a target surface. The described numerical experiment concerns the remote measurement of a large-scale aerial in the form of a frame with a parabolic shape. The experiment shows that the fitting process of a point cloud to a target surface converges in several linear steps. The method is applicable to the geometry remote measurement of large-scale objects in a contactless fashion.

Quantifying the Cosmic Web using the Shapefinder diagonistic

One of the most successful method in quantifying the structures in the Cosmic Web is the Minkowski Functionals. In 3D, there are four minkowski Functionals: Area, Volume, Integrated Mean Curvature and the Integrated Gaussian Curvature. For defining the Minkowski Functionals one should define a surface. We have developed a method based on Marching cube 33 algorithm to generate a surface from a discrete data sets. Next we calculate the Minkowski Functionals and Shapefinder from the triangulated polyhedral surface. Applying this methodology to different data sets , we obtain interesting results related to geometry, morphology and topology of the large scale structure