Discrete Normal Vector Field Approximation via Time Scale Calculus

Ömer Akgandüller,Sibel Paşalı Atmaca

doi:10.2478/amns.2020.1.00033

Abstract

Abstract The theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.

Highlights

The mimetic discretization of differential operators is a process that maintains the fundamental properties of continuous differential operators
A disadvantage of this method is that the selected discretization process may have little connection to the underlying physical problem
For the mimetic discretization process the theory of time scale calculus emerge as an efficient mathematical theory

Summary

Introduction

The mimetic discretization of differential operators is a process that maintains the fundamental properties of continuous differential operators. The main goal in this process is to ensure that the protected properties are maximized and, if not, to give up most of the properties. Geometric partial differential equations are very important to find discrete analogues of differential geometric operators such as mean curvature, Gaussian curvature and Laplace Beltrami, which are defined by surface normal. The de facto methods, such as finite differences and finite elements, are directly related to the discretization of the equation system. A disadvantage of this method is that the selected discretization process may have little connection to the underlying physical problem. Mimetic methods start with discrete analogue of the continuous theory underlying the problem.

Methods

Results

Conclusion