<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 ℝ<sup>3</sup> 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.</p></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.

Discrete Normal Vector Field Approximation via Time Scale Calculus

Faculty of Science, Department of Mathematics

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Discrete Normal Vector Field Approximation via Time Scale Calculus"}

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...

Discrete Normal Vector Field Approximation via Time Scale Calculus

Published Online: Mar 31, 2020

Page range: 349 - 360

Received: Jun 14, 2019

Accepted: Aug 02, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00033

Keywords
Time Scale Calculus, Symmetric Differential, Discrete Normal, Geometric Approximation

© 2019 Ömer Akgüller et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

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The procedure to obtain the bundle of ⋄-smooth surfaces on M.

Discrete Normal Vector Field Approximation via Time Scale Calculus

Published Online: Mar 31, 2020

Page range: 349 - 360

Received: Jun 14, 2019

Accepted: Aug 02, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00033

KeywordsTime Scale Calculus, Symmetric Differential, Discrete Normal, Geometric Approximation

© 2019 Ömer Akgüller et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

The procedure to obtain the bundle of ⋄-smooth surfaces on M.

Keywords
Time Scale Calculus, Symmetric Differential, Discrete Normal, Geometric Approximation