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Discrete Normal Vector Field Approximation via Time Scale Calculus

Ömer Akgandüller

Ömer Akgandüller

Muğla Sıtkı Koçman University, Faculty of Science, Department of MathematicsMuğla, Turkey
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Akgandüller, Ömer
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Sibel Paşalı Atmaca

Sibel Paşalı Atmaca

Muğla Sıtkı Koçman University, Faculty of Science, Department of MathematicsMuğla, Turkey
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Atmaca, Sibel Paşalı
  
31. März 2020
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Online veröffentlicht: 31. März 2020
Seitenbereich: 349 - 360
Eingereicht: 14. Juni 2019
Akzeptiert: 02. Aug. 2019
DOI: https://doi.org/10.2478/amns.2020.1.00033
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© 2019 Ömer Akgüller et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

A finite surface with the patch φ on 𝕋1 × 𝕋2.
A finite surface with the patch φ on 𝕋1 × 𝕋2.

Fig. 2

Eight neighbouring non-uniform data points of O = R(0, 0).
Eight neighbouring non-uniform data points of O = R(0, 0).

Fig. 3

The surfaces with the parameterizations φ1(t, s) (on the left) and φ2(t, s) (on the right), and the points sampled on them.
The surfaces with the parameterizations φ1(t, s) (on the left) and φ2(t, s) (on the right), and the points sampled on them.

Fig. 4

The Delaunay triangulations of the point set M sampled on φ1(t, s) and φ2(t, s).
The Delaunay triangulations of the point set M sampled on φ1(t, s) and φ2(t, s).

Fig. 5

The error in unit normal vectors for φ1(t, s).
The error in unit normal vectors for φ1(t, s).

Fig. 6

The error in unit normal vectors for φ2(t, s).
The error in unit normal vectors for φ2(t, s).

Fig. 7

The points where the unit normal is approximated better for φ1(t, s) and φ2(t, s).
The points where the unit normal is approximated better for φ1(t, s) and φ2(t, s).

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

Input: M
Build: G=(V,E), V=M
for i=1 to |M| do
  N(i) = {wi: wi is adjacent to i in G} ∪ i
  Πl : l−th coordinate function
   Fit quadratic z(x, y) in N(i)
   for j=1 to |N(i)| do
       for k=1 to | N(i) do
      𝒮 ← {Π1(wj), Π2(wk), z(wj, wk)
       end for
     end for
   end for
Output: Bundle S = ∪𝒮
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