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Research Article
Online Published: 04 Sep 2024
 


Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology

Md. Morshed Bin Shiraj, Md. Safik Ullah, Tozam Hossain, Md. Mizanur Rahman, Md. Masum Murshed, Nasima Akhter.


Abstract
Background: Visualization of complex mathematical surfaces, like the n-torus, is an open challenge. In this study, a double torus data generation process has been proposed using the deformation of a torus by a sphere.

Methods: It is proved that a torus $T(R,r)$ can be deformed into a $3$-sphere $S^{3}(R+r)$ with the same center and it is the smallest ball to cover the torus $T(R,r)$. Then 3D and 4D torus data have been generated from their parametric equations. After that, the generated data have been compared with the known knowledge of the shape of a torus using the persistent diagram. Then following the theoretical findings, an approximation to the torus-torus intersection has been computed to extract it from the union of two 4D torus data. Finally, the generated double torus data has been validated by explaining the hollowness in the intersection of two sampled torus introducing subtraction operation among the persistent diagrams of each of the generated structures on the proposed generation process.

Results: In the persistent diagrams, it is found that the 4D torus data gives more significant results than the 3D torus data which supports the claim of changing the original topology of a higher dimensional manifold by using lower dimensional reduction. The result shows that the double torus data with hollow intersection has been generated properly.

Conclusion: The successful generation of the double torus paves the way for creating more complex data with well-defined topology. This approach is particularly significant in scientific computation, especially for researchers focused on topological aspects.

Key words: Double torus data generation; torus-torus intersection; geometry processing; computer-aided geometric design; persistent homology; data visualization.


 
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How to Cite this Article
Pubmed Style

Shiraj MMB, Ullah MS, Hossain T, Rahman MM, Murshed MM, Akhter N, . Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. Eu J Sci Res Rev. Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644

Web Style

Shiraj MMB, Ullah MS, Hossain T, Rahman MM, Murshed MM, Akhter N, . Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. https://www.wisdomgale.com/ejsrr/?mno=211296 [Access: September 05, 2024]. doi:10.5455/EJSRR.20240725051644

AMA (American Medical Association) Style

Shiraj MMB, Ullah MS, Hossain T, Rahman MM, Murshed MM, Akhter N, . Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. Eu J Sci Res Rev. Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


Vancouver/ICMJE Style

Shiraj MMB, Ullah MS, Hossain T, Rahman MM, Murshed MM, Akhter N, . Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. Eu J Sci Res Rev, [cited September 05, 2024]; Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


Harvard Style

Shiraj, M. M. B., Ullah, M. S., Hossain, . T., Rahman, M. M., Murshed, M. M., Akhter, N. & (0) Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. Eu J Sci Res Rev, Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


Turabian Style

Shiraj, Md. Morshed Bin, Md. Safik Ullah, Tozam Hossain, Md. Mizanur Rahman, Md. Masum Murshed, Nasima Akhter, and . 0. Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. European Journal of Scientific Research and Reviews, Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


Chicago Style

Shiraj, Md. Morshed Bin, Md. Safik Ullah, Tozam Hossain, Md. Mizanur Rahman, Md. Masum Murshed, Nasima Akhter, and . "Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology." European Journal of Scientific Research and Reviews Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


MLA (The Modern Language Association) Style

Shiraj, Md. Morshed Bin, Md. Safik Ullah, Tozam Hossain, Md. Mizanur Rahman, Md. Masum Murshed, Nasima Akhter, and . "Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology." European Journal of Scientific Research and Reviews Online First: 04 Sep, 2024. Web. 05 Sep 2024 doi:10.5455/EJSRR.20240725051644


APA (American Psychological Association) Style

Shiraj, M. M. B., Ullah, M. S., Hossain, . T., Rahman, M. M., Murshed, M. M., Akhter, N. & (0) Approximating Torus-Torus Intersection for Hollow Double Torus Generation under the Supervision of Persistent Homology. European Journal of Scientific Research and Reviews, Online First: 04 Sep, 2024. doi:10.5455/EJSRR.20240725051644


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