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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