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[5] viXra:2512.0134 [pdf] submitted on 2025-12-29 00:32:22
Authors: Andy Zhuang
Comments: 18 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, we first introduce the Minimal Overlapping Circle Expansion (MOCE) problem. Solution to such a problem has real-world applications, such as finding the location to best communicate with a number of wireless devices, finding the quickest way for a number of vehicles to get to a rendezvous location etc. We present several algorithms to compute the solution with different running time and accuracy. The first uses enclosing square to get an approximate solution; the second only considers pair-wise overlap to approximate; the third uses the the results of the first two and a few other methods to speed up the computation. Our results show that (1) the approximate algorithm can be 1000 times faster than the accurate algorithm, and get to 99.9% of the correct value. (2) improvements can cut down the compute time by 50% for the accurate algorithm.
Category: Geometry
[4] viXra:2512.0126 [pdf] submitted on 2025-12-27 01:09:43
Authors: Himanshu M. Chavda
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references, list scientific references in a complete manner, and submit article written with AI assistance to ai.viXra.org)
"This paper introduces a novel recursive framework for approximating circular geometry and waveforms using discrete segment rotations. Traditional analytic methods, such as the classical circumference formula $C=2pi r$, rely on continuous functions that abstract away the geometric essence of rotation and introduce computational inefficiencies in discrete digital environments. By re-evaluating the 'Method of Exhaustion,' this work derives an original Discrete Radius Formula ($r = frac{C}{2n sin(Deltatheta/2)}$) that eliminates the inherent path-drift found in standard step-based systems. A recursive update algorithm is developed to reconstruct complex signals with $O(1)$ computational complexity, transforming global trigonometric evaluations into local iterative additions. Numerical validation demonstrates high-precision convergence to continuous limits, achieving an absolute error of approximately $7.97 times 10^{-9}$ at high resolution. The results establish a robust bridge between classical geometry and modern digital implementation, offering significant improvements in speed and accuracy for robotics, AI graphics, and signal processing."
Category: Geometry
[3] viXra:2512.0124 [pdf] replaced on 2026-04-18 17:38:05
Authors: Dan Howitt
Comments: 9 Pages. © 2026 Dan Howitt
The following metamathematical analyses of the mathematical concepts and language "zero dimensional", "one dimensional", "two dimensional", "extra dimensional", "infinity", "zero", "nothing", "pure geometry", and "curved space", demonstrate that they are arrived at via one or more of conceptual dissociation, conceptual association, and variations of linguistic alteration, and that they, as such, cannot represent facets of the universe. (The above and below language that are quoted are so because their meanings are in question).
Category: Geometry
[2] viXra:2512.0119 [pdf] submitted on 2025-12-26 00:02:27
Authors: Harish Chandra Rajpoot
Comments: 15 Pages. Original Research Work
A generalized framework from HCR's Theory of Polygon is presented for computing the solid angle subtended by an arbitrary polygonal plane, regular or irregular, at any point in three-dimensional space. The approach is unified and systematic, relying on a single master formula derived for a right triangular plane. This formula is simplified and equivalently expressed in terms of inverse trigonometric functions, including arcsine, arccosine, and arctangent. The variation of the solid angle with respect to the orthogonal sides of the triangle and the distance of the observation point is illustrated graphically. In addition, several corollaries are established for the solid angle subtended by planar surfaces, both polygonal and non-polygonal, at different coplanar locations of the observation point. The results are derived using the standard formula for right-triangle geometry and the concept of the angle of vision for observation of two-dimensional figures.
Category: Geometry
[1] viXra:2512.0091 [pdf] submitted on 2025-12-20 01:38:44
Authors: Peter Kugelmann
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This document presents a comprehensive mathematical framework for the Hopf-fibered 3-sphere $S^3$. We systematically derive the full geometric, topological, and analytic structure of $S^3$ equipped with its canonical round metric and Hopf fibration $S^1 hookrightarrow S^3 to S^2$. The framework establishes $S^3$'s uniqueness properties, rigidity theorems, and advanced geometric consequences emerging from combinations of its basic structures. All results are presented with complete proofs or references to standard mathematical literature. This article should be viewed as a comprehensive synthesis of canonical structures and standard results associated with the Hopf-fibered round 3-sphere, rather than a source of new classification theorems.
Category: Geometry
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