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[3] viXra:2511.0083 [pdf] submitted on 2025-11-17 16:07:30
Authors: Faiz Qamar
Comments: 64 Pages.
When faced with curves that defy standard expressions, those that one may regard as, in simpler terms, chaotic or random, this may prompt a question: Is there a way to describe them uniformly, without patching new rules each time? Herein, we will explore one such attempt.
Category: Geometry
[2] viXra:2511.0039 [pdf] submitted on 2025-11-10 18:35:27
Authors: Alex Wang
Comments: 14 Pages.
This report will be presenting a generalization to a previous method to solveEuclidean geometry problems which are parameterizable in one variable, known asthe Method of Moving Points". This method sometimes faces limitations, oftenunable to directly intersect or parameterize curves with degrees greater than onewithout tailored geometric analysis. We generalize this method through applying theVeronese map to be able to parameterize higher-degree moving curves, and extendthe notion of multiplicity of point-point concurrence to the degree of vanishing of adeterminant, to find effective bounds on the degree of higher-degree moving curves.Additionally, through an application of polynomial resultants, we bound the degreeof the locus of intersections of higher-degree moving curves. Finally, we present acollection of examples and applications of this theory to solving olympiad geometry problems involving moving circles and factoring their resultant bounds.
Category: Geometry
[1] viXra:2511.0037 [pdf] replaced on 2025-12-09 17:58:29
Authors: Absos Ali Shaikh, Uddhab Roy
Comments: 36 Pages.
The purpose of this article is to introduce the notion of constructing any arbitrary finite and infinite types of non-compact hyperbolic Riemann surfaces via (non-abelian) fundamental groups equipped with various types of classical Schottky structures, with limit sets as uncountable sets (but not necessarily Cantor sets), emphasising the cases in which the surfaces are of infinite hyperbolic areas. In particular, in this paper, the primary goal is to fabricate three different types of caconical non-compact infinite area Fuchsian polygons in the hyperbolic plane endowed with various kinds of classical Schottky structures. After that, we have initiated a structure of an arbitrary finite type non-compact hyperbolic Riemann surface with genus, conformal holes, cusps, and funnel ends by using the canonical Fuchsian Schottky polygons. Furthermore, in this manuscript, we have proposed the ideas of infinite types conformally compact and semi-conformally compact hyperbolic Riemann surfaces. Indeed, we have introduced four new and interesting types of infinite type hyperbolic Riemann surfaces (we call generalized flute surfaces) that are constructed from infinite sequences of infinite area hyperbolic pair of pants, each glued to the next along a common geodesic boundary with certain strategies.
Category: Geometry
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