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[8] viXra:2603.0105 [pdf] submitted on 2026-03-19 23:55:37
Authors: Lambros Stephanis
Comments: 2 Pages.
Proof Pythagorean Theorem using Euler's identity.
Category: Geometry
[7] viXra:2603.0081 [pdf] submitted on 2026-03-17 00:06:07
Authors: Harish Chandra Rajpoot
Comments: 13 Pages. 5 Figures (Note by viXra Admin: An abstract labled as such is required in the article; further repetitions of similar articles may not be accepted)
In this paper, generalized analytical expressions are derived for determining the internal (dihedral) angles between consecutive faces of an arbitrary tetrahedron and the solid angle subtended at any of its vertices. The derivation is based on HCR’s Inverse Cosine Formula in conjunction with HCR’s Theory of Polygon. The resulting relations provide a simple and systematic method for evaluating the dihedral angles and the vertex solid angle when the three apex angles between the edges meeting at a given vertex are known. Owing to their general form, the derived formulae are also applicable to configurations in which three faces meet at a vertex of regular and uniform polyhedra, thereby enabling efficient computation of vertex solid angles in such solids.
Category: Geometry
[6] viXra:2603.0068 [pdf] replaced on 2026-03-25 23:58:50
Authors: B. Wang
Comments: 10 Pages.
Let $X$ be a compact K"ahler manifold. We first define the positivity of homology classes in $H_{2k}(X;mathbb Q)$.From the positivity, we extract complex analytic cycles. Precisely, we show if $tauin H_{2k}(X;mathbb Q)$ is positive, i.e. $tau $ is represented by a closed, strongly positive current, then there are a complex analytic cycle $V$ with positive rational coefficients and a positive current $S$ of bidimension $(k, k)$ such thatbegin{equation}tau=[T_V+S]end{equation}where $T_{bullet}$ denotes the current of integration over the chain $bullet$, and $[bullet]$ denotes the homologyclass represented by $bullet$.
Category: Geometry
[5] viXra:2603.0019 [pdf] submitted on 2026-03-04 21:12:38
Authors: Ryo Takayama
Comments: 5 Pages. In Japanese
In this paper, we derive an explicit formula for the area of a particular quadrilateral using only the lengths of a pair of opposite sides and the four interior angles. The formula is based on known geometric relationships and is presented after algebraic simplification and manipulation. Furthermore, we clarify the conditions under which the formula is valid and elucidate its geometric background. As a result, we demonstrate its applicability not only to convex and concave quadrilaterals but also to certain self-intersecting quadrilaterals.
Category: Geometry
[4] viXra:2603.0008 [pdf] submitted on 2026-03-01 22:14:10
Authors: Harish Chandra Rajpoot
Comments: 8 Pages. 2 Figures
This paper presents the derivation of analytical formulae for estimating the solid angle subtended by a circular plane at an arbitrary point in space. The proposed relations are also employed to determine the geometric parameters of the elliptical projection of a circular plane when observed from an off-center position, including the major axis, minor axis, and eccentricity. All mathematical expressions are derived using the Approximation Formula for the Solid Angle of Symmetrical Planes, as developed by the author and presented in his book Advanced Geometry. The resulting formulae provide a simple and effective approach for the approximate evaluation of solid angles and associated projection parameters in related geometric problems.
Category: Geometry
[3] viXra:2603.0007 [pdf] submitted on 2026-03-01 22:13:48
Authors: Harish Chandra Rajpoot
Comments: 4 Pages. 1 Figure
This paper presents the derivation of the general formula to compute the area of the spherical triangle having each side as a great circle arc on the spherical surface when (1) the aperture angle subtended by each of the three sides at the center of the sphere is known, and (2) the arc length of each of the three sides is known. These formulas are applicable to any spherical triangle to compute the area on the sphere.
Category: Geometry
[2] viXra:2603.0006 [pdf] submitted on 2026-03-01 22:21:35
Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 4 Figures
In this paper, a new analytic formula governing all regular spherical polygons (having sides in the form of great circle arcs) has been derived to compute the important parameters such as solid angle, surface area & check the feasibility of the existence of a regular spherical polygon. A regular spherical polygon will exist only if it duly satisfies HCR's formula for a regular spherical polygon.
Category: Geometry
[1] viXra:2603.0005 [pdf] submitted on 2026-03-02 01:18:20
Authors: Harish Chandra Rajpoot
Comments: 10 Pages. 2 Figures
In this work, the principal geometric parameters of an icosidodecahedron comprising 20 congruent equilateral triangular faces and 12 congruent regular pentagonal faces of equal edge length are analytically determined. These parameters include the normal distances and solid angles subtended by the faces, as well as the inner radius, outer radius, mean radius, surface area, and volume. The calculations are carried out using HCR’s formula for regular polyhedra, a generalized dimensional formulation applicable to all five Platonic solids, namely the regular tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. The versatility of the proposed formula further enables its application to the analysis, design, and geometric modeling of truncated polyhedra.
Category: Geometry
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