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[24] viXra:2602.0129 [pdf] submitted on 2026-02-22 11:36:40
Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 3 Figures, Original Research
In this work, the great-circle distance formula is derived using hcr's inverse cosine formula. An analytic and the most generalized formula has been derived to accurately compute the minimum distance or great circle distance between any two arbitrary points on a sphere of finite radius which is equally applicable in the geometry of sphere. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This formula is the most power tool which is applicable for all the distances on the tiny sphere as well as the large sphere like giant planet assuming them the perfect spheres.
Category: Geometry
[23] viXra:2602.0128 [pdf] submitted on 2026-02-22 11:40:31
Authors: Harish Chandra Rajpoot
Comments: 13 Pages, 10 Figures, Original Research
All the articles discussed and analyzed in this work are related to the five Platonic solids. A geometrical problem involving a finite number of identical circles mutually touching one another on the entire surface of a sphere of given radius is considered. Using elementary geometric relations together with tabulated parameters corresponding to the five Platonic solids, all important quantities, including the flat radius and arc radius of each circle, the total surface area covered by the circles, and the percentage of spherical surface coverage, are systematically evaluated. The derived parameters are useful for accurately drawing identical circles on a spherical surface and for the design and modeling of the five Platonic solids with identical flat circular faces.
Category: Geometry
[22] viXra:2602.0127 [pdf] submitted on 2026-02-22 11:44:26
Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 3 Figures, Original Research
In this work, previously reported analytical formulations are systematically discussed and analyzed to evaluate key beam parameters, including the solid angle subtended by a beam at a point source, the total area intercepted by the beam on a spherical surface, and the cone angle of an equivalent beam with a circular cross section. These formulations enable the transformation of a beam with a rectangular profile into an equivalent circular profile, and vice versa, while preserving the total radiation energy or luminous flux associated with the original beam emitted by a uniformly radiating point source. The presented analysis is particularly useful for radiometric applications involving the evaluation of radiation energy and directional intensity of uniform point sources, as well as for photometric applications concerned with luminous flux and luminous intensity in specified directions. Furthermore, the results facilitate the replacement of rectangular apertures with circular apertures, and conversely, without altering the total transmitted radiation energy or luminous flux. Consequently, the discussed formulations provide a valuable theoretical framework for the analysis and design of optical and beam-emitting devices, such as laser systems, based on uniformly radiating point sources.
Category: Geometry
[21] viXra:2602.0126 [pdf] submitted on 2026-02-22 11:47:06
Authors: Harish Chandra Rajpoot
Comments: 3 Pages,
This paper presents a comprehensive tabulation of the solid angles subtended at the vertices of all thirteen Archimedean solids (convex uniform polyhedra). The solid angles are analytically evaluated using the standard solid angle formula in conjunction with tetrahedral decomposition. The resulting values constitute a consistent and complete set of reference data for the vertex geometry of the Archimedean solids, including the truncated tetrahedron, truncated hexahedron (cube), truncated octahedron, truncated dodecahedron, truncated icosahedron, cuboctahedron, icosidodecahedron, small rhombicuboctahedron, small rhombicosidodecahedron, snub cube, snub dodecahedron, great rhombicuboctahedron, and great rhombicosidodecahedron. These results provide useful quantitative tools for the geometric analysis and comparative study of uniform polyhedra.
Category: Geometry
[20] viXra:2602.0125 [pdf] submitted on 2026-02-22 11:49:33
Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 4 Figures
This paper presents an analytical derivation of the fundamental geometric parameters of a non-uniform tetradecahedron composed of two congruent regular hexagonal faces, twelve congruent trapezoidal faces, and eighteen vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, explicit closed-form expressions are obtained for the solid angles subtended at the center by the hexagonal and trapezoidal faces, as well as for the corresponding normal distances of these faces from the center. From these results, exact formulas for the inradius, circumradius, mean radius, total surface area, and enclosed volume of the polyhedron are systematically derived. The analytical framework developed herein provides a general and rigorous method for the geometric characterization of non-uniform polyhedra.
Category: Geometry
[19] viXra:2602.0124 [pdf] submitted on 2026-02-22 11:54:14
Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 3 Figures, Original Research
In this paper, analytical formulas are derived using HCR’s Inverse Cosine Formula in conjunction with HCR’s Theory of Polygon. These formulas provide a simple and practical method for computing the internal (dihedral) angles between consecutive lateral faces of an arbitrary tetrahedron at any of its four vertices, as well as the solid angle subtended by the tetrahedron at a vertex when the apex angles between consecutive lateral edges meeting at that vertex are known. The resulting expressions are fully generalized and can also be applied to configurations in which three faces meet at a vertex of various regular and uniform polyhedra, enabling the calculation of the solid angle subtended by the polyhedron at that vertex.
Category: Geometry
[18] viXra:2602.0123 [pdf] submitted on 2026-02-22 11:57:20
Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 5 Figures
In this paper, the solid angles subtended at the vertices by all five platonic solids (regular polyhedrons) have been calculated by the author using the standard formula of solid angle. These are the standard values of solid angles for all five platonic solids i.e. regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron useful for the analysis of platonic solids.
Category: Geometry
[17] viXra:2602.0122 [pdf] submitted on 2026-02-22 12:00:13
Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 2 Figures
In this work, all the articles have been derived using simple geometry & trigonometry. All the formulas are very practical & simple to apply in the case of any spherical rectangle to calculate all its important parameters, such as solid angle, surface area covered, interior angles, etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formulas can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the center of the sphere, such as normal height, angle between the consecutive lateral edges, area of the rectangular base, etc.
Category: Geometry
[16] viXra:2602.0119 [pdf] submitted on 2026-02-22 21:45:57
Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 3 Figures
In this paper, the principal geometric parameters of a spherical triangle are derived using elementary geometry and trigonometry. The resulting formulae are practical and straightforward to apply for computing key quantities such as the solid angle subtended at the center, the covered spherical surface area, and the interior angles. The analysis is further extended to the corresponding plane triangle obtained by joining the vertices of the spherical triangle with straight line segments, allowing for the evaluation of its geometric parameters. In addition, the derived relations are applied to the right pyramid formed by connecting the vertices of a spherical triangle to the center of the sphere, enabling analytical computation of parameters such as the normal height, angles between consecutive lateral edges, and the area of the base.
Category: Geometry
[15] viXra:2602.0106 [pdf] submitted on 2026-02-20 20:04:49
Authors: Harish Chandra Rajpoot
Comments: 15 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)
In this work, the principal geometric parameters of the great rhombicuboctahedron, an Archimedean solid, are analytically derived. This polyhedron consists of 12 congruent square faces, 8 regular hexagonal faces, and 6 congruent regular octagonal faces of equal edge length, with 72 edges and 48 vertices lying on a circumscribed spherical surface. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and octagonal face, as well as their corresponding normal distances from the center of the great rhombicuboctahedron. The formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The derived formulas are useful for the geometric analysis, design, and modeling of uniform (convex or non-convex) polyhedra.
Category: Geometry
[14] viXra:2602.0105 [pdf] submitted on 2026-02-20 20:04:30
Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)
In this paper, the principal geometric parameters of the great rhombicosidodecahedron, the largest Archimedean solid, are analytically derived. This polyhedron consists of 30 congruent square faces, 20 regular hexagonal faces, and 12 congruent regular decagonal faces, all of equal edge length, with 180 edges and 120 vertices lying on a circumscribed sphere. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and decagonal face, along with their corresponding normal distances from the center of the great rhombicosidodecahedron. The derived formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The resulting formulas are useful for the geometric analysis, design, and modeling of uniform polyhedra.
Category: Geometry
[13] viXra:2602.0047 [pdf] submitted on 2026-02-05 23:06:15
Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 15 Figures
In this paper, a new convex polyhedron is introduced, obtained by systematically truncating all 24 edges of a rhombic dodecahedron such that the newly generated 24 congruent vertices lie exactly on a common spherical surface. The resulting truncated rhombic dodecahedron is a non-uniform convex polyhedron composed of 12 congruent rectangular faces, 6 congruent square faces, and 8 congruent equilateral triangular faces, with a total of 48 edges and 24 identical vertices. Using HCR’s Theory of Polygon, closed-form analytical expressions are derived for the radius of the circumscribed sphere passing through all vertices, the normal distances of the rectangular, square, and equilateral triangular faces from the center of the polyhedron, as well as its total surface area and enclosed volume. In addition, analytical formulae are obtained for the solid angles subtended at the center by each type of face, the dihedral angles between any two faces meeting at each of the 24 vertices, and the solid angle subtended by the truncated rhombic dodecahedron at each of its vertices.
Category: Geometry
[12] viXra:2602.0046 [pdf] submitted on 2026-02-05 23:11:58
Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 9 Figures
In this paper, a comprehensive mathematical analysis of the rhombic dodecahedron is presented, and closed-form analytical expressions are derived for a polyhedron consisting of 12 congruent rhombic faces, 24 edges, and 14 vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, generalized formulae are obtained for the face angles and diagonals of the rhombic faces, as well as for the radii of the circumscribed sphere, inscribed sphere, and midsphere. Analytical expressions are further derived for the total surface area and enclosed volume in terms of the edge length. In addition, the solid angles subtended at the vertices and the dihedral angles between adjacent faces are evaluated. It is also shown that this convex polyhedron can be constructed by assembling twelve congruent right pyramids with rhombic bases and a specific normal height.
Category: Geometry
[11] viXra:2602.0045 [pdf] submitted on 2026-02-05 23:17:43
Authors: Harish Chandra Rajpoot
Comments: 18 Pages, 11 Figures
In this paper, the circumscribed radius of a rhombicuboctahedron is derived using an alternative geometrical approach based on HCR’s Theory of Polygon. An explicit analytical expression is obtained for the radius of the circumscribed sphere passing through all 24 congruent vertices of a rhombicuboctahedron with a given edge length. Using the same theoretical framework, closed-form formulae are subsequently derived for the normal distances of the equilateral triangular and square faces from the centre, the total surface area, and the enclosed volume. In addition, analytical expressions are presented for the solid angles subtended at the centre by each equilateral triangular face and each square face, the dihedral angles between any two faces meeting at a vertex, and the solid angle subtended by the rhombicuboctahedron at any of its 24 identical vertices.
Category: Geometry
[10] viXra:2602.0044 [pdf] submitted on 2026-02-05 23:22:30
Authors: Harish Chandra Rajpoot
Comments: 32 Pages, 22 Figures
This paper presents generalized analytical formulas for computing key geometric parameters of pyramidal flat containers with regular polygonal bases, right pyramids, and related polyhedra. The derived expressions include the V-cut angle (obtained using HCR’s Theorem), edge length of the open end, lateral edge length, dihedral angle (derived using HCR’s Corollary), surface area, and volume. These formulas provide a unified framework for the systematic analysis and construction of such structures for arbitrary regular polygonal bases. The results are applied to pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal, and octagonal bases, as well as to right pyramids and bipyramidal polyhedra. The underlying geometric construction is based on the rotation or folding of two coplanar planes about their intersecting straight edges, as formulated in HCR’s Theorem, and is illustrated through practical paper models.
Category: Geometry
[9] viXra:2602.0043 [pdf] submitted on 2026-02-06 00:08:09
Authors: Harish Chandra Rajpoot
Comments: 9 Pages, 5 Figures
This paper presents a proof of the angle between any two bonds in a molecule possessing a tetrahedral structure, such as the methane molecule, in which all four σ-bonds (corresponding to four hydrogen atoms bonded to a central carbon atom) are equally inclined in three-dimensional space. The bond angle is derived using two independent approaches. The first method involves formulating the geometry of a right pyramid with a regular n-gonal base, while the second employs HCR’s formula for regular polyhedra. Both approaches lead to the same result, thereby providing a simple and rigorous geometric justification of the tetrahedral bond angle.
Category: Geometry
[8] viXra:2602.0042 [pdf] submitted on 2026-02-06 00:14:36
Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 4 Figures
In this paper, all the important formulas have been generalized which are applicable to calculate the important parameters, of any non-uniform polyhedron having 2 congruent regular n-gonal faces, 2n congruent trapezoidal faces each with three equal sides, 5n edges & 3n vertices lying on a spherical surface, such as solid angle subtended by each face at the center, normal distance of each face from the center, inner radius, outer radius, mean radius, surface area & volume. These are useful for the analysis, designing & modeling of different non-uniform polyhedrons.
Category: Geometry
[7] viXra:2602.0041 [pdf] submitted on 2026-02-06 00:17:57
Authors: Harish Chandra Rajpoot
Comments: 14 Pages, 6 Figures
In this paper, the generalized formula have been derived to analytically compute the radii of the circles inscribed and circumscribed by three mutually tangent circles of known radii. The formulation and analysis of three external tangent circles packed in the smallest rectangle and the intersection circles have also been done. . The analytic formula derived here can also be used in case of three tangent spheres in three dimensions. These formula are also used for calculating any of three radii if rest two are known & computing the length of common chord, angle of intersection & area of intersection of two intersecting circles. The generalized formula derived here can also be used to derive the recurrence relations for circle packing over a plane which can further be extended into 3 dimensions for sphere packings.
Category: Geometry
[6] viXra:2602.0040 [pdf] submitted on 2026-02-06 00:20:43
Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 8 Figures
The generalized formula, derived here, are equally applicable on any n-gonal trapezohedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius. These formula have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the trapezohedrons having congruent right kite faces simply by setting n=3,4,5,6,7,u2026u2026u2026u2026u2026u2026upto infinity, to analytically compute all the important parameters such as ratio of unequal edges, outer radius, inner radius, mean radius, surface area, volume, solid angles subtended by the polyhedron at its vertices, dihedral angles between the adjacent right kite faces etc. These formula are very useful for the analysis, modeling & designing of various n-gonal trapezohedrons/deltohedrons.
Category: Geometry
[5] viXra:2602.0039 [pdf] submitted on 2026-02-06 00:26:58
Authors: Harish Chandra Rajpoot
Comments: 28 Pages, 6 Figures
In this paper, the generalized formula has been derived that is applicable to locate any sphere, with a certain radius, resting in a vertex (corner) at which n no. of edges meet together at angle α between any two consecutive of them such as the vertex of platonic solids, any of two identical & diagonally opposite vertices of trapezohedron (uniform polyhedron with congruent right kite faces) & the vertex of right pyramid with regular n-gonal base. These are also useful for filleting the faces meeting at the vertex of the polyhedron to best fit the sphere in that vertex. These are used to determine the distance of the sphere from the vertex, the distance of the sphere from the edges, the fillet radius of the faces, etc. The formula has been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Category: Geometry
[4] viXra:2602.0037 [pdf] submitted on 2026-02-06 19:53:44
Authors: Harish Chandra Rajpoot
Comments: 12 Pages. 4 Figures
In the present work, general analytical expressions are derived for the fundamental geometric properties of a disphenoid using three-dimensional coordinate geometry. Specifically, closed-form formulae are obtained for the volume and total surface area, as well as for the radii of the inscribed and circumscribed spheres, and solid angle subtended by disphenoid at its vertex. In addition, explicit coordinates of the four vertices are determined, together with the coordinates of the in-center, circum-center, and centroid of the disphenoid. The proof for in-center, circum-center and centroid to be coincident is presented and mathematical equation governing all disphenoids is also derived in a closed form.
Category: Geometry
[3] viXra:2602.0022 [pdf] submitted on 2026-02-03 11:26:38
Authors: Harish Chandra Rajpoot
Comments: 29 Pages, 8 Figures, Original Research
This paper presents a set of practically oriented analytical formulas and tables compiled from geometric data of various convex uniform polyhedra, listing the dihedral angles between two faces with or without sharing a common edge. These formulas and tables are specifically intended to support the physical and computational construction of convex uniform polyhedral shells composed of different regular polygonal faces. By using the tabulated dihedral angles, wire-frame and shell models of polyhedra can be constructed efficiently by successively joining adjacent planar faces at their common edges with the correct angular orientation. The presented data are particularly useful for applications in geometric modeling, structural design, educational model fabrication, and the development of computational algorithms for polyhedral assembly. Overall, the tables provide a convenient and reliable reference for the practical realization and analysis of convex uniform polyhedral structures.
Category: Geometry
[2] viXra:2602.0021 [pdf] submitted on 2026-02-03 21:00:09
Authors: Harish Chandra Rajpoot
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please don't name title, equation/formula etc after the author's name) 3 Figures.
An analytical method is developed for the exact evaluation of the solid angle subtended by a torus at a point lying on its axis of symmetry, i.e., on the line perpendicular to the mid-plane of the torus and passing through its centre. The method is based on a geometric enclosure of the torus between two coaxial right circular cones with a common apex at the observation point and with axes coincident with the torus axis. It is shown that the solid angle associated with the torus equals the algebraic sum i.e. the difference between the solid angles subtended by the outer and inner bounding cones at the apex [1,2,3]. This construction leads to closed-form expressions for the solid angle as a function of the torus radii and the axial distance of the observation point, without recourse to surface integration. The resulting formulation provides a concise geometric characterization of toroidal visibility and is well suited for applications in geometric modeling, where exact angular measures are required, and in photometry and radiative transfer, particularly for the evaluation of irradiance, flux, and angular response of axially symmetric toroidal sources and apertures.
Category: Geometry
[1] viXra:2602.0006 [pdf] submitted on 2026-02-01 01:41:36
Authors: Yerkebulan Bolat
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
A pencil of plane conics induces an involution on any transversal line (Desargues’ Involution Theorem). For cubics, the analogous construction yields a natural degree-3 correspondence on a line, which we call a trivolution. Although the underlying mechanism is classical (Cayley—Bacharach and genus—one geometry), we give a synthetic treatment focused on the projective picture and (when relevant) circular cubics. We describe the induced degree—3 covering of a line, its monodromy via discriminants, and how this explains the rigidity of involution methods in degree 2.As concrete consequences we include a projective butterfly theorem for 2—torsion points and a cubic—method collinearity theorem of Sakhipov.
Category: Geometry
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