Harish Chandra Rajpoot is B. Tech. graduate of Mechanical Engineering from Madan Mohan Malaviya University of Technology (Formerly M.M.M. Engineering College), Gorakhpur (UP) India. He did his high school from D.A.V. Inter College, Mahoba (UP) India & Intermediate from Oxford Model Inter College, Syam Nagar, Kanpur (UP) India. He made his best efforts more than two years for this academic research in Applied Physics based on mathematical derivations & formulations. He derived a formula on permutations of alphabetic words, positive integral numbers & all other linear permutations,. It had been certified by International Journal of Mathematics & Physical Sciences Research. Manuscript ID: 004022014A. Consequently, he, using his formula, proved that factorial of any natural number can be expanded as the sum of finite terms. This expansion named as 'HCR's Series'
He had been taught, guided & inspired by his renowned & well experienced teacher of Physics Mr Upendra Sir @ Oxford Model I. C. Kanpur. He derived a formula for all five platonic solids which is the simplest & the most versatile formula to calculate all the importamt parameters of regular polyhedra. He worked to analyse Goldberg polyhedra, Archimedean solids & truncated & expanded polyhedra using his Theory of Polygon & his formula for platonic solids.
He wrote his first book Advanced Geometry based on research articles in Applied Mathematics & Radiometry for higher education which was first published by Notion Press, Chennai, India.
Published Papers of the author by International Journals of Mathematics
“HCR’s Rank or Series Formula” IJMPSR March-April, 2014
“HCR’s Series (Divergence)” IOSR March-April, 2014
“HCR’s Infinite-series (Convergence)” IJMPSR Oct, 2014
“HCR’s Theory of Polygon” IJMPSR Oct, 2014
The author Mr H.C. Rajpoot derived some important formula for 2D-figures using simple geometry & trigonometry. The formula, derived here, are related to the triangle, square, trapezium & tangent circles (Archimedean twin circles), inscribed circles. These formula are very useful for case studies in 2D-Geometry to compute the important parameters of 2D-figures.
Here the author represents all the important mathematical derivations of a disphenoid (isosceles tetrahedron) of volume, surface area, vertical height, radii of inscribed & circumscribed spheres, solid angle subtended at each vertex, coordinates of vertices, in-centre, circum-centre & centroid of a disphenoid for the optimal configuration in 3D space.
The author H.C. Rajpoot has derived all the general formula to compute the volume & surface area of a slice cut from a right circular cone by a plane parallel to its symmetry axis. All the generalized formula can be used for computing the volume, area of curved surface & area of hyperbolic section of slice obtained by cutting a right circular cone with a plane parallel to its symmetrical axis
A circular permutation is equivalent to a linear permutation when leading element is singleton/non repetitive & the rank of such a circular permutation in which the first or leading element is singleton/non-repetitive is easily calculated by using HCR's Rank Formula applied on any linear permutation. Thus the rank of a circular permutation having singleton leading element is computed.fairly easily
Here is the derivation of an analytic formula using vectors to calculates the minimum distance or great circle distance between any two arbitrary points on the sphere of a finite radius. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This is a highly precision formula which gives the correct values.
This is the application of HCR's cosine formula to derive a symmetrical & analytic formula to calculate the minimum distance or great circle distance between any two arbitrary points on any sphere of the finite radius which is equally applicable for all the distances on the tiny as well as the large sphere like giant planet if the calculations are made precisely.
H. Rajpoot's cosine formula is a dimensionless formula which is independent of radius of the sphere & holds equally good for all the spherical surfaces to compute the angle between the chords of any two great arcs meeting or intersecting each other at a common end point at some angle. It is used to find out the minimum distance between any two points on any sphere of finite radius.
This conditional inequality has been derived from a set formula which holds true for any three positive real numbers under certain conditions. There are three possible cases out of which one is always satisfied by any three positive real numbers. These conditions comes from three externally touching circles in a plane.
This formula holds good for all the regular spherical polygons. It is a very important formula (mathematical relation) applicable on any regular spherical polygon having each of its sides as an arc of the great circle on a spherical surface. It is of crucial importance to find out any of the four important parameters i.e. radius of sphere, no. of sides, length of side, interior angle of polygon.
The formula derived here by the author are applicable on a certain no. of the identical circles touching one another at different points, centered at the identical vertices of a spherical polyhedron analogous to an Archimedean solid for calculating the different parameters such as flat radius & arc radius of each circle, total surface area covered by all the circles, percentage of surface area etc
The formula generalized by the author Mr. H.C. Rajpoot are used to determine the important parameters for snugly packing the spheres in the vertices of all five platonic solids such as the radius of Nth sphere, total volume packed by all the spheres, packing ratio, the maximum packed volume & percentage etc.
The generalized formula derived by the author are 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 uniform polyhedrons with congruent right kite faces & the vertex of right pyramid
All the articles discussed & analysed here for calculating the important parameters such as solid angle subtended by the beam at the point-source, total area intercepted by the beam with a spherical surface & cone angle of equivalent beam with circular section. These formula are very useful for replacing the rectangular profile by circular profile of a beam emitted by a uniform point-source.
All the articles discussed & analysed here are related to all five platonic solids. A certain no. of the identical circles are touching one another on a whole (entire) spherical surface having certain radius then all the important parameters such as flat radius & arc radius of each circle, total surface area & its percentage covered by all the circles on the sphere have been calculated.
All the articles are related to the reflection of any point about a line in 2-D co-ordinate system and about a line & a plane in 3-D co-ordinate system. Point of reflection about a line or a plane can be easily determined simply by applying the procedures explained or by using formula derived here. These formulas are also useful to determine the foot of perpendicular drawn from a point to a line o
The generalized formula, applicable on any uniform polyhedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius, have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the uniform polyhedrons (trapezohedrons) having congruent right kite faces simply by setting n=3,4,5,6,7,………………upto infinity.
Tables of solid angles subtended at the vertices by all 5 platonic solids (regular polyhedrons) & all 13 Archimedean solids (uniform polyhedrons) calculated by the author using standard formula of solid angle & formula of tetrahedron. These are the standard values of solid angles which are useful for the analysis of platonic solids & Archimedean solids.
Generalized formula of a tetrahedron have been derived by the author Mr H.C. Rajpoot by using HCR's Inverse cosine formula & HCR's Theory of Polygon. These formula are very practical & simple to apply in case of any tetrahedron to calculate the internal (dihedral) angles between the consecutive lateral faces meeting at any of four vertices & the solid angle subtended by it (tetrahedron) at vertex
These tables have been prepared by the author Mr H.C. Rajpoot by using his data tables of the various polyhedra for determining the dihedral angle between any two adjacent faces with a common edge of different uniform polyhedra or polyhedral shells. These are very useful for the construction & preparing the wire-frame models of the uniform polyhedral shells having different regular polygonal face
All the important parameters of a great rhombicosidodecahedron (the largest Archimedean solid), having 30 congruent square faces, 20 regular hexagonal faces, 12 congruent regular decagonal faces each of equal edge length, 180 edges & 120 vertices lying on a spherical surface with certain radius, have been derived by the author Mr. H.C. Rajpoot by applying "HCR's Theory of Polygon".
All the important parameters of a great rhombicuboctahedron (an Archimedean solid), having 12 congruent square faces, 8 regular hexagonal faces, 6 congruent regular octagonal faces each of equal edge length, 72 edges & 48 vertices lying on a spherical surface with certain radius, have been derived by the author Mr H.C. Rajpoot by applying "HCR's Theory of Polygon".
All the formula have been generalized by the author which are applicable to calculate the important parameters, of any 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 centre, normal distance of each face from the centre etc.
All the important parameters of a decahedron having 10 congruent faces each as a right kite have been derived by the author H.C. Rajpoot by applying HCR's Theory of Polygon to calculate normal distance of each face from the center, inscribed radius, circumscribed radius, mean radius, surface area & volume. The formula are very useful in analysis, designing & modeling of polyhedrons.
All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry.
All the articles (formula) are useful for calculating the approximate value of solid angle subtended by a circular plane at any point in the space. These are also applicable for calculating all the parameters such as major axis, minor axis & eccentricity of elliptical plane of projection of a circular plane when viewed from any off-center position in the space.
All the articles have been derived by the author by using simple geometry & trigonometry. These articles are related to the analysis of the elliptical path in the annular region between two circle, smaller inside bigger one & their centers separated by a certain distance. These formula are used to calculate minor axis, major axis, eccentricity & the radius of the third tangent circle.
All the articles have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the formula are very practical & simple to apply in case of any spherical rectangle to calculate all its important parameters such as solid angle, surface area covered, interior angles etc.
All the important parameters of a spherical triangle have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the articles (formula) are very practical & simple to apply in case of a spherical triangle to calculate all its important parameters such as solid angle, covered surface area, interior angles etc.
All the parameters of a regular spherical polygon such as solid angle subtended at the center, area, length of side, interior angle etc. have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the formula are very practical & simple to apply in case of any regular spherical polygon to calculate all its important parameters such as solid angle, surface area covered etc.
All the important parameters of a snub dodecahedron (an Archimedean solid having 80 congruent equilateral triangular & 12 congruent regular pentagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's Theory of Polygon & Newton-Raphson Method. It can als
All the important parameters of a truncated cube (having 8 congruent equilateral triangular & 6 congruent regular octagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional
All the important parameters of a truncated icosahedron such as normal distances & solid angles of the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids.
Applications of HCR's Formula for Regular Polyhedrons like regular tetrahedron, regular hexahedron, regular dodecahedron etc. to calculate inner radius, outer radius, mean radius, surface area & volume simply by counting the no. of faces, the no. of edges in one face & measuring the edge length. It is the easiest & simplest way to calculate all important parameters of any regular polyhedron.
HCR's Method of concentric conical surfaces is the simplest & most versatile method to find out the solid angle subtended by a torus at any point lying on the geometrical axis (i.e. vertical axis passing through the center of torus).
This book is dealing with the standard formula to be remembered for case studies & practical applications. All the standard formula from 'Advanced Geometry' by the author Mr H.C. Rajpoot have been included in this book. These formula are related to the solid geometry dealing with the 2-D & 3-D figures in the space & miscellaneous articles in Trigonometry & Geometry
This book deals with the challenging review questions of permutations of words and positive integral numbers. All the questions are based on HCR's Rank Formula derived by the author.
Research paper subjected to the peer review by
Dr. K. Srinivasa Rao (email@example.com)
(Dr.K. Srinivasa Rao, FNASc.,FTNASc.,
Senior Professor (Retd.), IMSc, Chennai-6000113;
Distinguished DST-Ramanujan Professor
This book deals mainly with the analysis of solid angle which makes it unique one in Radiometry. The application of solid angle is wide spread in analysis of radiation energy, emitted by uniform point-sources, striking 2-D & 3-D figures . It analyses, systematically & logically all the concepts of articles & their applications to enable the learners to comprehend & apply in Mathematics & Physics.