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 & the figures in 3-D space & miscellaneous articles in Trigonometry & Geometry. These are useful the standard formula 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. These are useful the standard formula to be remembered for case studies & practical applications. Although, all the formula for the plane figures (i.e. planes bounded by the straight lines only) can be derived by using standard formula of right triangle that has been explained in details in "HCR's Theory of Polygon" published with International Journal of Mathematics & Physical Sciences Research in Oct, 2014.
And the analysis of oblique frustum of right circular cone has been explained in his research paper 'HCR's Infinite-series' published with IJMPSR
Mathematical analysis of n-gonal trapezohedron with 2n congruent right kite f...Harish Chandra Rajpoot
The generalized formula are applicable on any n-gonal trapezohedronhaving 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,………………upto infinity, to calculate 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 trapezoherons/deltohedrons.
HCR's Formula for Regular n-Polyhedrons (Mathematical analysis of regular n-p...Harish Chandra Rajpoot
This formula was derived by H.C. Rajpoot to calculate all the important parameters of a regular n-polyhedron such as inner radius, outer radius, mean radius, surface area & volume. This formula is a generalized dimensional formula which can be applied on any existing n-polyhedron since it depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It can be used in analysis, designing & modelling of regular n-polyhedrons. It's named as "H. Rajpoot's Formula" by the author.
This theory had been proposed by H.C. Rajpoot @ his college presently M.M.M. University of Technology, GKP-273010 in Oct, 2013, for finding out the solid angle subtended by any polygonal plane at any point in the space. It gives the simplest, easiest & the most versatile methods for calculating the mathematically correct value of solid angle subtended by any plane bounded by the straight lines like triangle, quadrilateral like rectangle, square, rhombus, trapezium etc., any regular or irregular polygon like pentagon, hexagon, heptagon, octagon etc.) at any point in the space. It is the unified theory applied for any polygon by using one standard formula only. This can derive expression for solid angle subtended by any plane bounded by the straight lines. This theory is equally applicable for atomic distances & stellar distances in the Universe.
Solid angle subtended by a rectangular plane at any point in the space Harish Chandra Rajpoot
This is the most general case for any location of given point in the space which is derived by using basic formula taken from the book "Advanced Geometry by H.C. Rajpoot". Its derivation & detailed explanation has been given in author's book of research articles of 3-D Geometry.
Volume & surface area of right circular cone cut by a plane parallel to its s...Harish Chandra Rajpoot
All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.
The author has derived the formula to analytically compute all the important parameters of a disphenoid (isosceles tetrahedron with four congruent acute-triangular faces) such as 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 has also proved the important conclusions related to a disphenoid by mathematical derivations using 3D coordinate geometry.
Mathematical Analysis of Spherical Rectangle (Application of HCR's Theory of ...Harish Chandra Rajpoot
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. & 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 formula 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 plane rectangular base etc.
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. These are useful the standard formula to be remembered for case studies & practical applications. Although, all the formula for the plane figures (i.e. planes bounded by the straight lines only) can be derived by using standard formula of right triangle that has been explained in details in "HCR's Theory of Polygon" published with International Journal of Mathematics & Physical Sciences Research in Oct, 2014.
And the analysis of oblique frustum of right circular cone has been explained in his research paper 'HCR's Infinite-series' published with IJMPSR
Mathematical analysis of n-gonal trapezohedron with 2n congruent right kite f...Harish Chandra Rajpoot
The generalized formula are applicable on any n-gonal trapezohedronhaving 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,………………upto infinity, to calculate 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 trapezoherons/deltohedrons.
HCR's Formula for Regular n-Polyhedrons (Mathematical analysis of regular n-p...Harish Chandra Rajpoot
This formula was derived by H.C. Rajpoot to calculate all the important parameters of a regular n-polyhedron such as inner radius, outer radius, mean radius, surface area & volume. This formula is a generalized dimensional formula which can be applied on any existing n-polyhedron since it depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It can be used in analysis, designing & modelling of regular n-polyhedrons. It's named as "H. Rajpoot's Formula" by the author.
This theory had been proposed by H.C. Rajpoot @ his college presently M.M.M. University of Technology, GKP-273010 in Oct, 2013, for finding out the solid angle subtended by any polygonal plane at any point in the space. It gives the simplest, easiest & the most versatile methods for calculating the mathematically correct value of solid angle subtended by any plane bounded by the straight lines like triangle, quadrilateral like rectangle, square, rhombus, trapezium etc., any regular or irregular polygon like pentagon, hexagon, heptagon, octagon etc.) at any point in the space. It is the unified theory applied for any polygon by using one standard formula only. This can derive expression for solid angle subtended by any plane bounded by the straight lines. This theory is equally applicable for atomic distances & stellar distances in the Universe.
Solid angle subtended by a rectangular plane at any point in the space Harish Chandra Rajpoot
This is the most general case for any location of given point in the space which is derived by using basic formula taken from the book "Advanced Geometry by H.C. Rajpoot". Its derivation & detailed explanation has been given in author's book of research articles of 3-D Geometry.
Volume & surface area of right circular cone cut by a plane parallel to its s...Harish Chandra Rajpoot
All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.
The author has derived the formula to analytically compute all the important parameters of a disphenoid (isosceles tetrahedron with four congruent acute-triangular faces) such as 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 has also proved the important conclusions related to a disphenoid by mathematical derivations using 3D coordinate geometry.
Mathematical Analysis of Spherical Rectangle (Application of HCR's Theory of ...Harish Chandra Rajpoot
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. & 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 formula 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 plane rectangular base etc.
HCR's Infinite or Convergence Series (Calculations of Volume, Surface Area of...Harish Chandra Rajpoot
Three series had been derived by the author, using double-integration in polar co-ordinates, binomial expansion and β & γ-functions for determining the volume, surface-area & perimeter of elliptical-section of oblique frustum of a right circular cone. All these three series are in form of discrete summation of infinite terms which converge into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case studies & practical computations.
Mathematical Analysis of Rhombicuboctahedron (Application of HCR's Theory)Harish Chandra Rajpoot
The author H C Rajpoot has derived the radius of circumscribed sphere passing through all 24 identical vertices of a rhombicuboctahedron with given edge length by applying ‘HCR’s Theory of Polygon’ & subsequently derived various formula to analytically compute the normal distances of equilateral triangular & square faces from the centre of rhombicuboctahedron, radius of mid-sphere, surface area, volume, solid angles subtended by each equilateral triangular face & each square face at the centre by using ‘HCR’s Theory of Polygon’, dihedral angle between each two faces meeting at any of 24 identical vertices (i.e. truncated rhombic dodecahedron), solid angle subtended by truncated rhombic dodecahedron at any of its 24 identical vertices.
Mathematical Analysis of Regular Spherical Polygons (Application of HCR's The...Harish Chandra Rajpoot
This document presents a mathematical analysis of regular spherical polygons. It derives a characteristic equation relating the radius of the sphere, length of each side, interior angle, and number of sides. This equation can be used to calculate any parameter when the other three are known. The document also discusses calculating the solid angle, area, and other properties of both regular spherical polygons and their corresponding planar polygons. It provides examples of applying the analysis to polygons on both general spheres and the unit sphere.
Mathematical analysis of truncated tetrahedron (Application of HCR's Theory o...Harish Chandra Rajpoot
All the important parameters of a truncated tetrahedron 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 polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of truncated octahedron (Applications of HCR's Theory)Harish Chandra Rajpoot
All the important parameters of a truncated octahedron 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 polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCRHarish Chandra Rajpoot
All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
Mathematical Analysis of Tetrahedron (dihedral angles between the consecutive...Harish Chandra Rajpoot
All the articles 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 the vertex when the angles between the consecutive edges meeting at the same vertex are known. These are the generalized formula which can also be applied in case of three faces meeting at the vertex of various regular & uniform polyhedrons to calculate the solid angle subtended by polyhedron at its vertex.
Mathematical analysis of great rhombicosidodecahedron (the largest Archimedea...Harish Chandra Rajpoot
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 by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular decagonal face & their normal distances from the center of great rhombicosidodecahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula are very useful in analysis, designing & modeling of various uniform polyhedrons.
Mathematical analysis of small rhombicosidodecahedron (Archimedean solid) by hcrHarish Chandra Rajpoot
This document provides a mathematical analysis of the small rhombicosidodecahedron, an Archimedean solid. It begins by describing how the solid can be generated from a regular icosahedron or dodecahedron. It then uses HCR's formulas for regular polyhedrons to derive equations for key parameters of the small rhombicosidodecahedron, including the angle between faces, outer radius, normal distances of different faces from the center, and solid angles subtended by each face. The analysis shows that the pentagonal faces are closest to the center and define the inner radius of the solid.
All the important parameters of a cuboctahedron (Archimedean solid having 8 congruent equilateral triangular & 6 congruent square 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
1) The document analyzes the mathematical properties of a truncated hexahedron (cube) which is generated by truncating the vertices of a regular cube.
2) It derives formulas to calculate key properties like edge length, normal distances, solid angles, volumes, surface area, radii and more using principles of right pyramids and HCR's theory of polygons.
3) Two methods are described for constructing a physical truncated hexahedron either from elementary right pyramids or by machining a solid sphere.
Mathematical analysis of truncated rhombic dodecahedron (HCR's Polyhedron)Harish Chandra Rajpoot
1) The document describes H.C. Rajpoot's analysis of a truncated rhombic dodecahedron using his "Theory of Polygon".
2) Key results include formulas for the radius of the circumscribed sphere, normal distances to different face types, surface area, volume, and dihedral angles between faces.
3) The analysis involves deriving the truncated polyhedron from a rhombic dodecahedron and establishing relationships between their geometric properties.
Mathematical analysis of decahedron with 10 congruent faces each as a right k...Harish Chandra Rajpoot
All the important parameters of a decahedron having 10 congruent faces each as a right kite have been derived by the author 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.
Mathematical analysis of truncated icosahedron & identical football by applyi...Harish Chandra Rajpoot
All the important parameters of a truncated icosahedron (Goldberg polyhedron, G(1,1)) such as normal distances & solid angles of 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 i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It is also used for finding out all the important parameters of a football similar to a truncated icosahedron. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Derivations of inscribed & circumscribed radii for three externally touching ...Harish Chandra Rajpoot
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. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles. Here is also the derivation of a general formula for computing the length of common chord of two intersecting circles.
All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 regular decagonal 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of polyhedrons.
Mathematical Analysis & Modeling of Pyramidal Flat Container, Right Pyramid &...Harish Chandra Rajpoot
This document derives generalized formulas to compute important parameters like the V-cut angle, edge length, dihedral angle, surface area, and volume of pyramidal flat containers, pyramids, and polyhedrons with regular polygonal bases. It applies HCR's Theorem and Corollary to develop formulas for a pyramidal container with a regular n-gonal base in terms of the base side length, slant height, and face inclination angle. Steps for constructing paper models of such containers are also outlined.
HCR's method of concentric cones (solid angle subtended by a torus at any poi...Harish Chandra Rajpoot
HCR's Method of concentric cones is the simplest & most versatile method to find out the solid angle subtended by a torus at any point lying on its geometrical axis.
This document summarizes 9 formulas derived by Harish Chandra Rajpoot related to geometry of 2D shapes like squares, triangles, trapezoids, and circles. The derivations use basic geometry and trigonometry. Formula 1 finds the angle subtended by a point inside a square. Formula 2 finds the area of a quadrilateral formed inside a square. Subsequent formulas find radii or lengths related to combinations of these basic shapes. Diagrams and step-by-step workings are provided for each formula derivation.
Mathematical derivations of some important formula in 2D-Geometry H.C. RajpootHarish Chandra Rajpoot
Here are some important formula in 2D-Geometry derived by the author Mr H.C. Rajpoot using simple geometry & trigonometry. The formula, derived here, are related to the triangle, square, trapezium & tangent circles. These formula are very useful for case studies in 2D-Geometry to compute the important parameters of 2D-figures.
Mathematical analysis of sphere resting in the vertex of polyhedron, filletin...Harish Chandra Rajpoot
The generalized formula derived here 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 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 sphere from the vertex, distance of sphere from the edges, fillet radius of the faces etc. The formula have been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Mathematical analysis of great rhombicuboctahedron (an Archimedean solid) (Ap...Harish Chandra Rajpoot
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 by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular octagonal face & their normal distances from the center of great rhombicuboctahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula are very useful in analysis, designing & modeling of various uniform polyhedra.
HCR's Infinite or Convergence Series (Calculations of Volume, Surface Area of...Harish Chandra Rajpoot
Three series had been derived by the author, using double-integration in polar co-ordinates, binomial expansion and β & γ-functions for determining the volume, surface-area & perimeter of elliptical-section of oblique frustum of a right circular cone. All these three series are in form of discrete summation of infinite terms which converge into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case studies & practical computations.
Mathematical Analysis of Rhombicuboctahedron (Application of HCR's Theory)Harish Chandra Rajpoot
The author H C Rajpoot has derived the radius of circumscribed sphere passing through all 24 identical vertices of a rhombicuboctahedron with given edge length by applying ‘HCR’s Theory of Polygon’ & subsequently derived various formula to analytically compute the normal distances of equilateral triangular & square faces from the centre of rhombicuboctahedron, radius of mid-sphere, surface area, volume, solid angles subtended by each equilateral triangular face & each square face at the centre by using ‘HCR’s Theory of Polygon’, dihedral angle between each two faces meeting at any of 24 identical vertices (i.e. truncated rhombic dodecahedron), solid angle subtended by truncated rhombic dodecahedron at any of its 24 identical vertices.
Mathematical Analysis of Regular Spherical Polygons (Application of HCR's The...Harish Chandra Rajpoot
This document presents a mathematical analysis of regular spherical polygons. It derives a characteristic equation relating the radius of the sphere, length of each side, interior angle, and number of sides. This equation can be used to calculate any parameter when the other three are known. The document also discusses calculating the solid angle, area, and other properties of both regular spherical polygons and their corresponding planar polygons. It provides examples of applying the analysis to polygons on both general spheres and the unit sphere.
Mathematical analysis of truncated tetrahedron (Application of HCR's Theory o...Harish Chandra Rajpoot
All the important parameters of a truncated tetrahedron 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 polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of truncated octahedron (Applications of HCR's Theory)Harish Chandra Rajpoot
All the important parameters of a truncated octahedron 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 polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCRHarish Chandra Rajpoot
All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
Mathematical Analysis of Tetrahedron (dihedral angles between the consecutive...Harish Chandra Rajpoot
All the articles 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 the vertex when the angles between the consecutive edges meeting at the same vertex are known. These are the generalized formula which can also be applied in case of three faces meeting at the vertex of various regular & uniform polyhedrons to calculate the solid angle subtended by polyhedron at its vertex.
Mathematical analysis of great rhombicosidodecahedron (the largest Archimedea...Harish Chandra Rajpoot
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 by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular decagonal face & their normal distances from the center of great rhombicosidodecahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula are very useful in analysis, designing & modeling of various uniform polyhedrons.
Mathematical analysis of small rhombicosidodecahedron (Archimedean solid) by hcrHarish Chandra Rajpoot
This document provides a mathematical analysis of the small rhombicosidodecahedron, an Archimedean solid. It begins by describing how the solid can be generated from a regular icosahedron or dodecahedron. It then uses HCR's formulas for regular polyhedrons to derive equations for key parameters of the small rhombicosidodecahedron, including the angle between faces, outer radius, normal distances of different faces from the center, and solid angles subtended by each face. The analysis shows that the pentagonal faces are closest to the center and define the inner radius of the solid.
All the important parameters of a cuboctahedron (Archimedean solid having 8 congruent equilateral triangular & 6 congruent square 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
1) The document analyzes the mathematical properties of a truncated hexahedron (cube) which is generated by truncating the vertices of a regular cube.
2) It derives formulas to calculate key properties like edge length, normal distances, solid angles, volumes, surface area, radii and more using principles of right pyramids and HCR's theory of polygons.
3) Two methods are described for constructing a physical truncated hexahedron either from elementary right pyramids or by machining a solid sphere.
Mathematical analysis of truncated rhombic dodecahedron (HCR's Polyhedron)Harish Chandra Rajpoot
1) The document describes H.C. Rajpoot's analysis of a truncated rhombic dodecahedron using his "Theory of Polygon".
2) Key results include formulas for the radius of the circumscribed sphere, normal distances to different face types, surface area, volume, and dihedral angles between faces.
3) The analysis involves deriving the truncated polyhedron from a rhombic dodecahedron and establishing relationships between their geometric properties.
Mathematical analysis of decahedron with 10 congruent faces each as a right k...Harish Chandra Rajpoot
All the important parameters of a decahedron having 10 congruent faces each as a right kite have been derived by the author 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.
Mathematical analysis of truncated icosahedron & identical football by applyi...Harish Chandra Rajpoot
All the important parameters of a truncated icosahedron (Goldberg polyhedron, G(1,1)) such as normal distances & solid angles of 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 i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It is also used for finding out all the important parameters of a football similar to a truncated icosahedron. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Derivations of inscribed & circumscribed radii for three externally touching ...Harish Chandra Rajpoot
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. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles. Here is also the derivation of a general formula for computing the length of common chord of two intersecting circles.
All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 regular decagonal 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 formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of polyhedrons.
Mathematical Analysis & Modeling of Pyramidal Flat Container, Right Pyramid &...Harish Chandra Rajpoot
This document derives generalized formulas to compute important parameters like the V-cut angle, edge length, dihedral angle, surface area, and volume of pyramidal flat containers, pyramids, and polyhedrons with regular polygonal bases. It applies HCR's Theorem and Corollary to develop formulas for a pyramidal container with a regular n-gonal base in terms of the base side length, slant height, and face inclination angle. Steps for constructing paper models of such containers are also outlined.
HCR's method of concentric cones (solid angle subtended by a torus at any poi...Harish Chandra Rajpoot
HCR's Method of concentric cones is the simplest & most versatile method to find out the solid angle subtended by a torus at any point lying on its geometrical axis.
This document summarizes 9 formulas derived by Harish Chandra Rajpoot related to geometry of 2D shapes like squares, triangles, trapezoids, and circles. The derivations use basic geometry and trigonometry. Formula 1 finds the angle subtended by a point inside a square. Formula 2 finds the area of a quadrilateral formed inside a square. Subsequent formulas find radii or lengths related to combinations of these basic shapes. Diagrams and step-by-step workings are provided for each formula derivation.
Mathematical derivations of some important formula in 2D-Geometry H.C. RajpootHarish Chandra Rajpoot
Here are some important formula in 2D-Geometry derived by the author Mr H.C. Rajpoot using simple geometry & trigonometry. The formula, derived here, are related to the triangle, square, trapezium & tangent circles. These formula are very useful for case studies in 2D-Geometry to compute the important parameters of 2D-figures.
Mathematical analysis of sphere resting in the vertex of polyhedron, filletin...Harish Chandra Rajpoot
The generalized formula derived here 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 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 sphere from the vertex, distance of sphere from the edges, fillet radius of the faces etc. The formula have been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Mathematical analysis of great rhombicuboctahedron (an Archimedean solid) (Ap...Harish Chandra Rajpoot
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 by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular octagonal face & their normal distances from the center of great rhombicuboctahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula are very useful in analysis, designing & modeling of various uniform polyhedra.
Mathematical Analysis of Spherical Triangle (Spherical Trigonometry by H.C. R...Harish Chandra Rajpoot
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. & also useful for calculating all the parameters of the corresponding plane triangle obtained by joining all the vertices of a spherical triangle by straight lines. These formula can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical triangle to the center of sphere such as normal height, angle between the consecutive lateral edges, area of base etc.
HCR’s Inverse Cosine Formula derived by Mr H.C. Rajpoot is a trigonometric relation of four variables/angles. It is applicable for any three straight lines or planes, either co-planar or non-coplanar, intersecting each other at a single point in the space. It directly co-relates the internal angles (i.e. angles between the consecutive lateral faces) & the face angles (i.e. angles between the consecutive lateral edges) of any tetrahedron. This formula is very useful to find out all the unknown internal angles if all the face angles of any tetrahedron are known & vice versa.
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) if other three are given (known) for any regular spherical polygon. It also concludes that any three of the four parameters are self-sufficient to exactly represent a unique regular spherical polygon. A regular spherical polygon having three known parameters can be created or drawn only on a unique spherical surface of a radius which is easily found out by HCR's formula.
Mathematical Analysis of Rhombic Dodecahedron by applying HCR's Theory of Pol...Harish Chandra Rajpoot
In this paper, the author Mr H C Rajpoot mathematically analyses & derives analytic formula for a rhombic dodecahedron having 12 congruent faces each as a rhombus, 24 edges & 14 vertices lying on a spherical surface with a certain radius. ‘HCR’s Theory of Polygon’ is used to derive formula to analytically compute the angles & diagonals of rhombic face, radii of circumscribed sphere, inscribed sphere & midsphere, surface area & volume of rhombic dodecahedron in terms of edge length, solid angles subtended at the vertices and dihedral angle between adjacent faces. This convex polyhedron can be constructed by joining 12 congruent elementary-right pyramids with rhombic base & certain normal height.
There are five regular polyhedrons called platonic solids that have congruent faces of regular polygons. The document lists these five solids and provides a table calculating their inner radius, outer radius, mean radius, surface area, and volume based on their edge length. The calculations are attributed to Harish Chandra Rajpoot.
The document provides information on isometric and perspective projections in engineering graphics. It defines isometric projection as a type of pictorial projection that shows the actual sizes of all three dimensions of a solid in a single view. It also defines perspective projection as representing how an object would appear to the eye from a fixed position. The document then discusses principles, scales, views and methods of isometric projection. It provides examples of isometric views of basic geometrical shapes. It also discusses the principles and methods of perspective projection like visual ray and vanishing point methods.
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...Harish Chandra Rajpoot
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 centre, normal distance of each face from the centre, inner radius, outer radius, mean radius, surface area & volume. These are useful for the analysis, designing & modeling of different non-uniform polyhedra.
FACE RECOGNITION ALGORITHM BASED ON ORIENTATION HISTOGRAM OF HOUGH PEAKSijaia
In this paper we propose a novel face recognition algorithm based on orientation histogram of Hough Transform Peaks. The novelty of the approach lies in utilizing Hough Transform peaks for determining the orientation angles and computing the histogram from it. For extraction of feature vectors first the images are divided into non overlapping blocks of equal size. Then for each of the blocks the orientation histograms are computed. The obtained histograms are combined to form the final feature vector set. Classification is done using k nearest neighbor classifier. The algorithm has been tested on the ORL
database, Yale B Database & the Essex Grimace Database.97% Recognition rates have been obtained for
ORL database, 100% for Yale B and 100% for Essex Grimace database
The document is a lecture on photogrammetry prepared by Dr. Ahmed Elhadary of Benha University's Faculty of Engineering. It discusses topics like image scale, flight planning, stereoscopic parallax, relief displacement on vertical photographs, and calculating object heights from aerial images. Examples are provided to illustrate key concepts and formulas for determining factors like the number of photographs needed to cover an area, relief displacement, and object heights.
Similar to Hcr's hand book (Formula of Advanced Geometry by H.C. Rajpoot) (11)
Mathematical analysis of a non-uniform tetradecahedron having 2 congruent reg...Harish Chandra Rajpoot
All the important parameters of a non-uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with a certain radius, have been derived by the author by applying "HCR's Theory of Polygon" to calculate solid angle subtended by each regular hexagonal & trapezoidal face & their normal distances from the center of non-uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formulas are very useful in the analysis, designing & modeling of various non-uniform polyhedra.
Regular N-gonal Right Antiprism: Application of HCR’s Theory of PolygonHarish Chandra Rajpoot
A regular n-gonal right antiprism is a semiregular convex polyhedron which has 2n identical vertices all lying on a sphere, 4n edges, and (2n+2) faces out of which 2 are congruent regular n-sided polygons, and 2n are congruent equilateral triangles such that all the faces have equal side. The equilateral triangular faces meet the regular polygonal faces at the common edges and vertices alternatively such that three equilateral triangular faces meet at each of 2n vertices. This paper presents, in details, the mathematical derivations of the generalized and analytic formula which are used to determine the different important parameters in terms of edge length, such as normal distances of faces, normal height, radius of circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, solid angle subtended by each face at the centre, and solid angle subtended by polygonal antiprism at each of its 2n vertices using HCR’s Theory of Polygon. All the generalized formulae have been derived using simple trigonometry, and 2D geometry which are difficult to derive using any other methods.
The document presents mathematical derivations of parameters for a regular pentagonal right antiprism. It derives analytic formulas for the antiprism's normal heights, normal distances from faces to the center, radius of the circumscribed sphere, surface area, volume, and other values in terms of the antiprism's edge length. All formulas are derived using trigonometry and 2D geometry applied to the transformation of a regular icosahedron into the antiprism shape.
Derivation of great-circle distance formula using of HCR's Inverse cosine for...Harish Chandra Rajpoot
The author derives the great-circle distance formula 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.
The circumscribed and the inscribed polygons are well known and mathematically well defined in the context of 2D-Geometry. The term ‘Circum-inscribed Polygon’ has been proposed by the author and used as a new definition of the polygon which satisfies the conditions of a circumscribed polygon and an inscribed polygon together. In other words, the circum-inscribed polygon is a polygon which has both the inscribed and circumscribed circles. The newly defined circum-inscribed polygon has each of its sides touching a circle and each of its vertices lying on another circle. The most common examples of circum-inscribed polygon are triangle, regular polygon, trapezium with each of its non-parallel sides equal to the Arithmetic Mean (AM) of its parallel sides (called circum-inscribed trapezium) and right kite. This paper describes the mathematical derivations of the analytic formula to find out the different parameters in terms of AM and GM of known sides such as radii of circumscribed & inscribed circles, unknown sides, interior angles, diagonals, angle between diagonals, ratio of intersecting diagonals, perimeter, area, and distance between circum-centre and in-centre of circum-inscribed trapezium. Like an inscribed polygon, a circum-inscribed polygon always has all of its vertices lying on infinite number of spherical surfaces. All the analytic formulae have been derived using simple trigonometry and 2-dimensional geometry which can be used to analyse the complex 2D and 3D geometric figures such as cyclic quadrilateral and trapezohedron, and other polyhedrons.
HCR's theorem (Rotation of two co-planar planes, meeting at angle bisector, a...Harish Chandra Rajpoot
In this theorem, the author derives a mathematical expression to analytically compute the V-cut angle (δ) required for rotating through the same angle (θ) the two co-planar planes, initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed).
This theorem is very important for creating pyramidal flat containers with polygonal (regular or irregular) base, closed right pyramids & polyhedrons having two regular n-gonal & 2n congruent trapezoidal faces.
The author has also presented some paper models of pyramidal flat containers with regular pentagonal, heptagonal and octagonal bases
How to compute area of spherical triangle given the aperture angles subtended...Harish Chandra Rajpoot
The author Mr H.C. Rajpoot has derived 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.) aperture angle subtended by each of three sides at the center of sphere are known 2.) arc length of each of three sides is known. These formula are applicable for any spherical triangle to the compute area on the sphere.
Mathematical analysis of identical circles touching one another on the spheri...Harish Chandra Rajpoot
The formula, derived here by the author H.C. Rajpoot, 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 covered etc. These formula are very useful for tiling, packing the identical circles in different patterns & analyzing the spherical surfaces analogous to all 13 Archimedean solids. Thus also useful in designing & modelling of tiled spherical surfaces.
Mathematical analysis of identical circles touching one another on the whole ...Harish Chandra Rajpoot
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 easily calculated by using simple geometry & table for the important parameters of all five platonic solids by the author Mr H.C. Rajpoot. These parameters are very useful for drawing the identical circles on a spherical surface & for designing & modeling all five platonic solids having identical flat circular faces.
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...Harish Chandra Rajpoot
The document discusses methods for calculating the reflection of points about lines and planes in 2D and 3D coordinate systems. It provides general expressions to find the coordinates of a reflected point given a point and line/plane of reflection. For a point reflected about a line or plane, the midpoint between the original and reflected point must lie on the line/plane and the line connecting the points must be perpendicular/parallel to the line/plane. Using geometry relationships, the author derives formulas to calculate the coordinates of the reflected point and foot of the perpendicular from the original point to the line/plane.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
2. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
CONTENTS
Topics Page No
Fundamental & Integral Theorems of Solid Angle 3
Basic Definition & Formulation
Solid Angle subtended by Plane Figures 4
Right Triangular Plane
Rectangular Plane
Square Plane
Regular Polygonal Plane
Approximate Analysis of Solid Angle 8
Solid Angle subtended by a Symmetrical Plane
Elliptical Plane
Circular Plane
Regular Polygonal Plane
Rectangular Plane
Rhombus like Plane
Solid Angle subtended by 3-D Figures 10
Sphere
Circular Cylinder
Right Circular Cone
Right Pyramid
Right Prism
Torus, Ellipsoid & Paraboloid
Solid State (arrangements of identical spheres) 16
Simple Cubic Cell, B.C.C. Unit Cell, F.C.C. Unit Cell
Photometry/Radiometry 17
Luminous Flux, Intensity, Radiation Energy, Lambert‟s Formula
Space Science (Celestial Bodies) 18
Energy of Interception by Earth & Moon emitted by Sun
Analysis of Frustums 20
Sphere, Right Circular Cone, Circular Cylinder
Analysis of Miscellaneous Articles 22
Vector‟s Cosine Formula, HCR‟s Cosine Formula
Axioms of Tetrahedron, Concurrent Vectors, Triangle
Spiked Regular Polygon, Right Pyramid & Analogy of a Right Cone
HCR’s Theory of Polygon (Graphical Method) 25
Solid Angle subtended by a Polygonal Plane at any Point in the Space
Axiom of Polygon & Right Triangle
Comparison of Analytical & Graphical Methods
3. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Fundamental & Integral Theorems of Solid Angle
1. Solid angle: “The three dimensional angle subtended by an object at a certain point
in the space”. It is also known as Angle of View for 2-D & 3-D objects.
It is usually denoted by (Omega). Its unit is Steradian (in brief „sr‟).
Mathematically, it is given by the following expression (Basic Definition)
( ( ) )
Where, distance ‘r’ from the given point (observer) to all the points on visible surface of
the object is constant. Solid angle subtended by a point & a straight line is zero.
Although the distance „r‟ from the given point to all the points of visible surface is no more
constant except in case of a spherical surface. But still the distance „r‟ may be assumed to be
constant for an infinitesimal small area „dA‟ on the visible surface of the given object.
This concept derives the fundamental theorem which is the integration of solid angle sub-
tended by infinitesimal small area „dA‟ on visible surface at the given point in the space.
2. The solid angle subtended by an object at a given point (observer) in the space is
given by Fundamental Theorem
∫ ( )
Where, distance „r‟ is constant within limit i.e. distance (r) from the given point is constant
for all the points of infinitesimal small area „dA‟ of visible surface of the object.
3. The solid angle subtended by a right circular cone with apex angle at the apex
point is given as
( )
If normal height of the right cone is „H‟ & radius of the base is „R‟ then the apex angle is
related as
√
(
√
)
4. The solid angle subtended by an object at a given point is given by „Integral
Theorem’ (in form of spherical co-ordinates)
4. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
∫ |∫ | [ ] [ ]
Where, ( )
a. Geometrical shape of the object &
b. Orientation of shape w.r.t. to the given point (observer) in the space
Solid Angle Subtended by Plane Figures
5. The solid angle subtended by a plane-figure, in the first quadrant w.r.t. foot of
perpendicular drawn to the plane from a given point in the space, is given by the
„HCR’s Corollary of Plane-figure’ (proposed by HCR)
∫
( )
√( ( ))
Where, „h‟ is the normal height of the point from the plane
( ) ( ) ( ) is the equation of boundary-curve of the given
plane in the first quadrant w.r.t. foot of perpendicular assuming as the origin &
is variable & .
Note: Foot of perpendicular drawn from the point must lie within the boundary of plane &
( ) must be differential at each point of the curve in the first quadrant i.e. ( ) is
the equation of single-boundary of the given plane..
Standard formulae:
Standard formulae to be remembered are following
6. The solid angle subtended by a right triangular plane with base „b‟ & perpendicular
„p‟ at any point lying at a height „h‟ on the vertical axis passing through the acute
angled vertex common to the perpendicular ‘p’ & hypotenuse is given as
,
√
- ,(
√
) (
√
)- ( )
If the vertical axis passes through the vertex common to the base ‘b’ & hypotenuse, the solid
angle is given as
,
√
- ,(
√
) (
√
)- ( )
5. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
The above formulae are extremely useful which can be used to find out the solid angle
subtended by any polygonal plane at any point in the space. (HCR’s Invention-2013)
7. The solid angle subtended by a right triangular plane with orthogonal sides „a‟ &
„b‟ at any point lying at a height „h‟ on the vertical axis passing through the right
angled vertex is given as
,
( √ √ )
( )
-
8. The solid angle subtended by a rectangular plane with sides „ ‟ & „b‟ at any point
lying at a height „h‟ on the vertical axis passing through the centre is given as
(
√( )( )
)
9. The solid angle subtended by a rectangular plane with sides „l‟ & „b‟ at any point
lying at a height „h‟ on vertical axis passing through one of the vertices is given as
(
√( )( )
)
10. The solid angle subtended by a square plane with each side „a‟ at any point lying at
a height „h‟ on the vertical axis passing through the centre is given as
( )
11. The solid angle subtended by a square plane with each side „a‟ at any point lying at
a height „h‟ on the vertical axis passing through one of the vertices is given as
( )
12. The solid angle subtended by a regular polygon having „n‟ number of the sides each
of length „a‟, at any point lying at a height „h‟ on the vertical axis passing through the
centre is given as
(
√
)
13. The solid angle subtended by a circular plane with a radius „R‟ at any point lying at a height
„h‟ on the vertical axis passing through the centre is given as
(
√
)
6. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
14. The solid angle subtended by an infinitely long rectangular plane having width „b‟ at any
point lying at a height „h‟ on the vertical axis passing through the centre is given as
(
√
)
15. The solid angle subtended by an infinitely long rectangular plane having width „b‟ at any
point lying at a height „h‟ on the vertical axis passing through the vertex is given as
(
√
)
16. Solid angle subtended by a triangular plane at any point lying on the vertical axis
passing through any vertex is determined by drawing a perpendicular from the same
vertex to its opposite side i.e. by dividing the triangular plane into two right triangular
planes & using standard formula (from eq(1) or (2) above) as follows
,
√
- ,(
√
) (
√
)-
17. The solid angle subtended by a regular pentagonal plane with each side „a‟ at any
point lying at a normal height h from one of the vertices is given as
[ (
√
) (
√
)
(
√
)]
⇒ ( )
⇒
18. The solid angle subtended by a regular hexagonal plane with each side „a‟ at any
point lying at a normal height h from one of the vertices is given as
* (
√
) (
√
√
) (
√
)+
⇒ ( )
⇒
19. The solid angle subtended by a regular heptagonal plane with each side „a‟ at any
point lying at a normal height h from one of the vertices is given as
7. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
[
(
( )
√ ( ))
(
( )
√( ( ) ( ))
)
(
( )
√ ( )) (
( )
√ ( ))
(
( )
√ ( ) ( ))
]
⇒ ( )
⇒
20. The solid angle subtended by a regular octagonal plane with each side „a‟ at any
point lying at a normal height h from one of the vertices is given as
[
(
√
)
(
√( √ ) ( ( √ ))
)
(
√
√
)
(
√ ( √ )
)
(
√( √ ) ( ( √ ))
)]
⇒ ( )
⇒
8. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Approximate Analysis of Solid Angle
21. Approximate value of the solid angle subtended by a symmetrical plane at any
point in the space is given by „Approximation-theorem’ as follows
[
√( )
]
( )
Where is the area of the given symmetrical plane
is the angle between the normal to the plane & the line joining the given point to the
centre of the plane &
is the distance of the given point from the centre of the plane
Limitations:
This formula is applicable only for the planes satisfying the following two conditions
I.) The plane should have at least two axes of symmetry i.e. a point of symmetry &
II.) The factor of circularity ( ) should be close to the unity i.e.
If the above conditions are satisfied then the error involved in the results obtained from
Approximation-formula will be minimum & permissible.
22. The solid angle subtended by an elliptical plane with major & minor axes „a‟ & „b‟ at
any point lying at a normal distance „r‟ from the centre is given as
[
√( )]
Factor of circularity of elliptical plane,
√ 0 ( ) √ 1
23. The solid angle subtended by a circular plane with a radius „R‟ at any point lying at a
normal distance „r‟ from the centre is given as
9. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
[
√( )
]
Factor of circularity of circular plane,
24. The solid angle subtended by a regular polygonal plane with „n‟ number of the sides
each of length „a‟ at any point lying at a normal distance „r‟ from the centre is given
as
[
√ ( )
]
( )
Factor of circularity of regular polygonal plane,
[ ⇒ ( )]
Higher is the number of sides higher will be the factor of circularity of a regular polygon.
25. The solid angle subtended by a rectangular plane with sides „l‟ & „b‟ at any point
lying at a normal distance „r‟ from the centre is given as
[
√( )]
Factor of circularity of rectangular plane,
√
[ ⇒ ( )]
26. The solid angle subtended by a rhombus-like plane with diagonals at any
point lying at a normal distance „r‟ from the centre is given as
[
√( )]
Factor of circularity of rhombus-like plane,
10. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
√
[ ⇒ ( )]
27. The solid angle subtended by a circular plane at any point in the space is given as
[ , (
√
)- ]
The value of is found out from the auxiliary equation given as follows
⇒ ( ) ( )
Where
(
( )
) (
( )
)
(
√ √
) √
28. The solid angle subtended by an elliptical plane with major axis „a‟, minor axis „b‟ &
eccentricity „e‟, at a distance „d‟ from the centre to a particular point from which it
appears as a perfect circular plane is given as
⇒ * (
√
) (
√ √
)+
( )
Where, is the angle between normal to the plane & the line joining the particular point to
the centre of the plane.
Solid Angle subtended by 3-D figures
* Solid angle subtended by the Universe at any point in the space is
Sphere:
29. The solid angle subtended by a closed surface at any point completely inside its
boundary is always ( )
30. The solid angle subtended at any point lying on the plane by a cap fully covering the
plane is always
31. The solid angle subtended by a sphere with a radius „R‟ at any point lying at a
distance d from the centre is given as
11. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
*
√
+
32. The solid angle subtended by a hemispherical shell with a radius „R‟ at any internal
point lying on the axis at a distance „x‟ from the centre is given as
[
√
]
33. The solid angle subtended by a hemispherical shell with a radius „R‟ at any external
point lying on the axis at a distance „x‟ from the centre of circular-opening is given as
[
√
]
34. The solid angle subtended by a hemispherical shell at any point lying inside the
shell on the plane of circular opening, is always
35. The solid angle subtended by a hemispherical shell at the peak-point of the surface
lying inside the shell, is always irrespective of the radius.
36. The solid angle subtended by a hemispherical shell at the centre of mass is always
irrespective of the radius.
Circular Cylinder:
37. The solid angle subtended by the solid cylinder at any point lying on the longitudinal
axis at a distance „x‟ from the centre of front-end is given as
[
√
]
38. The solid angle subtended by a cylinder with radius „R‟ & length „L‟ at any point
lying on the transverse axis at a distance „x‟ from the centre (neglecting the effect of
curvature of circular ends) is given as
*
√ ( )
+ [ ]
39. The solid angle subtended by an infinitely long cylinder with a radius „R‟ at any
point lying on the transverse axis at a distance „x‟ from the centre is given as
( ) [ ]
40. The solid angle subtended by a hollow cylindrical shell with radius „R‟ & length „L‟
at any internal point lying on the longitudinal axis at a distance „x‟ from the centre of
one of the ends is given as
12. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
*
√
( )
√( )
+ [ ]
41. The solid angle subtended by a hollow cylindrical shell with radius „R‟ & length „L‟
at any external point lying on the longitudinal axis at a distance „x‟ from the centre of
front end is given as
*
( )
√( ) √
+ [ ]
42. The solid angle subtended by a hollow cylindrical shell with radius „R‟ & length „L‟
at the centre is given as
[
√
]
Right Circular Cone:
43. The solid angle subtended by a right cone with normal height „H‟ & radius „R‟ at the
apex point is given as
[
√
]
44. The solid angle subtended by a hollow right conical shell with normal height „H‟ &
radius „R‟ at any internal point lying on the axis at a distance „x‟ from the centre of
base is given as
[
√
]
45. The solid angle subtended by a hollow right conical shell with normal height „H‟ &
radius „R‟ at any external point lying on the axis at a distance „x‟ from the centre of
base is given as
[
√
]
46. The solid angle subtended by a hollow right conical shell with normal height „H‟ &
radius „R‟ at the apex point inside the shell is given as
[
√
]
47. The solid angle subtended by a hollow right conical shell with normal height „H‟ &
radius „R‟ at the centre of mass is given as
[
√
]
13. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Right Pyramid:
48. The solid angle subtended by a right pyramid with base as a regular polygon with
„n‟ number of the sides each of length „a‟, normal height „H‟ & angle between
consecutive lateral edges , at any internal point lying on the axis at a distance „x‟
from the centre of base is given as
(
√
)
( √ )
Where, the relation between is given as
√ ( )
49. The solid angle subtended by a right pyramidal shell with base as a regular polygon
with „n‟ number of the sides each of length „a‟ & normal height „H‟, at any internal
point lying on the axis at a distance „x‟ from the centre of base is given as
(
√
)
50. The solid angle subtended by a right pyramidal shell with base as a regular polygon
with „n‟ number of the sides each of length „a‟ & normal height „H‟, at any external
point lying on the axis at a distance „x‟ from the centre of base is given as
(
√
)
51. The solid angle subtended by a right pyramid with base as a regular polygon with „n‟
number of the sides each of length „a‟, normal height „H‟ & angle between
consecutive lateral edges at apex point inside the shell is given as
(
√
)
( √ )
52. The solid angle subtended by a regular tetrahedron at each of the vertices
(externally), is 0.55 sr.
The solid angle subtended by a regular tetrahedron at each of the vertices (internally),
is 6.83 sr.
14. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
53. The solid angle subtended by each of the octants of orthogonal planes at the origin
point is 0.5 sr.
Right Prism:
54. The solid angle subtended by a solid right prism with cross-section as a regular
polygon with „n‟ number of the sides each of length „a‟ & length „L‟, at any point
lying on the longitudinal axis at a distance „x‟ from the centre of front end is given as
(
√
)
In this case, value of solid angle is independent of the length of right prism.
55. The solid angle subtended by a hollow right prismatic shell with cross-section as a
regular polygon with „n‟ number of the sides each of length „a‟ & length „L‟, at any
internal point lying on the longitudinal axis at a distance „x‟ from the centre of one
of the ends is given as
[ (
√
) (
( )
√ ( )
)]
[ ]
56. The solid angle subtended by a hollow right prismatic shell with cross-section as a
regular polygon with „n‟ number of the sides each of length „a‟ & length „L‟, at any
external point lying on the longitudinal axis at a distance „x‟ from the centre of front
end is given as
[ (
( )
√ ( )
) (
√
)]
57. The solid angle subtended by a hollow right prismatic shell with cross-section as a
regular polygon with „n‟ number of the sides each of length „a‟ & length „L‟ at the
centre is given as
(
√
)
15. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
58. The solid angle subtended by a hollow right prismatic shell with cross-section as a
regular polygon with „n‟ number of the sides each of length „a‟ & length „L‟ at the
centre of one of the open ends is given as
(
√
)
59. The solid angle subtended by a right pyramidal shell with one end open & another
closed, cross-section as a regular polygon with „n‟ number of the sides each of length
„a‟ & length „L‟, at any internal point lying on the longitudinal axis at a distance „x‟
from the centre of open end is given as
(
√
)
[ ]
60. The solid angle subtended by a right pyramidal shell with one end open & another
closed, cross-section as a regular polygon with „n‟ number of the sides each of length
„a‟ & length „L‟ at any internal point lying on the longitudinal axis at a distance „x‟
from the centre of front end (open or closed) is given as
(
√
)
[ ]
Torus, Ellipsoid & Paraboloid:
61. The solid angle subtended by a torus with inner & outer radii „r‟ & „R‟ respectively at
any point lying at a height „h‟ on the vertical axis passing through centre is given as
(
( )
) [ )
62. The solid angle subtended by a torus with inner & outer radii „r‟ & „R‟ respectively at
the centre is given as
( )
63. The solid angle ( ) subtended by an ellipsoid generated by rotating an ellipse
⁄ ⁄ about the major axis at any point lying on the major axis at a
distance d from the centre is given as
16. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
. √
( )
( )
/
64. The solid angle ( ) subtended by an ellipsoid generated by rotating an ellipse
⁄ ⁄ about the minor axis at any point lying on the minor axis at a
distance d from the centre is given as
. √
( )
( )
/
65. The solid angle ( ) subtended by a paraboloid generated by rotating the parabola
about the axis at any point lying on the axis at a distance d from the vertex
is given as
. √ /
Solid State (arrangement of identical spheres):
66. The solid angle subtended by ‘n’ number of the identical spheres each with a radius
„R‟ arranged, touching one another, in a complete circular fashion at a point lying on
the vertical axis at a height „h‟ from the centre of reference plane is given as
[
√
]
[ ]
67. The solid angle subtended by „n‟ number of the identical spheres each with a radius
„R‟ arranged, touching one another, in a complete circular fashion at the centre of
reference plane is given as
( ) [ ]
68. The solid angle subtended by the spheres forming a simple cubic cell at the centre is
. Each of the spheres subtends
69. The ratio of radius (r) of the largest sphere fully trapped in a simple cubic cell to the
radius (R) of spheres forming the cell, is ⁄
70. The solid angle subtended by the largest sphere fully trapped in a simple cubic cell
at the centre of each of the spheres forming the cell is
71. The solid angle subtended by the spheres forming a tetrahedral void at the centre is
Each of the spheres subtends
17. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
72. The ratio of radius (r) of the largest sphere fully trapped in a tetrahedral void to
the radius (R) of spheres forming the void, is ⁄
73. The solid angle subtended by the largest sphere fully trapped in a tetrahedral void
at the centre of each of the spheres forming the void is
74. The solid angle subtended by the spheres forming a octahedral void at the centre is
. Each of the spheres subtends
75. The ratio of radius (r) of the largest sphere fully trapped in a octahedral void to the
radius (R) of spheres forming the void, is ⁄
76. The solid angle subtended by the largest sphere fully trapped in an octahedral void
at the centre of each of the spheres forming the void is
* About of tetrahedral void & of octahedral void is exposed to their
outside (surrounding) space. Thus an octahedral void is more packed than a tetrahedral
void. (Mr HC Rajpoot)
Photometry/Radiometry
77. The radiation (visible light or other E.M. radiation) energy emitted by uniform point-
source located at the centre of a spherical surface is uniformly distributed over the
entire the surface.
78. The radiation (visible light or other E.M. radiation) energy emitted by uniform
linear-source coincident with the axis of a cylindrical surface is uniformly distributed
over the entire the surface.
79. The distribution of radiation energy, emitted by a uniform point-source, over a
plane surface is no more constant. Energy density at the foot of perpendicular drawn
from the point-source to the plane is maximum & decreases successively at the points
lying away from point of maxima.
80. The total luminous flux, emitted by a uniform point-source with luminous intensity I,
intercepted by an object with visible surface „S‟, is given as
∫
Where, is the solid angle subtended by the same object at the given point-source.
81. The total radiation energy, emitted by a uniform point-source with luminosity L,
intercepted by an object with visible surface „S‟, is given as
( ) ( ) ∫
Where, is the solid angle subtended by the same object at the given point-source.
18. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
82. Density of luminous flux (or luminosity) emitted by a uniform point-source with
intensity I, over a plane-surface (S) with a total area „A‟ (assuming uniform
distribution of intercepted flux over the entire plane), is given as
( )
( )
( ) ∫
83. Lambert’s cosine formula:
Density of luminous flux (or luminosity (E)) emitted by a uniform point-source with intensity
„I‟ over a plane-surface with area „A‟ (assuming uniform distribution of intercepted flux over
the entire plane), is given
Where, „r‟ is the distance of point-source from the centre of the plane &
is the angle between normal to the plane & line joining point-source to the centre.
Limitation: The above formula is applicable for smaller plane surfaces and longer distances
for minimum error involved in the results obtained.
Space Science (Celestial Bodies):
84. The solid angle subtended by a celestial body with a mean radius at any point in the
space (assuming that the body has a perfect spherical shape) is given as
*
√
+ ( )
85. The solid angle subtended by the Sun at the centre of the Earth is about
.
86. The solid angle subtended by the Earth at the centre of the Sun is about
.
87. The solid angle subtended by the Sun at the centre of the Earth is about 11891.42
times that subtended by the Earth at the centre of the Sun.
88. The total radiation energy, emitted by the Sun, incident on the Earth is about
⁄ & energy density over the intercepted surface of the Earth is
about ⁄ (Assuming no part of the radiation, emitted by the Sun,
travelling to the Earth is absorbed or reflected).
89. The solid angle subtended by the Moon at the centre of the Earth is about
.
19. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
90. The solid angle subtended by the Earth at the centre of the Moon is about
.
91. The solid angle subtended by the Earth at the centre of the Moon is about 13.45 times
that subtended by the Moon at the centre of the Earth.
92. The luminosity (L) of a star with a mean radius „ ‟, effective temperature „ ‟
(absolute) & emissivity „e‟ is given as
Where,
If the star is assumed as a perfect black body i.e. e = 1 then we have
93. Brightness (B) of a star with a mean radius „ ‟, effective temperature „ ‟ (absolute)
& emissivity „e‟ at a point lying at a distance „x‟ from the centre is given as
( )
If the star is assumed as a perfect black body i.e. e = 1 then we have
( )
94. The relation between luminosity (L) & brightness (B) of a star at a particular point
lying at a distance „x‟ from the centre, is given as
95. Energy emitted by a uniform spherical-source with mean radius , luminosity „L‟,
effective temperature „ ‟ & emissivity „e‟, intercepted by a spherical body with a
radius „R‟ lying at a distance „d‟ between their centres, is given as
( ) (
√
) (
√
)
If the spherical-source is assumed as a perfect black body i.e. emissivity, e = 1 then we have
( ) (
√
) (
√
)
96. Area of interception of a spherical body with a radius „R‟ w.r.t. a given point lying at
a distance d from the centre is given as
20. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
( ) [ ]
97. If radiation energy, emitted by a uniform spherical-source with mean radius ,
effective temperature „ ‟, emissivity „e‟ & luminosity „L‟, is uniformly distributed
over the area of interception of a spherical body with a radius „R‟ & lying at a
distance d from the centre of the source then the Energy Density (E.D.) is given as
( ) (
√
) ( ) (
√
) [ ( ) ]
The above expression is applicable for uniform spherical-source like a star. If the source is
assumed as perfect black body, take e =1 in the above expression.
Analysis of Frustums
Frustum of Sphere:
98. A frustum, of the sphere with a radius „R‟, has a single circular-section then the radius
(r) of circular-section, surface area (A) & volume (V) are given as
√
( ) ( )
Where, „h‟ is the radial thickness of the frustum.
If the angle subtended by the circular-section of the frustum at the centre of parent sphere
with radius „R‟ is then we have
( )
( ) [ ]
99. A frustum, of the sphere with a radius „R‟, has two parallel circular-sections with
radii subtending angles respectively at the centre of parent sphere then
radii ( ) of circular-sections, surface area (A) & volume (V) of the frustum are
given as
( )
21. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
[ ( ) ( )]
( ⇒ [ ) ( ] )
100. A frustum of the sphere with a radius „R‟ has two parallel & identical circular-
sections each subtending an angle at the centre of parent sphere then radius „r‟ of
circular-sections, surface area (A) & volume of the frustum are given as
( ) * +
Frustum of Right Circular Cone:
101. Major axis ( ), minor axis (2b) & eccentricity (e) of an elliptical section,
generated by cutting a right cone with apex angle with a smooth plane inclined at
an angle with the axis & at a normal distance „h‟ from the apex point, are given as
√
( ⇒ )
102. Angular shift ( ): Angle between axis of the right cone & line joining the
centre of elliptical section (generated by cutting right cone obliquely with the axis) to
the apex point. It is given as
( ) ( ⇒ )
103. Perimeter (P) of elliptical section
( )
Where
∑ *,
( ) ( )
- +
104. Volume (V) of oblique frustum with elliptical section
22. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
( )
( )
Where
∑ *( )( ) ,
( ) ( )
- +
105. Surface area (S) of oblique frustum with elliptical section
( )
Where
∑ *( ) ,
( ) ( )
- +
106. Radius of the circular section, generated by cutting a right cone with apex
angle with a smooth plane normal to the axis & at a normal distance „h‟ from the
apex point, is given as
( )
Frustum of Circular Cylinder:
107. Major axis ( ), minor axis (2b) & eccentricity (e) of an elliptical section,
generated by cutting a circular cylinder with a diameter „D‟ with a smooth plane
inclined at an angle with the axis, are given as
( )
Analysis of Miscellaneous Articles
108. Vectors’ Cosine-formula: (Derived by the author of the book)
If „ ‟ is the angle between two concurrent vectors and one of them, keeping other stationary,
is rotated by an angle „ ‟ about the point of concurrency in a plane inclined at an angle „ ‟
23. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
with the plane containing both the vectors in initial position then the angle ( ) between them
in final position is given by the following expression
⇒ ( ) ( ( ) )
109. HCR’s Cosine-formula:
If the angle between two concurrent vectors in the shifted position is known then the angle
between the plane of rotation & the plane containing the vectors in initial position is given
as
( ) ( ( ) )
110. Axiom of tetrahedron: If are the angles between consecutive lateral
edges meeting at a vertex of a tetrahedron then the tetrahedron will exist only and if
only the sum of any two angles out of is always greater than third one i.e.
( ) ( ) ( ) ( )
111. Internal angles , between lateral faces respectively opposite to the
angles between consecutive lateral edges meeting at a vertex of a
tetrahedron, are given as
( ) ( )
( )
112. Axiom of three concurrent-vectors: The perpendicular, drawn from the point
of concurrency of three equal vectors to the plane of a triangle generated by joining
the heads of all the vectors, always passes through the circumscribed-centre of the
triangle.
113. Axiom of triangle: The distance between circumscribed & inscribed centres
in a triangle is given as
⇒ √{( ) ( ) ( )}
Where,
24. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
114. The distance, between the circumscribed & inscribed centres of an isosceles
triangle having each of the equal angles , is given as
√( ) ( ) ( ) ( ( ))
( ( ))
115. Spiked/star-like regular-polygon: Regular-polygon obtained by the straight
lines joining each vertex to the end-points of its opposite side in a regular-polygon
having odd no. of the sides is called a spiked regular-polygon or a star-like regular-
polygon.
116. Spike angle ( ) of spiked regular polygon is given as
117. Area ( ) of a spiked regular polygon is given as
( )
( ⇒ )
118. Radius of the circle passing through tips of all the spikes of a spiked regular
polygon with „n‟ number of the spikes & span „a‟, is given as
( ⇒ )
119. Right pyramid with base as a regular polygon: Right pyramid with base as
a regular polygon having „n‟ number of the sides each of length „a‟, angle between
consecutive lateral edges & normal height „H‟
Area of lateral surface,
√
Area of regular polygonal base,
Volume,
( )
25. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Relation between the angle (between consecutive lateral edges) & acute angle (between
any of lateral edges & axis of the right pyramid)
√ ( )
Relation between normal height „H‟ & length „a‟ of side of the base
√
120. Analogy:
Volume of a right conical body with cross-section varying linearly with its normal height
(length) is given as
( )( )
121. Angle ( ) between any two bonds in a molecule having regular tetrahedral
structure (like molecule) is given as
(
√
)
122. Angle between any two lines joining the centre of a tetrahedral void to the
centres of spheres (forming the void) is equal to (
√
⁄ )
Note: In some of the books, it is given equal to which is very closer to the most
correct value
HCR’s Theory of Polygon
Axiom of Polygon
For a given point in the space, each of the polygons can be divided internally or externally or
both (w.r.t. F.O.P.) into a certain number of the elementary triangles all having a common
vertex at the foot of perpendicular (F.O.P.) drawn from the given point to the plane of
polygon, by joining all the vertices of polygon to the F.O.P. by extended straight lines.
1. Polygon is externally divided if the F.O.P. lies outside the boundary
2. Polygon is divided internally or externally or both if the F.O.P. lies inside or on the
boundary depending on the geometrical shape of polygon (i.e. angles & sides)
(See the figures (1), (2), (3), (4), (5) & (6) below)
Axiom of Right Triangle
26. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
If the perpendicular, drawn from a given point in the space to the plane of a given triangle,
passes through one of the vertices then the triangle can be divided internally or externally
(w.r.t. F.O.P.) into two elementary right triangles having common vertex at the foot of
perpendicular, by drawing a normal from the common vertex (i.e. F.O.P.) to the opposite side
of given triangle.
1. An acute angled triangle is internally divided w.r.t. F.O.P. (i.e. common vertex)
2. A right angled triangle is internally divided w.r.t. F.O.P. (i.e. common vertex)
3. An obtuse angled triangle is divided
a. Internally if and only if the angle of common vertex (F.O.P.) is obtuse
b. Externally if and only if the angle of common vertex (F.O.P.) is acute
(As shown in the figures (1) & (2) below)
Fig 1: are different locations Fig 2: are different locations
Graphical Method: (For polygonal-planes)
This method is based on above two axioms (proposed by the author) for analysis of polygonal
planes. It is the most versatile method for finding out the solid angle subtended by a
polygonal plane at any point in the space. This method overcomes all the limitations of
Approximation formula used for symmetrical planes. Although, it necessitates the drawing of
a given polygonal plane but it involves theoretically zero error. Still due to computational
error some numerical error may be involved in the results obtained by this method.
Graphical method eliminates a lot of calculations in finding out the solid angle subtended by
a given plane if the location of foot of perpendicular, drawn from a given point in the space,
is known w.r.t. the given plane.
Working Steps:
STEP 1: Trace/draw the diagram of a given polygon (plane) with known geometrical
dimensions.
STEP 2: Specify the location of foot of perpendicular drawn from a given point to the plane
of polygon.
27. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
STEP 3: Join all the vertices of polygon to the foot of perpendicular by the extended straight
lines. Thus the polygon is divided into a number of elementary triangles, all having a
common vertex at the foot of perpendicular.
STEP 4: Now, consider each elementary triangle & divide it into two sub-elementary right
triangles to find the solid angle subtended by each elementary triangle.
STEP 5: Solid angle subtended by each individual sub-elementary right triangle is
determined using the standard formula of right triangle given as
,
√
- ,(
√
) (
√
)-
STEP 6: Now, solid angle, subtended by polygonal plane at the given point, will be the
algebraic sum of solid angles of its individual sub-elementary right triangles as given
Let‟s us consider a polygonal plane (with vertices) & a given point say ( ) at
a normal height from the given plane (i.e. plane of paper) in the space.
Now, specify the location of foot of perpendicular say „O‟ on the plane of polygon which
may lie ( ( ) )
1. Outside the boundary
2. Inside the boundary
3. On the boundary
a. On one of the sides or b. At one of the vertices
Let‟s consider the above cases one by one as follows
1. F.O.P. outside the boundary:
Let the foot of perpendicular „O‟ lie outside the boundary of polygon. Join all the vertices of
polygon (plane) 123456 to the foot of perpendicular „O‟ by the extended straight lines
(As shown in the figure 3)
Thus the polygon is divided into elementary
triangles (obtained by the extension lines), all
having common vertex at the foot of
perpendicular „O‟. Now the solid angle
subtended by the polygonal plane at the
given point „P‟ in the space is given
By Element-Method
Figure 3: F.O.P. lying outside the boundary
28. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
( )
Where,
Thus, the value of solid angle subtended by the polygon at the given point is obtained by
setting these values in the eq. (I) as follows
⇒ ( ) ( ) ( ) ( )
Further, each of the individual elementary triangles is easily be divided into two sub-
elementary right triangles for which the values of solid angle are determined by using
standard formula of right triangle.
2. F.O.P. inside the boundary:
Let the foot of perpendicular „O‟ lie inside the boundary of polygon. Join all the vertices of
polygon (plane) 123456 to the foot of perpendicular „O‟ by the extended straight lines
(As shown in the figure 4)
Thus the polygon is divided into elementary triangles
(obtained by the extension lines), all having common
vertex at the foot of perpendicular „O‟. Now the solid
angle subtended by the polygonal plane at the given
point P in the space is given
By Element-Method
( )
Where,
Thus, the value of solid angle subtended by the polygon at the given point is obtained by
setting these values in the eq. (II) as follows
⇒ ( )
Figure 4: F.O.P. lying inside the boundary
29. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Further, each of the individual elementary triangles is easily divided into two sub-elementary
right triangles for which the values of solid angle are determined by using standard formula
of right triangle.
3. F.O.P. on the boundary: Further two cases are possible
a. F.O.P. lying on one of the sides:
Let the foot of perpendicular „O‟ lie on one of the sides say „12‟ of polygon. Join all the
vertices of polygon (plane) 123456 to the foot of perpendicular „O‟ by the straight lines
(As shown in the figure 5)
Thus the polygon is divided into elementary triangles
(obtained by the straight lines), all having common
vertex at the foot of perpendicular „O‟. Now the solid
angle subtended by the polygonal plane at the given
point „P‟ in the space is given By Element-Method
( )
Further, each of the individual elementary triangles is easily divided into two sub-elementary
right triangles for which the values of solid angle are determined by using formula of right
triangle.
b. F.O.P. lying at one of the vertices:
Let the foot of perpendicular lie on one of the vertices say
„5‟ of polygon. Join all the vertices of polygon (plane)
123456 to the foot of perpendicular (i.e. common vertex „5‟)
by the straight lines
(As shown in the figure 6)
Thus the polygon is divided into elementary triangles
(obtained by the straight lines), all having common vertex at the foot of perpendicular „5‟.
Now the solid angle subtended by the polygonal plane at the given point „P‟ in the space is
given
By Element-Method
( )
Figure 5: F.O.P. lying on one of the sides
30. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Further, each of the individual elementary triangles is easily divided into two sub-elementary
right triangles for which the values of solid angle are determined by using standard formula
of right triangle.
Comparison of Approximation Method & Graphical Method
Approximation Method Graphical Method
1. It is applicable only for symmetrical plane-
figures.
2. It is not useful for finding the precise/exact
value of solid angle i.e. a certain error is
included in the results.
3. It depends on the geometrical shape of
symmetrical plane & its configuration (factor
of circularity).
4. It requires mere mathematical calculations
which are easier as compared to that of
graphical method.
1. It is applicable only for polygonal plane-
figures.
2. It gives theoretical zero error in the values if
computations/calculations are done correctly.
3. It necessitates exact location of foot
perpendicular (F.O.P.) drawn from given point
to the plane which is not easy to specify.
4. It necessitates drawing since mathematical
calculations are easy to find all the dimensions
of elementary planes.
Illustrative Examples of HCR’s Theory of Polygon
Example 1: Let‟s find out solid angle subtended by a pentagonal plane ABCDE having sides
at a point „P‟ lying at a
normal height 6cm from a point „O‟ internally dividing the side AB such that & calculate the
total luminous flux intercepted by the plane ABCDE if a
uniform point-source of 1400 lm is located at the point „P‟
Sol: Draw the pentagon ABCDE with known values of the
sides & angles & specify the location of given point P by
( ) perpendicularly outwards to the plane of paper &
F.O.P. „O‟ (as shown in the figure 7)
Divide the pentagon ABCDE into elementary triangles
by joining all the vertices of
pentagon ABCDE to the F.O.P. „O‟. Further divide each of the
triangles in two right triangles
simply by drawing a perpendicular to the opposite side in the
respective triangle. (See the diagram)
It is clear from the diagram, the solid angle subtended by
pentagonal plane ABCDE at the point „P‟ is given by Element
Method as follows
( )
Fig 7: Point P is lying perpendicularly outwards to
the plane of paper. All the dimensions are in cm.
31. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
From the diagram, it‟s obvious that the solid angle subtended by the pentagon ABCDE is expressed as
the algebraic sum of solid angles of sub-elementary right triangles only as follows
Now, setting the values in eq(I), we get
( ) ( ) ( ) ( ) ( )
Now, measure the necessary dimensions & set them into standard formula-1 to find out above values of solid
angle subtended by the sub-elementary right triangles at the given point „P‟ as follows
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
32. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
Hence, by setting the corresponding values in eq(II), solid angle subtended by the pentagonal plane at the given
point „P‟ is calculated as follows
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
Calculation of Luminous Flux: If a uniform point-source of 1400 lm is located at the given point „P‟ then the
total luminous flux intercepted by the pentagonal plane ABCDE
( )
It means that only 86.63434241 lm out of 1400 lm flux is striking the pentagonal plane ABCDE & rest of
the flux is escaping to the surrounding space. This result can be experimentally verified. (H.C. Rajpoot)
Example 2: Let‟s find out solid angle subtended by a quadrilateral plane ABCD having sides
at a point „P‟ lying at a normal height 4cm from a point „O‟
outside the quadrilateral ABCD such that & calculate the total luminous flux
intercepted by the plane ABCD if a uniform point-source of 1400 lm is located at the point „P‟
33. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Sol: Draw the quadrilateral ABCD with known
values of the sides & angle & specify the location of
given point P by ( ) perpendicularly outwards
to the plane of paper & F.O.P. „O‟ (See the figure 8)
Divide quadrilateral ABCD into elementary triangles
by joining all the vertices of
quadrilateral ABCD to the F.O.P. „O‟. Further divide
each of the triangles in two
right triangles simply by drawing perpendiculars OE,
OG & OF to the opposite sides AB, AD & CD in the
respective triangles. (See the diagram)
It is clear from the diagram, the solid angle subtended
by quadrilateral plane ABCD at the given point „P‟ is
given by Element Method as follows
( ) ( ) ( )
Now, replacing areas by corresponding values of solid angle, we get
( ) ( ) ( ) ( )
Now, draw a perpendicular OH from F.O.P. to the side BC to divide into right triangles &
express the above values of solid angle as the algebraic sum of solid angles subtended by the right triangles only
as follows
Now, setting the above values in eq(I), we get
( ) ( ) (
)
( )
Now, measure the necessary dimensions & set them into standard formula-1 to find out above values of solid
angle subtended by the sub-elementary right triangles at the given point „P‟ as follows
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
⇒ ,
√( ) ( )
- ,(
√( ) ( )
) (
√( ) ( )
)-
Fig 8: Point P is lying perpendicularly outwards to the
plane of paper. All the dimensions are in cm.
35. Mr H.C. Rajpoot (B Tech, ME) @M.M.M. University of Technology, Gorakhpur-273010 (UP) India
Hence, by setting the corresponding values in eq(II), solid angle subtended by the pentagonal plane at the given
point „P‟ is calculated as follows
Calculation of Luminous Flux: If a uniform point-source of 1400 lm is located at the given point „P‟ then the
total luminous flux intercepted by the quadrilateral plane ABCD
( )
It means that only 34.56302412 lm out of 1400 lm flux is striking the quadrilateral plane ABCD & rest of
the flux is escaping to the surrounding space. This result can be experimentally verified. ( H.C. Rajpoot)
Thus, all the mathematical results obtained above can be verified by the experimental results. Although, there
had not been any unifying principle to be applied on any polygonal plane for any configuration & location of the
point in the space. The symbols & names used above are arbitrary given by the author Mr H.C. Rajpoot.
Note: This hand book has been written by the author in good faith that it will help to the learners to
memorise the formulae from the book “Advanced Geometry”. The derivations & detailed explanation
have been given in the book.
Author: H.C. Rajpoot
Email: rajpootharishchandra@gmail.com
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
Profile URLs of the author:
https://notionpress.com/author/HarishChandraRajpoot
http://www.hcrajpoot1991.blogspot.com