He made independent research on 'Solid Angle' & applied it in Radiometry/Photometry in Fundamental Physics. He applied it for the point-sources of radiation emitting energy uniformly in all the directions. He modified Lambert's Cosine Formula to apply for higher accuracy & to analytically compute Illuminance. He proposed 'Approximation Theorem of Solid Angle' for symmetrical 2-D figures. His Approximation-formula give the results very closer to the values calculated by analytic method.
He proposed 'Theory of Polygon' which is applicable on all the polygons to analytically & graphically compute the solid angle subtended at any arbitrary point in the space. He consequently, applying his theory of polygon derived a generalized formula for all five regular polyhedra & analysed all 13 Archimedean solids to compute all the important parameters or dimensions i.e. inner radius, outer radius, mean radius, surface area & volume.
He derived & proved HCR's Theorem to analytically compute the V-cut angle for folding two co-planar, meeting at angle bisector, about their intersecting straight edges so as to coincide their new edges. This theorem is very useful for making Pyramidal Flat Containers with Regular Polygonal Base, Right Pyramids & Polyhedrons with polygonal & trapezoidal faces using sheet of paper, polymer, metal or alloy which can be easily bent and butt-joined at the mating edges. He, using his theorem, also derived HCR's Corollary to analytically compute di-hedral angle between two folded planes.
He derived a unique formula on permutations of alphabetic words, positive integral numbers & all other linear permutations to calculate the position (hierarchical rank) of any linear permutation in the ordered sequence. It was certified by International Journal of Mathematics & Physical Sciences Research in March, 2014. Consequently, he, using HCR's Rank Formula-1 proved that the factorial of any number can be expanded as a sum of finite terms. This series was named as HCR's Series certified by International Organization of Scientific Research. He applied his formula-1 for computing rank of circular permutations He further gave HCR's Rank Formula-2 for computing rank of any linear permutations when repetition of articles is permitted.
1. formula for regular polyhedra (all five platonic solids)
2. formula for infinite class of uniform polyhedra with regular n-gonal & trapezoidal faces
3. formula for infinite class of trapezohedrons (uniform polyhedrons with congruent right kite faces)
4. a formula for all regular spherical polygons & derived formula to compute radii of circles internally as well as externally touching three external tangent circles.
5. a formula to compute the distance between any two points on the globe .
Mr H. C. Rajpoot wrote his first book Advanced Geometry based on his research articles in Applied Mathematics & Radiometry for higher education which was first published by Notion Press, Chennai, India in April, 2014.
This book is based on research in higher education to put the advanced articles in the practice of education.
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