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Mandan, Sahib Ram
- Uni- and Demi-Orthocentric Simplexes, II
Authors
1 Indian Institute of Technology, Kharagpur, IN
Source
The Journal of the Indian Mathematical Society, Vol 26, No 1-2 (1962), Pagination: 5-11Abstract
a. Sr(n_r) is defined in [5] to denote a Uni-orthocentric Simplex, or briefly UoS, in an n-space such that its r altitudes from its r vertices concur at its r-orthocentre Hr and n - r others at Hn-r.- Orthic Axes of Triangles of a Simplex
Authors
1 Indian Institute of Technology, Kharagpur, IN
Source
The Journal of the Indian Mathematical Society, Vol 26, No 1-2 (1962), Pagination: 13-24Abstract
The orthic axis of a triangle T is the axis of perspectivity of T and its orthic triangle formed of the feet of its altitudes [5]. The purpose of this paper is to study the distribution and location of the orthic axes of the triangles of a simplex in a space of n dimensions, or briefly in an n-space, or of an n-simplex, according as it is Semi-Uni-or Demi-orthocentric ([6], [11], [15], [20]-[23]).- On Four Intersecting Spheres
Authors
1 Indian Inst, of Tech., Kharagpur, IN
Source
The Journal of the Indian Mathematical Society, Vol 23, No 3-4 (1959), Pagination: 151-167Abstract
The interesting results 'On Three Intersecting Circles' obtained by Prof. N.A. Court[4] led me to the present paper. The following results of some interest have been arrived at:
The radical tetrahedron of four intersecting spheres coincides with a diagonal tetrahedron of the desmic system of intersection of those spheres. The pairs of opposite vertices of the system referred to this tetrahedron form pairs of conjugate points for the orthogonal sphere of the given four spheres and are the centres of similitude of the tetrad of associated spheres.
The planes of perspectivity of the eight pairs of complementary tetrahedra of intersection of four intersecting spheres form two tetrahedra desmic with their radical tetrahedron, and are the radical planes of the corresponding pairs of complementary spheres of intersection, and are the planes of similitude of the associated tetrad of spheres.
The diagonal tetrahedra of a desmic system of intersection of four intersecting spheres form the other desmic system of intersection of those spheres.
The eight centres of similitude (other titan the orthogonal centre of four intersecting spheres) of the eight pairs of complementary spheres of intersection of the four given spheres form two tetrahedra desmic with their central tetrahedron and thus form a set of eight associated points.
The desmic system of intersection of four intersecting spheres is inscribed in the one conjugate to that of centres for those spheres and reciprocally their other desmic system of intersection is circumscribed to that of centres for them.
The perpendicular from the orthogonal centre of four intersecting spheres upon the Newtonian plane of their associated tetrad of spheres passes through the circumcentre of their radical tetrahedron.
Each sphere of anti-similitude of the associated tetrad of spheres is orthogonal to eight of the spheres of intersection and to two of the four given intersecting spheres.
- Uni- and Demi-Orthocentric Simplexes
Authors
1 Indian Institute of Technology, Kharagpur, IN
Source
The Journal of the Indian Mathematical Society, Vol 23, No 3-4 (1959), Pagination: 170-184Abstract
The study of the altitudes of a simplex in an n-space grows more and more complex as n increases, nevertheless it forms an interesting theme and thus makes it worthwhile to have some glimpses of these new unseen highlands.- Semi-Inverse Simplexes
Authors
1 Indian Institute of Technology, Kharagpur, IN