Special triad of tetrahedrons and s-transformation in a three-dimensional projective space

Panda, Golak Bihari; Misra, Saroj Kanta

doi:10.47974/JIM-2320

Open Access Research Article

Special triad of tetrahedrons and s-transformation in a three-dimensional projective space

^*Golak Bihari PandaCorresponding authorgolakmath@gmail.comDepartment of MathematicsSiksha ‘O’ Anusandhan (Deemed to be University)Institute of Technical Education and ResearchBhubaneswar, Odisha, 751030, IndiaView full profile → , Saroj Kanta Misradrsaroj@silicon.ac.inDepartment of Basic Sciences & HumanitiesSilicon Institute of TechnologyBhubaneswar, Odisha, 751024, IndiaView full profile →

* Corresponding author · click or hover a name for details

pp. 1849–1859Vol. 29Issue 6June 2026DOI: 10.47974/JIM-2320XML

Received:: 01 Mar 2025
Published Online:: 30 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2320
Pages:: 1849–1859

Abstract

Given a pair of perspective tetrahedron (A) and (B) where the vertex Bi is on the opposite face of Ai the center and plane of perspectivity are called the pole and polar of each other respectively for the reference tetrahedron (A). Konully studied a pair of mutually self polar tetrahedrons and Sanyal [9] studied the polar tetrahedrons (C) of (B) for some reference (A). Misra and Sanyal [7] studied a conjugate pair where each of the pair is the polar tetrahedron of the other for (A). In the present study a triad of double perspective tetrahedrons is obtained which under special condition form a special triad where every pair is a conjugate pair for the third. Using a transformation called s-transformation a family of special triad is obtained, which is fixed for a fixed reference (A) and a point P not on the face of (A). The different centers of perspectivities of the double perspective pairs of this family form desmic systems.

Keywords

Homogeneous coordinates Perspective tetrahedrons Pole and polar Cross-ratio Conjugate tetrahedrons

Subject Classifications

51N15

References

[1] N. A. Court, Modern Pure Solid Geometry. New York: Chelsea Publishing Company (1964).
[2] H. S. M. Coxeter, Real Projective Plane. Cambridge: Cambridge University Press (1961).
[3] L. Gerber, “Associated and perspective simplexes,” Trans. Amer. Math. Soc., vol. 201, pp. 43–55 (1975).
[4] A. O. Konnully, “Simplexes self polar for a simplex,” J. Aust. Math. Soc., vol. 12, pp. 309–314 (1971).
[5] S. R. Mandan, “Polarity for a simplex,” Czechoslovak Mathematical Journal, vol. 16, no. 2, pp. 307–313 (1966).
[6] E. A. Maxwell, General Homogeneous Coordinates in Space of Three Dimensions. Cambridge: Cambridge University Press (1951).
[7] S. K. Misra and A. Sanyal, “Quaternions and conjugate system of tetrahedra,” Indian J. Pure Appl. Math., vol. 24, pp. 97–105 (1993).
[8] A. Sanyal and S. K. Misra, “Veronesian pair and associated desmic system,” J. Math. Phys. Sci., vol. 23, pp. 161–169 (1989).
[9] A. Sanyal, “Associated set of (n+1) lines in n-dim. projective space,” Progress of Mathematics, vol. 11, pp. 111–114 (1977).
[10] G. B. Panda and S. K. Misra, “Use of block Toeplitz matrix in the study of Möbius pairs of simplexes in higher-dimensional projective space,” Arnold Mathematical Journal, vol. 6, no. 2, pp. 189–197 (2020).
[11] N. Y. K. Yahya, “Applications geometry of space in PG(3, P),” Journal of Interdisciplinary Mathematics, vol. 25, no. 2, pp. 285–297 (2022).
[12] F. Shaikh, M. S. Pervez, S. S. Banait, S. Goswami, A. Kumar, and U. C. Patkar, “Algorithmic approaches to real and complex geometry in computer vision applications,” Journal of Interdisciplinary Mathematics, vol. 28, no. 6, pp. 2349–2357 (2025).
[13] D. Tondini, “(11, 3)-arc configurations in PG (2, 5),” Journal of Interdisciplinary Mathematics, vol. 26, no. 8, pp. 1717–1731 (2023).
[14] D. Tondini, “An excursion in finite geometry,” Journal of Interdisciplinary Mathematics, vol. 22, no. 8, pp. 1589–1595 (2019).

Views: 173Downloads: 136Citations: 0