Perspective Graves Cycle of Triangles in Finite Projective Planes: A Matrix Representation Study
Keywords:
Projective Plane, Homogeneous coordinates, Perspective Triangles, Graves Triad of Triangles 2000 MSC: 51E20, 51E15, 51N15Abstract
TThe study of the Graves Triad, a cyclic triad of triangles each circumscribing the next, was done in the Euclidean plane and projective plane. This current study identifies cyclic sequences of triangles called Graves cycles in finite projective planes, where each triangle in the sequence circumscribes the next one, and any two triangles from the sequence form a four-fold perspective pair with one center of perspectivity common to all. The length of such cycles depends on the order of the field. The study uses the matrix representation of a triangle by homogeneous coordinates with respect to a reference frame.
References
E. Maxwell, General Homogeneous coordinates in space of three dimen- sions, Cambridge University Press, Cambridge, 1961.
J. Todd, Projective and Analytical Geometry, Pitman, London, 1960.
H. Havlicek, S. M. Odehnal, and B., Mobius Pairs of Simplices and Commuting Pauli Operators, Math. Panonm. 21 (2009) 115–128.
doi:10.48550/arXiv.0905.4648.
