The Structure of n Harmonic Points and Generalization of Desargues’ Theorems

In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective compl...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-05, Vol.9 (9), p.1018
Main Authors: Thaqi, Xhevdet, Aljimi, Ekrem
Format: Article
Language:English
Subjects:
complete plane n-point
Correspondence
Food science
generalization of Desargues’ theorems
Geometry
harmonic points
perspectivity
projective transformations
set of H-points rank k
Theorems
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Summary:In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective complete n-points, which are used to implement the definition of the generalization of harmonic points. We present new findings regarding the uniquely determined and constructed sets of H-points and their structure. The well-known fourth harmonic points represent the special case (n = 4) of the sets of H-points of rank 2, which is indicated by P42.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9091018