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Perspectivity of two complete n-points from a line.

Perspectivity of two complete n-points from a line.

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Figure 1. The meaning of Veblen-Young Axiom.
Figure 2. Four Harmonic points.
Figure 3. Perspectivity of two complete n-points from a line.
Figure 4. Perspectivity of two n-points from a point. The collinear...
+1 Figure 5. The line is incident with two diagonal points of complete...
The Structure of n Harmonic Points and Generalization of Desargues' Theorems
Article
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  • Apr 2021
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 perspec-tivity, we note that it is necessary to conduct a generalization of the Desargues' theorems for pro-jective complete n-points, which are used to implement the de...
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... there be two given coplanar (or noncoplanar) complete plane í µí±› -point, í µí°´íµí°´í µí°´…µí°´… í µí°ánd í µí°µ í µí°µ … í µí°µ . (Figure 3). i) For í µí±› = 3, the GCD theorem is equivalent to the Theorem 2 (the converse of Desargues' theorem). ...

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... Source: compiled by the author based on (13) . ...
... This perspective shows the inclusion of the Euclidean plane in a wider projective plane. Source: compiled by the author based on (13) . ...
... furthermore (13,14): ...
Universal tools for analysing structures and interactions in geometry
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This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions.
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