Invariant and Preserving Transforms for Cross Ratio of 4-Points in a line on Desargues Affine Plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909572908711936 |
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| author | Zaka, Orgest Peters, James F. |
| author_facet | Zaka, Orgest Peters, James F. |
| contents | This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic's and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, \emph{Invariant} and \emph{Preserving} transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural Dilation, Mobiüs Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is \emph{Preserved} under transforms: parallel projection, translations and dilation's in the Desargues affine plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16055 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invariant and Preserving Transforms for Cross Ratio of 4-Points in a line on Desargues Affine Plane Zaka, Orgest Peters, James F. General Mathematics 51-XX, 51Axx, 51A30, 51E15, 51N25, 30C20, 30F40 This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic's and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, \emph{Invariant} and \emph{Preserving} transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural Dilation, Mobiüs Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is \emph{Preserved} under transforms: parallel projection, translations and dilation's in the Desargues affine plane. |
| title | Invariant and Preserving Transforms for Cross Ratio of 4-Points in a line on Desargues Affine Plane |
| topic | General Mathematics 51-XX, 51Axx, 51A30, 51E15, 51N25, 30C20, 30F40 |
| url | https://arxiv.org/abs/2211.16055 |