The Fourth Geometry II: From Angle Axioms to Metric Foundations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910182764707840 |
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| author | Nakazato, Masanori |
| author_facet | Nakazato, Masanori |
| contents | This paper is a sequel to arXiv:2511.01024 (Base 1), where an axiomatic framework for angles and the foundations of difference-angle geometry were introduced.
In difference-angle geometry, where the difference of slopes of lines is treated as a primary angular quantity (the difference angle), we reconstruct the focal structure of parabolas from a difference-angle-theoretic viewpoint and develop the associated algebraic and analytic structures. First, we introduce the difference-angle focal function and define the focus of a parabola constructively as its zero set. This approach yields a formulation of the parabolic power that differs from that presented in Base 1.
Next, by interpreting the power as a classical representation of an inner product, we derive a difference-angle version of the parallelogram theorem via a polarization identity, and thereby define the difference-angle inner product as a pseudo-inner product. The robustness of this structure is substantiated by deriving a difference-angle version of Stewart's theorem based solely on computations involving the difference-angle inner product.
Furthermore, we define the parabolic trigonometric functions cosp(theta) and sinp(theta) (together with related functions) associated with a difference angle (theta), and show that they satisfy identities corresponding to the first and second cosine laws in Euclidean geometry. Finally, we reexamine the Cayley-Klein angle and distance derived from Laguerre's formula, and in particular verify that the existing Cayley-Klein angle satisfies the axiomatic system for angles introduced in Base 1. We then show that, in the parabolic limit of the absolute conic, the difference angle and the difference-angle norm arise naturally as the linear degeneration of the logarithmic cross ratio. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_00001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Fourth Geometry II: From Angle Axioms to Metric Foundations Nakazato, Masanori General Mathematics This paper is a sequel to arXiv:2511.01024 (Base 1), where an axiomatic framework for angles and the foundations of difference-angle geometry were introduced. In difference-angle geometry, where the difference of slopes of lines is treated as a primary angular quantity (the difference angle), we reconstruct the focal structure of parabolas from a difference-angle-theoretic viewpoint and develop the associated algebraic and analytic structures. First, we introduce the difference-angle focal function and define the focus of a parabola constructively as its zero set. This approach yields a formulation of the parabolic power that differs from that presented in Base 1. Next, by interpreting the power as a classical representation of an inner product, we derive a difference-angle version of the parallelogram theorem via a polarization identity, and thereby define the difference-angle inner product as a pseudo-inner product. The robustness of this structure is substantiated by deriving a difference-angle version of Stewart's theorem based solely on computations involving the difference-angle inner product. Furthermore, we define the parabolic trigonometric functions cosp(theta) and sinp(theta) (together with related functions) associated with a difference angle (theta), and show that they satisfy identities corresponding to the first and second cosine laws in Euclidean geometry. Finally, we reexamine the Cayley-Klein angle and distance derived from Laguerre's formula, and in particular verify that the existing Cayley-Klein angle satisfies the axiomatic system for angles introduced in Base 1. We then show that, in the parabolic limit of the absolute conic, the difference angle and the difference-angle norm arise naturally as the linear degeneration of the logarithmic cross ratio. |
| title | The Fourth Geometry II: From Angle Axioms to Metric Foundations |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2605.00001 |