arXiv subjects

Masanori Nakazato

Publications and source records attributed to Masanori Nakazato.

3 recordsLinked to original sources

arXiv2026

Affine Angles via Area Cross Ratio and Isoptic Hyperbolas

Affine geometry is usually regarded as a framework in which metric notions such as distance and angle are absent. However, just as projective geometry produces various metric geometries by introducing additional structures on the line at infinity, affine geometry can also serve as a natural basis for an angular geometry once certain directions at infinity are fixed. In this paper we introduce an affine angle determined by two fixed directions on the line at infinity and defined via an area cross ratio. This quantity is invariant under affine transformations preserving the chosen directions. We show that the locus of points from which a fixed segment is seen under a constant affine angle is a hyperbola whose asymptotes are parallel to the chosen directions. This provides an affine analogue of the classical fact that in Euclidean geometry the isoptic curve of a segment is a circle. Furthermore, we establish that this angle arises as a parabolic degeneration of the Cayley--Klein angle, and that the same quantity naturally appears in a power theorem associated with hyperbolas. These results provide a unified perspective linking affine angles, isoptic hyperbolas, and hyperbolic power through the area cross ratio.

math.MG↗

arXiv2026

The Fourth Geometry II: From Angle Axioms to Metric Foundations

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.

math.GM↗

arXiv2025

The Fourth Geometry I: Difference--Angle Geometry Beyond Euclid, Hyperbolic, and Elliptic

In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the fundamental question ``What is an angle?'', and shows that an angle can be defined independently of circles or rotations, as a primary geometric notion. Within this geometry, one can define difference-angle triangles, norms, bisectors, perpendiculars, and inner products. Several characteristic properties not seen in existing geometries emerge, together with behaviors analogous to those in Euclidean geometry: the triangle inequality always holds with equality, the sum of the interior angles of any triangle is $0$, and a Miquel point exists even for parabolas. In particular, the concurrency of the parabolic Miquel configuration was suggested by Weiss and Odehnal, and our main theorem provides the first explicit and rigorous confirmation of this assertion. We also point out that many classical Euclidean configurations (including Brocard-type configurations) naturally reappear in the setting of difference-angle geometry. These results indicate that difference-angle geometry is a promising candidate for a ``Fourth Geometry'' following the Euclidean, hyperbolic, and elliptic geometries.

math.MG↗