Abstract
Cayley and Klein discovered in the 19th century that Euclidean and non-Euclidean geometries can be introduced as geometries living inside of a projective-metric space. We present a new definition of projective-metric coordinate planes which allows a uniform analytic representation of Cayley–Klein geometries over ordered fields.
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Notes
We consider in this paper Cayley–Klein geometries of dimension 2.
In Klein’s famous model of hyperbolic geometry the points of the hyperbolic plane \({\mathcal {H}}\) are the projective points which are interior of a non-degenerate conic \(\kappa \), and the lines of \({\mathcal {H}}\) are the secants of \(\kappa \). The hyperbolic distance between points A and B is (up to a multiple constant) the natural logarithm of the cross ratio of A, B and the two points of \(\kappa \) which are incident with the joining line of A and B. The measurement of angles is obtained by a dualization of the measurement of distances (this requires, obviously, the extension of the field of real numbers to the field of complex numbers). In the Euclidean case the absolute conic is a pair of imaginary points, known as the ‘circular points at infinity’, and the measures of distances and angles are obtained from the hyperbolic case by a ‘passage to the limit’ (see Behnke et al. [7, p. 494]).
Equivalently, an isomorphism is a map** \(\varphi : {\mathcal {P}} \cup {\mathcal {L}} \rightarrow {\mathcal {P}} \cup {\mathcal {L}}\) with \({\mathcal {P}} \varphi = {\mathcal {P}}\) and \({\mathcal {L}} \varphi = {\mathcal {L}}\) which preserves the incidence relation (cp. Bachmann [4, p. 85]).
We note that if a projectivity \(\sigma \) is induced by a linear map** \(\varphi \) which is represented by a matrix M (in point coordinates) then the transpose \((M^{-1})^{t}\) of \(M^{-1}\) represents \(\sigma \) in line coordinates [u, v, w].
In the literature such substructures are sometimes called Eigentlichkeitsbereiche (see Bachmann [4]).
We note that the motions of \({\mathfrak {C}}(K, p, q)\) are the motions of \({\mathfrak {P}}(K, p, q)\), restricted to the set of points and lines of \({\mathfrak {C}}(K, p, q)\). The groups of motions of \({\mathfrak {C}}(K, p, q)\) and of \({\mathfrak {P}}(K, p, q)\) are isomorphic (basically since they are generated by the same set of reflections).
Please note that if (A, a) is any pole-polar pair with non-isotropic elements A and a then either A or a is an element of the hyperbolic plane.
by the map** which associates to a point (x, y, z) of \({\mathfrak {P}}(K, -1, 1)\) the point (z, y, x) of \({\mathfrak {P}}(K, 1, -1)\) and to a line [u, v, w] of \({\mathfrak {P}}(K, -1, 1)\) the line [w, v, u] of \({\mathfrak {P}}(K, 1, -1)\)
by the map** which associates to a point (x, y, z) of \({\mathfrak {P}}(K, -1, -1)\) the point (x, z, y) of \({\mathfrak {P}}(K, 1, -1)\) and to a line [u, v, w] of \({\mathfrak {P}}(K, -1, -1)\) the line [u, w, v] of \({\mathfrak {P}}(K, 1, -1)\)
Please note that the order structure of a projective plane is a cyclic one. If the points and lines of the co-Minkowskian plane are defined equivalently by the conditions \(q_{_{{{\mathcal {P}}}}}(x, y, z) < 0\) and \(q_{_{{{\mathcal {L}}}}}(x, y, z) < 0\) then the strip consists of all points (x, y, z) of \({\mathfrak {P}}(K, p, q)\) with \(-1< z < 1\).
The table should be read as follows “The set of points of an elliptic plane is the set of points of a projective plane”, and so on.
According to Sylvester’s law of inertia the type of a Cayley–Klein coordinate plane is a projectively invariant notion.
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Struve, H., Struve, R. Cayley–Klein geometries and projective-metric geometry. J. Geom. 113, 32 (2022). https://doi.org/10.1007/s00022-022-00646-2
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DOI: https://doi.org/10.1007/s00022-022-00646-2
Keywords
- Cayley–Klein geometries
- Cayley–Klein coordinate plane
- projective-metric geometry
- projective-metric coordinate plane