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Note on the Geometries in which Straight Lines are represented by Circles
Published online by Cambridge University Press: 20 January 2009
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In a recent paper read before the Society, Professor Carslaw gave an account, from the point of view of elementary geometry, of the well-known and beautiful concrete representation of hyperbolic geometry in which the non-Euclidean straight lines are represented by Euclidean circles which cut a fixed circle orthogonally. He also considered the case in which the fixed circle vanishes to a point, and showed that this corresponds to Euclidean geometry. The remaining case, in which the fixed circle is imaginary and which corresponds to elliptic or spherical geometry, is not open to the same elementary geometrical treatment, and Professor Carslaw therefore omitted any reference to it. As this might be misleading, the present note has been written primarily to supply this gap. It has been thought best, however, to give a short connected account of the whole matter from the foundation, from the point of view of analysis, omitting the detailed consequences which properly find a place in Professor Carslaw's paper.
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- Copyright © Edinburgh Mathematical Society 1909
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page 82 note * When the fundamental circle is real we may employ another artifice and consider only the points in the interior (or exterior) of the fundamental circle. When the fundamental circle vanishes to a point, this point may be considered as the representative of all the points at infinity. (See §7.)
page 82 note † Stromquist, C. E., in a paper “On the geometries in which circles are the shortest lines,” New York, Trans. Amer. Math. Soc, 7 (1906), 175–183,Google Scholar has shown that “the necessary and sufficient oondition that a geometry be such that extremals are perpendicular to their transversals is that the geometry be obtained by a conformal transformation of some surface upon the plane.” The language and his methods are those of the calculus of variations. The extremals are the curves along which the integral which represents the distance function is a minimum, i.e. the curves which represent shortest lines; and the transversals are the curves which intercept between them arcs along which the integral under consideration has a oonstant value. Thus in ordinary geometry, where the extremals are straight lines, the transversals to a one-parameter system of extremals are the involutes of the curve which is the envelope of the system. In particular, when the straight lines pass through a fixed point the transversals are concentric circles.
page 83 note * Cf.Liebmann, , Nichteuklidische Geometric (Leipzig, 1905), §§8, 11.Google Scholar
page 83 note * When, as is often taken to be the case, the fundamental circle is the x-axis the conditions are simply that the coefficients α, β, γ, δ be all real numbers.
page 87 note * Cf. Darboux, Théorie des surfaces, vii., chap. XI.
Also Klein, Nichteuklidische Geometrie, Vorlesungen.
page 88 note * Cf.Wellstein, Weber u., Encyklopädie der Elementar-Mathematik (2 Aufl. Leipzig, 1907),Google Scholar Bd. 2, Abschn. 2. Also, Fricke, Klein u., Vorlesungen über die Theorie der automorphen Functionen (Leipzig, 1897), Bd. 1.Google Scholar
page 91 note * It may be noticed that the line-element can be expressed in terms of x′, y′ alone. Thus expressing z′, dz′ in terms of x′, y′ by means of the equation x ′2+y ′2+z ′2=R2, we have
Here 
are the so-called Weierstrass' coordinates. Let the position of a point P on the sphere be fixed by its distances ξ, η from two fixed great circles intersecting at right angles at Ω, and let ΩP=ρ, all the distances being measured on the sphere along arcs of great circles. Then
On the pseudosphere the circular functions become hyperbolic functions. (See Killing, , Die nichteuklidischen Raumformen, Leipzig 1885, p. 17.)Google Scholar
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