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Supplement A. Nine Plane Geometries Supplement A. Nine plane geometries This book is essentially devoted to a comparative study of two geometric systems that can be introduced in the (ordinary or affine) plane, namely, the familiar Euclidean geometry and the simpler Galilean geometry which, in spite of its relative simplicity, confronts the uninitiated reader with many surprising results. In Section 12, we learned of a third geometric system, namely, Minkowskian geometry. Finally, most readers of this book are probably aware of the existence of, or are familiar with, a fourth geometric system, namely Bolyai-Lobachevskian (hyperbolic) geometry. In this Supplement, we propose to describe a number of plane geometries including Euclidean, Galilean, Minkowskian, and Bolyai­ Lobachevskian. Since these geometries were first introduced in 1871 by Klein, I who used the earlier work of Cayley,2 it seems best to call them Cayley-Klein geometries (rather than non-Euclidean geometries; the latter term is obviously not applicable to Euclidean geometry and is applicable to a great many geometries not dealt with in this Supplement). Following Cayley and Klein, we distinguish three fundamentally differ­ ent geometries on a line o. They are Euclidean geometry, elliptic geometry, and hyperbolic geometry. Euclidean geometry is based on the familiar rule for measuring the length of segments. According to this rule we choose a unit of length DE on 0 and define the distance between points A and Bon o by the formula3 (I) This distance is called parabolic (hence the letter P in dj~». If C is a point ·See Klein [73] (first published in 1871) or Klein [56]. 2For an historical account of the discovery of these geometries, see Sec. 7, Chap. X of Klein [56] or the very interesting history of mathematics by Klein [79]. 3The ratio of two segments on 0 can be defined in affine geometry without developing Euclidean geometry. Supplement A Nine plane geometries 215 of the segment AB, then dj2 + dZ/ = dj~) (Fig. l84a). The Euclidean motions of 0, i.e., the transformations of 0 that preserve the distance dj~ between points, are (a) translations (along 0); (b) reflections (of 0 about its points). To introduce elliptic geometry on 0, we pick a point Q not on 0 and define the elliptic distance dj~) between A and B on 0 as the usual Euclidean measure of LAQB: dji)= LAQB (Fig. l84b). If A, B, and C are three consecutive points on 0, then djf] + d~i) = dji) (Fig. l84b). The elliptic motions of 0 which preserve the elliptic distance dj~) are: (a) transformations induced by rotating the pencil with center Q (i.e., the lines through Q) through a fixed angle a; such a transformation takes A to the point A' such that LAQA'=a; (b) transformations induced by reflecting the pencil with center Q about a line I of the pencil; such a transformation takes A to the point A' such that! bisects LAQA'. Hyperbolic geometry is introduced on 0 by fixing two of its points, I and J, and defining the hyperbolic distance dj~) between A and Bon 0 by the formula (H)_ AI/AJ dAB - klog BI/ BJ (lb) (Fig. l84c). The choice of k is arbitrary, and changing the value of k (with the base b of the logarithm fixed, say b = 10) amounts to a change of the unit used in measuring distances (lengths of segments). It is convenient to assume that the segments on the right-hand side of the cross ratio AI/AJ (A,B; I,J) = BI/ BJ (2) of A,B,I,J are directed. In view of the fact that a real-valued logarithm is defined only for positive reals, we must ensure that the cross ratio (2) is a A. -11-11 -0:>--B 0 B A /J -_---0-__ t Q 11 0- I A. I G o £ C A C B :r Figure 184a Figure 184b Figure 184c 216 Supplement A. Nine plane geometries Figure 185a Figure 185b positive. One way of doing this is to assume that A and B are points of the segment IJ. Then the segment IJ represents the whole hyperbolic line. It is easy to show that if A,C,B are three consecutive points on IJ (cf. Fig. 184c), then d(H) d(H)- d(H). AC + C B- AB, indeed, AI/AJ CI/CJ (A,C; I,J)'(C,B; I,J)= CI/CJ' BI/BJ AI/AJ = BI/BJ =(A,B; I,J), and so 10g(A,C; I,J)+log(C,B; I,J)=log(A,B; I,J). Another fact is that the cross ratio has the same value for two sets of quadruples on 0 related by a central projection.4 It follows that the hyperbolic motions of 0 (more precisely, of the hyperbolic line, i.e., the segment IJ) which preserve the hyperbolic distance dl1:) are generated by (a) reflections about the midpoint S of IJ [clearly, if such a reflection takes A,B to A',B', then (A,B; I,J)=(A',B'; J,l)]; (b) central projections of IJ to itself, where IJ is first projected to some segment IJI from a point 0 1, and IJI is in tum projected to IJ from a point O2 on the line JIJ (Fig. 185a). Figure 185b shows a sequence of equal (in the sense of hyperbolic geometry) segments AB,BBI'BIB2'B2B3' .... This figure shows that if, beginning with some point A of the (hyperbolic) line IJ, we layoff a sequence of equal (in the sense of hyperbolic geometry) segments AB=BBI =BIB2=B2B3=···, 4See. for example. [121. [191. [321. or [671. Supplement A. Nine plane geometries 217 we remain within the segment IJ, i.e., the hyperbolic line IJ is infinite. [This can also be seen from Fig. 184c. Indeed, if B tends to I, then (A,B; I,J) = (AI/ AJ)/(BI/ BJ)~oo (since BI~). Consequently, dj~)~oo.] Just as there are three ways of measuring length that give rise to three geometries on a line 0, so, too, are there three ways of measuring angles that give rise to three geometries in a pencil of lines with center O. The usual measure of angles in the pencil with center 0 given by the formula 8~)= LaOb (3a) (Fig. 186a; the symbol L is used here as well as below in the usual Euclidean sense) is called elliptic. The so-ca11edparabolic measure of angles is very different from that used in Euclidean geometry; it can be defined by choosing a line q not passing through 0 and putting 8J[ ) = AB (= djIJ , the signed Euclidean length of AB ), (3) where A = a n q and B = b n q (Fig. 186b). It is clear that our pencil contains a unique line "'ilq which forms an "infinitely large angle" with every other line of the pencil. Finally, the hyperbolic measure of angles is introduced in our pencil as follows: we fix two lines i andj of the pencil, and for any two other lines a and b we put 8(H)=kl (sin L(a,i)/sin L(aJ») (3b) ab og sin L(b,i)/sin L(bJ) (Fig. 186c). The constant k depends on the choice of unit. It is convenient to regard the angles on the right-hand side of (3b) as directed. Define the cross ratio (a,b; i,)) of four lines a,b,iJ as the quantity ( b.· .)- sin L(a,i)/sin L(aJ) (2') a, ,l,j - sin L(b,i)/sin L(bJ) . For 133/) to make sense we must ensure that (a,b; i,))>O. One way of doing this is to assume that the lines a, b,. .. belong to a definite pair of vertical angles formed by i andj. Since the cross ratio (2') of four lines of the pencil is equal to the cross ratio of the four points in which these lines meet an arbitrary fixed line q (not passing through the center of the Figure 186a Figure 186b Figure 186c 218 Supplement A. Nine plane geometries Table I Nine Cayley-Klein geometries in the plane Measure of length Measure of angles Elliptic Parabolic Hyperbolic elliptic elliptic Euclidean hyperbolic geometry geometry geometry parabolic (Euclidean) co-Euclidean Galilean co-Minkowskian geometry geometry geometry hyperbolic cohyperbolic Minkowskian doubly geometry geometry hyperbolic geometry pencil),S we could also define Bgj) as f>(H)=d(H) Uab AB , (3c) where A and B are the points in which the lines a and b meet an arbitrary fixed line q (see Fig. 186c). The equality (3c) implies that (in the hyperbolic measure of angles) each of the lines iJ forms an "infinitely large angle" with every other line. The Cayley-Klein scheme yields nine plane geometries. Specifically, we can choose one of three ways (parabolic, elliptic, or hyperbolic) of measur­ ing length on a line and one of three ways of measuring angles in a pencil with center o. This gives nine ways of measuring lengths and angles and thus the nine plane geometries listed in Table I . We are familiar with the geometries of Euclid, Galileo, and Minkowski (middle column of Table I) in all of which the metric on the line is parabolic; the length of a segment AB is defined as a ratio AB/ OE, where OE is a "unit segment" on the line. In Euclidean geometry, the endpoints of all unit segments issuing from a point 0 lie on a (Euclidean) circle S (Fig. 187a). Angles in a pencil are measured in the elliptic, i.e., in the usual, way, which amounts to measur­ ing the length of the arc of S cut off on S by the appropriate lines: in Figure 187a, Bab= LaOb=arcAB.
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