122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin

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121 6,.,,,,,,. 2, 1. 6.1,.., M, A(2 R).,. 49.. Oxy ( ' ' ), f Oxy, O 1 x 1 y 1 ( ' ' ). A (p q), A 0 (p q). y q A q q 0 y 1 q A O 1 p x 1 O p p 0 p x 6.1: ( ), 6.1,

122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin ; q cos + q 0 6.1 6.2.,,,. 31, ( x 0 = ax + by + x 0 (6.3) y 0 = cx + dy + y 0, (x y) (x 0 y 0 ). 6.3 a, b, c, d. (1 1) )., c d (1 0) (0 1), 6.3, (a c) (b d), 2 ad ; bc.,,.,. a b, 6.2 = 1 1=2. c d 0 1, ab ; bc 6= 0 (ad ; bc a b 6.2: 32, ad ; bc 6= 0. A(2 R).

6.2. 123. O, GL(2 R). 33 ( x 0 = ax + by (6.4) y 0 = cx + dy. GL(2 R) ad ; bc 6= 0. 132. R 2., A(2 R)=R 2 = GL(2 R)., (p.13 ). C AB k : l, C C 0 k : l A 0 B 0. A(2 R),,.,.,,,., 1 : 3 1, 2:1 ( 10 ).,.. 133., 4 (p.15).,, 1. x 7! ax,, x 7! ax + b. a 6= 0, GL(1 R), A(1 R). R,... GL(1 C ), A(1 C ) (6.3 ). 134. (a) G = GL(2 Z 2 )., G. (b) A(1 Z 3 ). 6.2.,,., A AO (O, ), A 0 ( 6.3).,., ` '.

124 6 6.3:,.,. 34 l, l 0 2, S ( 6.4a)., A 2 l 2 SA, l 0 A 0 p : l! l 0. 2 1, 2 S, p : 1! 2 ( 6.4b ). A A B B S a b 6.4: (a) (b) 35,... (, )..,,., 4., 2, 3., 6.4, B A C, B 0 A A 0 C C 0., 4 4 p.

6.2. 125 36 4 A, B, C, D, (A B C D) = AC BC : AD BD :, 4 AC, BC, AD, BD,,. 13,, (A B C D) =(A 0 B 0 C 0 D 0 ). 2, 6.5, S 4SAC = 1 2 h AC = 1 SA SC sin \ASC 2 S 4SBC = 1 2 h BC = 1 SB SC sin \BSC 2 S 4SAD = 1 2 h AD = 1 SA SD sin \ASD 2 S 4SBD = 1 2 h BD = 1 SB SD sin \BSD: 2 6.5:,, (A B C D) = AC BC : AD BD = S 4SAC S 4SBC : S 4SAD = = S 4SBD SA SC sin \ASC SB SD sin \BSD SB SC sin \BSC SA SD sin \ASD sin \ASC sin \BSC : sin \ASD sin \BSD : (A 0 B 0 C 0 D 0 )= sin \A0 SC 0 : sin \A0 SD 0 sin \B 0 SC 0 sin \B 0 SD 0., (A B C D) =(A 0 B 0 C 0 D 0 ).

126 6 50 A, B, C, D, ( ) 4, A 0, B 0, C 0, D 0 A, B, C, D., A 0 B 0 =6cm, B 0 C 0 =2 cm., C 0 D 0.. C 0 D 0 = x (A B C D) = AC BC : AD BD = 2 1 : 3 2 = 4 3 : (A 0 B 0 C 0 D 0 )= A0 C 0 : A0 D 0 = 8 B 0 C 0 B 0 D 0 2 : x +8 x +2 : (A B C D) =(A B 0 C 0 D 0 ),, x =1. 8(x +2) 2(x +8) = 4 3 :,. p : l! l 0 l l 0, x l M, p M M 0 2 l 0 x 0., l 3 A B C, a, b, c.,, (A B C M) =(A B 0 C 0 M 0 ) c ; a c ; b : x ; a x ; b = c0 ; a 0 c 0 ; b 0 : x0 ; a 0 x 0 ; b 0 :, p(a) =a 0, p(b) =b 0, p(c) =c 0, p(d) =d 0., x x 0. p. x 0 = mx + n px + q : (6.5), m, n, p, q a, b, c, a 0, b 0, c 0.. 6.5, mq ; np 6= 0 (mq ; np, m n ). f : x 7! mx + n px + q p q, 1 1., p 6= 0, x = ;q=p f.

6.2. 127 135. f. 1 1 ( ),, R = R [1 (p.104 ) a 0 = 1 for 8a 6= 0 ( m 1+ n m p 1+ q = p if p 6= 0, 1 if p =0., mq ; np 6= 0 f(x) = mx+n px+q R 1 1. 136., R ff : R! Rjf (x) = mx+n mq; np 6= 0 m n p q 2 Rg, px+q. ( ), PGL(1 R). 137. 6.5, R 4. 6:5, 3, 3., 3,., 2 x 7! 1=x, x 7! 1 ; x ( 74, 43 ),. 138. 2 x 7! 1=x, x 7! (x ; 1)=(x +1), D 4 8. 139..,.,, PGL(2 R)., A(2 R). 140. A(2 R), PGL(2 R).,. 8>< >: x0 = 14, ( ),. a 1 x + b 1 y + c 1 y 0 = a 0 x + b 0 y + c 0 2x + b 2 y + c (6.6) 2 a 0 x + b 0 y + c 0

128 6 15 PGL(2 R), 3 4.,.,,,.,.,,.,.. D E F B C A 6.6: Pappus 141. Pappus ( 6.6 ): 3 A, B, C 3 D, E, F, 3 AE \ BD, AF \ CD, BF \ CE. 6.3 37, 1.,,. 37,.,.. ;;! 38 A, k 6= 0 HA k, M AM 0 = k ;;! AM M 0 ( 6.7 ).,. z, z 7! kz., k 0., 0 R?.

6.3. 129 A 6.7: 142. GL(2 R)=R? = PGL(1 R). =.,. 51 ABC.. 3 ABC ( 6.8 KLMN ). 6.8: 6.8, A, KLMN 3 ABC. HA k N, BC E k. 6.8. 143. ABC 3 ABC. 1,., l l..

130 6 6.9: 52 S 2 M, N. MN 1 g MN S 1, S 2, ( 6:9). A i = S i \ g MN, Bi = S i \ MN, A 1 B 1, A 2 B 2,, 1.. A i B i C., C 6.9, MN, S l. A 1, k = OA 1 : O 1 A 1 h 1. h 1 S 1 ( O 1 ) S., S 1 MN MN, S l. B 1 = S 1 \ MN, h 1 (B 1 )=C., S i.. 144. 145. 146. 2, 2 4 A, B, C, D, AB =2BC = CD.,, 2 3, 1. 9 ( ) Euler ABC, 9 ABC (3 ), (3 ), KA, KB, KC (3 ),, K ABC.,. 39, ( 6.10 ).,,. a, jaj, arg a (2.4)..

6.3. 131 A 6.10: 6.11: ABC 53 ABC, P, Q \AP B = \BQC = 90, \ABP = \CBQ = 4ABC 2 ( 6.11)., 4PQK., K AC.. 2 F P = H k P R d P F Q = H 1=k Q R d Q., d =90 k = PB : PA = QB : QC. F P (A) =B F Q (B) =C, (F Q F P )(A) = C., 2,, ( )., F = F Q F P 180. F (A) =C, K. F (K) = K. F P (K) = K 1, F Q (K 1 ) = K 2 KPK 1, QKK 1 K 1., 2 ( 6.12)., PQ? KK 1, \KPQ = \KQP =., 2, 2,., 2..,.

132 6 6.12: 16, z, z 7! pz + a (6.7) z 7! pz + a (6.8). p, a, p 6= 0. 2 6.7, 6.8 (i.e. ), (i.e. )., w = pz + a (jpj =1) 2.3., F, F k (, ),, H k., H ;1 F.,, w = pz + a (jpj =1)., F = H (H ;1 F ) w = k(pz + a).., pz + a, k = jpj (, ), j(pz 1 + a) ; (pz 2 + a)j = jpjjz 1 ; z 2 j: 6.8 6.7 z 7! z,. 6.1, 16,, A(1 C )., C., 2 pz + a(, p 6= 1), 1.

6.4. 133 pz +a, (, )., pz 0 + a = z 0 z 0. p 6= 1, 1 z 0 = a=(1 ; p).,., pz + a = p z ; a 1 ; p + a 1 ; p, 1=(1 ; p), jpj, arg p.,., 53 F 1, F 2, F 1 F 2 (F 1 )(F 2 ), F 1 F 2 (F 1 )+(F 2 )., w = pz + a, u = qw + b 2, u = q(pz + a)+b = pqz +(aq + b):, jpqj = jpjjqj, arg(pq) = arg p + arg q. 147. 2., 1, 1,, 2. 148. ;! ;;! AB 6= CD 4 A, B, C, D, ABE CDE E. 149. ABC,. M, N, P AB, BC, CA, NP? CM, jnpj = jcmj. 6.4 54 S, 2 A, B S 1, S 2., AB, S 1 S 2.. AB O 1 O 2 K ( 6:13 ). K S 1 S 2.,. f, M KM KM 0 = KA KB = const M 0 2 KM,, f(a) =B, f(b) =A.

134 6 6.13: 3 S, f., (, ) S, K K 2 L, L 0 S l, jkljjkl 0 j, l. M 2 S 2. M 0 = f 0 (M), KM KA KM 0 KB = KM : M 1 S 2 KM., KM KM 1 = C = const KM 0 = KA KB C KM 1 :, M 0 K M 1!, f S 2,. S 0. S 2 2 B, B S, S 0 2 A A S. S 0 = S 2 1., S 1 S 2, K., 54 f. 40 O, R T, M OMOM 0 = R 2 OM M 0., T, T., T T. H. Petard A contribution to the mathematical theory of big game hunting.,

6.4. 135 6.14: (H. Petard )..,.,., 1 1., O, 1 1. M O, M M 0 O., p.127, 1 1 (\ ")., 1 1.., z z zz = jzj 2., 0, r z 7! r 2 =z. 150. O. 54, f, f. 151. 152...,.,. ( ).,.. 55 6.15a, 4 2., 4 1.

136 6 6.15: 4. A( 1 ) ( ) f.,. S 0 i = f(s i)(n =1 2 3 4). S 0 1 S0 2, S0 3 S0 4. S 0 i, S i., S 0 1 S0 2, S 0 2 S0 3, S0 3 S0 4, S0 4 0 S 1., S 0 1 S0 2., S 1 S 2 1,., f 6.15a, 6.15b., 3 B 0, C 0, D 0, 3 B 0, C 0, D 0 B, C, D A. 3 B 0, C 0, D 0. S 3 S 4 S 0 1 S 0, 2 S0 1 S0 2 M, N, MC 0 D 0 NC 0 B 0 2 (i.e. MD 0 = MC 0 NB 0 = NC 0 ), \M = \L., \D 0 C 0 M = 1=2(180 ; \D 0 MC) = 1=2(180 \C 0 NB) = \B 0 C 0 N:, 3 B 0, C 0, D 0.. 54, 55, 2,. 147. 153. 2, 2., 2. 6.5.,.., 2 :

6.5. 137 PGL(1 C )(p.127 ),.. 56 C O OB, A OB., 2 C, A.,,,.. 2. O 0, B 1., A 1=2. A, f 1 (z) =1; z:, z w jzjjwj =1,argz = arg w., C, f 2 (z) =1=z: 2 f 1, f 2., f 1 f 2, f 1 f 2. z, z f 1, f 2, f 1, z, 1 ; z, 1=(1 ; z), z=(z ; 1), 1 ; 1=z, 1=z, z, 1; z, 1=(1 ; z), z=(z ; 1), 1 ; 1=z, 1=z., 1=z f 2 z,.,, 12 (G ), G 12. 12. 154..,. G, G. f 1 C C 0, f 2 C 0 MM 0 ( 6.16 ). OB. OB 2 z 7! z (2 G)., 12.,, G. 12, 1 G., G 1.

138 6 6.16: z 7! 1 ; z z 7! 1=z. 6.16 4, G. 12., 6. 3 3 2 (1=2 2 ;1 0 1 1) 2 1 (1=2 i p 3=2 M, M 0 ). 6.16 4, 18. G, 6 3., 3., (, PGL(1 C ), z z ). 17 8s, b, c, d 2 C s:t: ad ; bc 6= 0, 1. w = az + b cz + d ( ),. 2. w = az + b cz + d ( ),. c =0, 2 a 1 z + b 1.,. 1 a 1 z + b 1, z, z! z.

6.6. 139 c 6= 0, az + b cz + d az + b 1 cz + d = p + z z ; 0 + r z 0., z 0 = ; d=c, p =(bc ; ad)c 2, r = a=c ; pz 0., z 7! z + z 0 ( O, 1 ).,, z 0, 1., z 7! pz + r,. (az + b)=(cz + d), z 7! z,.,.,,.,,. 155. ( ) G,., H, G, G=H. 17,., ( ).,.. 1,.,.,,., P (z)=q(z)., P, Q. 156., w az = + b cz + d a b c d 2 R ad ; bc > 0 (6.9) w az = + b cz + d a b c d 2 R ad ; bc < 0: (6.10) 6.6 6.9, 6.10 w, y>0. z = x + iy, w = u + iv, 6.9, u = v = (ax + b)(cx + d)+acy2 (cx + d) 2 + y 2 (ad ; bc)y (cx + d) 2 + y : 2

140 6, v y. 6.10. L H = f(x y)jy >0g 6.9, 6.10. H Lobachevsky, L H.., L, (, ). H, L., ( ) Ox ( ) ( 6.17 ). H 2 1, L-, Euclid. 6.17: L; L, Euclid.,., L- (L- ) L,., Euclid,. Euclid, 2,. a a A, A a 1. 6.18. L- l L- A. A 4 a 1 ( l), a 3 ( k, n, m)., Lobachevsky,,. Lobachevsky. 2 L- Euclid (, L- L- ). 57 74, 3 A, B, C (0 7), (4 3), (0 5)., Lobachevsky ABK., L; BK O, L; AB M.

6.6. 141 6.18: 2 L; 6.19:. L; ABK 6.14. AK Oy. KB O. AB M. 157. M. K., \K =90 : 2 B., \B = \OBM: \A = \OMA: M tan \OMA = 7 3 tan \OBM = tan(\bob 1 ; \BMB 1 )= 9 37 tan(\a + \B) = 7 + 9 3 37 1 ; 7 9 3 37 = 143 24 :

142 6 tan \A + \B > 0, \A + \B < 90., L- ABK 180. i.e. \A + \B + \K < 180., Lobachevsky ( ),., ABC ABK 2, ABC ABK., 1 - U U = az + b cz + d j a b c d 2 Z : U, 2 S : z 7! ;1=z, T : z 7! 1+z: 158. S 2 = ST 3 = id. 6.20, U =fz = x + yijjzj1 jxj 1 2 g T, S, T ;1, TS, ST. 6.20:, L- ( Lobachevsky, )., U.,, U Lobachevsky.,.