k i MA i = 0. OM = n OM = 1 (a OA + b f( u + v ) = f( u ) + f( v ) i=1 i=1

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2 (, ) (, ) (D, ) D () (D) = D (, ) (, ) (, ) k v (, ) k v (, ) () = k (, ) ( i ) i 1 n (k i ) i k i M n k i M i = 0. i=1 O M 1 n OM = n i=1 k k ioi i a b M OM = 1 (a O + b a+b O) (, a) (, b) (, c) (, a) (M, b + c) M (, b) (, c) f f ( ) = f()f() f( u + v ) = f( u ) + f( v ) u (, ) u i=1

3 O k OM = k OM M M E E E E (M + u ) + v = M + ( u + v ) (M, N) u M + u = N u MN E = 2 u v u v u v u u ( v + w ) = u v + u w =. 2 = 2 = Â 2 = ( ) 2

4 (x, y) z = x + iy D a b c d [0, 1, t, ] = t [,,, D] = c a b d c d b a a b c z [a, b, c, z] z az+b cz+d z az + b a a k 2 M M (OM) OM OM = k 2 O z k 2 / z p P p/p O Γ P Γ O p Γ O p/p P M N (OM) Γ OM ON = P M N O Π M N M N M N M N OMN OM N M N = Π MN OM ON OM OM = ON ON = P OM ON = ON OM M N MN = OM ON = Π OM ON.

5 Z Y X D D D D + D D [,,, D] 1 D D X Y Z XZ XY + Y Z X Y Z D XZ = D D XY = D D Y Z = D D XZ XY + Y Z D D D E (, ) 2 =

6 P G N M M M M G M N P [] [] [] GM GM GN GN GP GP M M M = M M M [] [] H H H H H H H H

7 O O O O O O O O H α β γ Ĉ Â H = π/2 γ Ĥ = π/2 α ÂH = π H Ĥ = π β S H () ÂS = α π Â π S [] S H P S H H O N H M S

8 P H P P P P P P P P P P P () () () P P P P P P P P P P = P P P P = P 180 P P S H () P H P P P () P H = H = π/2 β P H = π/2 β P P P P P P = 2 P P P = 2β P P P d i = d j d k k d i + d j = 0 d 1 = d 2 = d 3 d 1 = d 2 = d 3 d 1 = d 2 = d 3 d 1 = d 2 = d 3 I I I I I I

9 I K I M L I I () M () I I II [I] I Ĉ I β Î π/2 β [] M K L II [H] [H] [H] [OH] OH = 3 OG = O + O + O. O G H h G 1/2 R/2 MNP O MNP H O OH = 3 OG [OH] H 1/2 R/2 [OH] H [H] [H] [H] ι M [] M

10 S H P H S H G O N H M S H I Ω O H K M MΩ Ω [OH] M Â M ι () G 1/2 HO OMΩ ÔMΩ = ĤO = α 2(π/2 β) = β γ. D ÂD ÂD β γ (MΩ) (OM) () ι(h ) = D D MΩ MK 2 = MD MH H K I 2 = D () D/ = / I = D Â D = Â = α/2

11 I S I L H S H P K H H S I G Ω K K O M N L L I K K K L L L

12 M M O N P M O 2π/3 2π/3 d(m, ) + d(m, ) + d(m, ) = d(m, ) + d(p, ) + d(n, ) = 3d(O, ) M M+M M M M N N M M + M = MN + N M NM N [M] ÂN = ÂM = 2π/3 M M+M+M ÂM = M = ĈM 2π/3 T T T M+M +M M+MT T M M 2π/3 F T T ÂF = F = 2π/3 ĈF 2π/3 F ÂF T = T F = π/3 F T T T F F + F + F = T = T = T

13 T T F T M (M) (M) (M) (M) (M) (M) N M (O, H) (x, y, z) (yz, zx, xy) M N [MN] M M M () () N N N M MM N NN M N = M N M M N N N M M N N M M M N N

14 N N N M M M M N P (P ) (P ) (P ) P P Q P P P P P P () () Q P (P ) P P P P P = QP (Q) (P ) QP = P P P P P QP (Q) (P P ) P 2 (R) [x : y : z] [a 2 yz : b 2 zx : c 2 xy] a b c x + y + z = 0 a 2 xy + b 2 yz + c 2 zx = 0.

15 M () (M) () [1 : 0 : 0] [ : y : z] [y : z]

16 P P P P O () ( ) (O) () ( ) X O O ( ) (O ) X Γ Γ (O) (O ) Γ Γ X O P D [,,, D] = D O D O D [ 1, 2, 3, 4 ] O D O O D O = ÂO O Ĉ D ĈO O = ĈOD OD

17 ĈO OD [,,, D] = ĈOD O D D D (M) (M) (M) (MD) M M [,,, D] = D D M (M) (M) (M) (MD) ĈM ĈMD MD M. ĈM = /D D M λ 1 λ 1 1/λ 1 1/λ λ 1 λ λ /2 D /D = /D D O (O) (OD) O / = O/O D/D = O/O O (O) O O [D] (OD)

18 P P D E F O (O) (O) (O) / = EF /ED O = O O O O f I J K p I J K f(i) f(j) f(k) f p p 1 f X Y (XY ) (f(x)f(y )) h X Y f(x) f(y ) f h X Y X Y (XY ) X Y 0 X 1 Y f f : R R m M f(m) f(m) f f f (x, y) (f(x), f(y))

19 f f(xy) = f(x)f(y) [1, f(x)] [f(y), f(xy)] f( x) = f(x) x y x y (x + y, y) (x + y) f(x + y) = f(x) + f(y) f : R R P 2 P GL 3 [x : y : z] (x, y, z) [x : y : 1] [x : y : 0] x = a 11x + a 12 y + a 13 y = a 21x + a 22 y + a 23 a 31 x + a 32 y + a 33 a 31 x + a 32 y + a 33 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

20 OM OM = R 2 X Y OX OY = R 2 X OX OY = R 2 Y P Q Q P (MN) M N P P M O P N M N P OM OP = OM 2 = R 2 OMP M N P (MN) D X Y Z () (D) X Y Z (Y Z) (ZX) (XY ) T () (XY ) U (XY ) V (XY ) Z (XY ) Z () (D) (XY ) () (XY ) U = V

21 D Z U V Y T X D U Y T X T X V Y [U, X, T, Z] = [,,, D] = [T, Z, V, X] = [V, X, T, Z] U = V (D) X Y Z (,, ) (, D, ) X Y U (,, ) (D, E, F ) (E) (D) (F ) (E) (D) (F ) Z Y X F D E X (F ) (E) Y (D) (F ) Z (E) (D) (XZ) (F ) (D) Y 1 (F ) (XZ) Y 2 (D) (XZ) x 2 + y 2 z 2 x 2 + y 2 + z 2

22 z = 1 q 2 q(v = (x, y, z)) = ax 2 + by 2 + cz 2 + 2dyz + 2ezx + 2fxy b(v, v ) = axx + byy + czz + d(yz + zy ) + e(zx + xz ) + f(xy + yx ) v v q b(v, v ) = 0 x 2 + y 2 + z 2 = 0

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