008 9 5
.. O A, B AB AB A, B. A, B A, B AB
O A D B AB AB AB. AB A, B.4 AB, D E AE : EB E D E A B D A : B = AE : EB E AEB AD : AB = AE : EB ED AEB EB = EF EBF = EFB E FB EB = EBF, GE = EFB F G E A B D.
. ab 0 {(x, R ax + b = c} {(x, R x + = } a 0 = ax + bx + c. {(x, R x a + b =} (x 0, 0, (x, (x x 0 +( 0. a>b>0, f = a b (±f,0 a (> f (x f + + (x + f + =a ( (x + f + =(a (x f + (x + f + =4a 4a (x f + +(x f + (x, 4a (x f + =4a 4xf 4 a (x f + = a xf a xf f f a {x xf + f + } =(a xf = a 4 xa + x f a {x + f + } = a 4 + x f 4
f = a b a {x + a b + } = a 4 + x (a b a b b x + a = a b x a + b =. a>b>0, f = a b, c = a f, e = f e <, f<a<c a ( f,0, x = c, e (< (x + f + = e x + c (x, c f f (x + f + = e (x + c f = ce x +fx+ f + = e x +e cx + e c x + f + = e x + e c ( e x + = e c f = a f = b e = a f a x a + b = = b a 5
.4. {(x, R x a b =}.5 a>b>0, f = a + b (±f,0 a (< f. f f. a, b>0, f = a + b, c = a f, e = f e >, c<a<f, a ( f,0, x = c, e (>. f c f
.7 f>0 (f,0, x = f,. f f x, =0. A A = {(x, x, } A x p, B A (x, X = x + p, Y = + (X, Y B B = {(X, Y X p, Y } (a, b {(x, (x a +( b = } x p, (a + p, b + {(X, Y (X p a +(Y b = }. A A = {(x, x, } 7
A k B A (x, X = kx, Y = k (X, Y B B = {(X, Y X k, Y k } (a, b {(x, (x a +( b = } k (ka, kb k {(X, Y X ( k a +( Y k b = } {(x, x a + b =} {(x, x + =} x a b. sin θ, cosθ x + i (x, (x + i+(x + i=(x + x +( + i 0, x + i,(x + x +( + i, x + i x + x +( + i x + i x + i x + i x (x + i(x + i=(x + ix +(x + i i =(x + ix +( + x i 8
(x + i(x + i + x i ( + x i x + i + x i x + i (x + ix x + i + x i x + i x + i x + x x + (x + ix +( + x i (x + i(x + i 0, x, x + i 0, (x + ix,(x + i(x + i x x + i θ x + x + i x + i x + i = x + x x + i θ x + i.4 e θi x + i = x + i = x + θ 0=0+0i π, θ e θi e θi, θ ( e θ i ( e θ i =( e (θ +θ i e θi e x θi e x x 9
.5 e θi =cosθ + i sin θ i sin θ e θi =cosθ + i sin θ θ cos θ θ cos θ, sin θ e θ i e θ i = e (θ +θ i e αi e βi =(cosα + i sin α(cos β + i sin β =(cosα cos β sin α sin β+i(sin α cos β +cosα sin β e (α+βi =cos(α + β+i sin(α + β cos(α + β =cosα cos β sin α sin β sin(α + β =sinα cos β +cosα sin β z =sinθ θz θ π = 0.5 z 0.5 z =cosθ θz z =sinθ π 0
4 ax + bx + c 4. ax +bx +c x+i = e θi x i = e θi x = eθi + e θi, = eθi e θi i ax + bx + c = a ( eθi + e θi = a ( eθi ++e θi 4 = a c bi 4 + b ( eθi + e θi ( eθi e θi i +b ( eθi + e θi 4i e θi + a c + bi e θi + a + c 4 +c ( eθi e θi i +c ( eθi +e θi 4 a c bi a c bi =0 a = c, b =0 4 4 ax + bx + c = a(x + a c bi 0 4 K = a c bi (a c + b = 4 4 a c + bi 4 a c bi 4 = Ke αi = Ke αi ax + bx + c = K e αi e θi + K e αi e θi + a + c = K e (θ+αi + K e (θ+αi + a + c =K cos(θ + α+ a + c cos(θ + α θ π θ = α θ = α ± π 4. (x, α (X, Y z = x + i α Z = X + Yi α e α i Z = ze α i X + Yi= Z = ze α i = e θ e α i = e θ+ α i X Yi= e (θ+ α i
e (θ+αi =(e (θ+ α i =(X + Yi, e (θ+αi =(e (θ+ α i =(X Yi ax + bx + c = K e (θ+αi + K e (θ+αi + a + c = K(X + Yi + K(X Yi + a + c = K(X + Yi + K(X Yi + a + c (X + Y =(K + a + c X +(K a + c Y AX + BY ax + bx + c =0 α AX + BY =0 α (X, Y (x, 4. A, B A, B A, B A, B 0 a, b, c 4.4 z = x + i Z = X + Yi Z = ze α i e ± α i =cos α ± i sin α X, Y x, x, X, Y
4.5 A {(x, x, } A θ B (x, θ (x, θ (X, Y X = x cos θ sin θ Y = x sin θ + cos θ x = X cos θ + Y sin θ = X sin θ + Y cos θ B {(X, Y X cos θ + Y sin θ, X sin θ + Y cos θ } 5 5. x, ax + bx + c + dx + e + f =0 ax + bx + c + dx + e + f =0 5. ax + bx + c ax + bx + c + dx + e + f =0 4.5 x, x = X cos θ + Y sin θ = X sin θ + Y cos θ X, Y 4. AX + BY AX + BY + DX + EY + F =0 AX + BY + DX + EY + F =0 A = B =0 D, E 0 A 0 A 0,B =0 E 0 Y = A E X D E X F E
E =0 AX + DX + F =0 A 0,B 0 AX + BY + DX + EY + F = A(X + D A + B(Y + E B + F D 4A E 4B G = F + D 4A + E 4B AX + BY = G X D A, Y E B 5. n n x,..., x n,. x, x ax +b +c ax +b+c h ax +b +c = h ax +b+c = h f(x, =ax +b +c, f(x, =ax +b+c z = ax + b + c, z = ax + b + c 4
. f(x, =ax + b + c z = ax + b + c xz a z = ax + c z b z = b + c z c z = ax + b + c z = f(x, z = f(x, c z z = ax + b + c xz z = b + c xz = x z = b 0 = a b x c b ab 0 f(x, =ax + b z = ax + b a, b x a b f(x, =x + f(x, =x, f(x, = x +, f(x, = x z = x +, z = x, xz z = x, z = x z z = x, z = x + z =x z =x x = =. ax + b + cz = (abc 0 ax + b + cz = x a b z c ±x ± ± z = ± x + + z =,x + z =, x + z =, x z = 5
z 平面上の直線 z = ± を x 軸に沿って平面 x = 上に平行移動した直線は {(,, ± R} のようにも書かれるが これを z 軸の周りに回転すると 曲面 x + = z + すなわち１葉双曲面となる １葉双曲面も線織面である 7 円錐曲線 7. 歴史 楕円 放物線 双曲線は 円錐と平面の共通線として表される 歴史的には 円錐と平面の交わりとして得られる曲線として最初に認識された 日中に 地面に立てた棒の太陽による影を地面に記録していくと 棒の先端は 双 曲線を描くことがわかる ギリシャ ローマ時代の幾何学の発展において 例えば 実用的な日時計を作成する こと 太陽光を鏡で集める武器を製作すること 面積体積を測りうまく分割すること などへの応用が重要であったようである 公理から始めて命題の証明を演繹するユー クリッドの原論の成立には 幾何学のこのような問題に対する有効性が 一度証明さ れたものは常に正しいことが保証されるということがあったようである ユークリッ ドの原論が書かれる前から 大難問と呼ばれる問題が定式化されていた 円積問題 立方倍積問題 角の３等分がそれである 円錐曲線として理解されていた２次曲線を使って 立方倍積問題 角の３等分を解 くことが考えられている 数学の歴史について 日本語ならば hp://.com.mie-u.ac.jp/kanie/osm/humanind/jinmei.hm 英語では The actuo iso of ahemaics achive hp://-hiso.mcs.s-andes.ac.uk/hiso/index.hml などのウェブページが参考になる ここで 円錐曲線に関係する数学者の名前を挙げると 以下のようになる １ ターレス (4 B 547 B, エジプト ギリシャ ピタゴラス (59 B 475 B, イタリア ユークリッド (5 B 5 B, アレクサンドリア アルキメデス (87 B B, シシリア アポロニウス ( B 90 B, トルコ エジプト プトレマイオス (85 5, アレクサンドリア パップス (90 50, アレクサンド リア
(54 0, (54 4, (57 0, (59 50, (59, (, (4 77, (777 855, (790-88, (849 95, (894 97, (95 997 (9 (95 (95 7. l 0, l A l l 0 A l 0 l l 0 A P l 0 c c A S A O A A A l 0 AO A l 0 n m A m B l 0 B L, L m m l 0 l, l m m l l l, l 7
7. m l l, l, m l 0 n m n a m F P P a PF PA T PF PT PT Q Q ap P a 7.4 m l, l 0, l l, l, m m F, l 0 n m F, l 0 n P PF PF n n PA n T PF PT PT Q PA n T PF PT PT Q Q a A T n F l l m a Q P A T n F P F n T l l PF + PF = PT + PT = Q + Q = m n a a P ap PF Q F P : ap = Q : ap 8
7.5 m l 0, l, l l 0, l, l l 0, l, l l, l, m F m T n F, l 0 a A n m F, l 0 T F n n P PF PF n n P Q PA n T PF PT PT Q PA n T PF PT PT Q PF PF = PT PT = Q Q = m n a a P ap PF Q F P : ap = Q : ap 7. 9
8 8. 8. 9 9. S S = { (x,, z x + + z = } S N(0, 0, x P(x,, z S \{(0, 0, } NP x Q P(x,, z Q(X, Y, 0 z>0 0
N(0, 0, O P(x,, z E E E E E E E E E E E E E E E E E R XXXXXXXXXXXXX E E Q(X, Y, 0 NP // NQ λ NP = λ NQ (x,, z = λ(x, Y, x = λx, = λy, z = λ λ = z X = R p : S \{(0, 0, } R, x z, Y = z (x,, z ( x z, z p : S \{(0, 0, }, (x,, z x + i z
O, N, P N P Q QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ O R Q Q x Y N P R Q X z<0 9.
fi fi fi fi fi fi 9.. N ρ ρρ ρ ρρ ρ ρρ ρ ρρ ß ρd ρρ fi ß DDD fi ß fi ß D fi ß DDD fi ß fi ß D DDD fi ß fi ß D fi ß DDD fi ß fi ß D DDD fi ß ß D D ß DDD D DDD D DDD D DDD D DDD D S N ρ ρρ ρ ρρ ρ ρρ ρ ρρ ρd ρρ l DDD D DDD D DDD D DDD D DDD fi fi fi fi fi fi fi fi fi fi fi fi D d S d l d l l d =, d l = 9.. s N(0, 0, s ssss s ssss s ssss s ss s ssss s ssss s ssss s N(0, 0, : = d : l :s = l : s = l d =, s = d l =
9.. N N 9..4 fi fifi fi fifi fi fifi fi fi fifi 4
9. 9.. cos π 5 α =cosπ 5 +sinπ 5 i α =cos π 5 +sin π 5 i, α =cos 4π 5 α =cos π 5 +sin 4π 5 i, +sin π 5 i, α 4 =cos 8π +sin 8π 5 5 i, α 5 =cos 0π 5 +sin0π 5 i = α 5 =0 α 0 α 5 =(α (α 4 + α + α + α + α 4 + α + α + α +=0 α + α 4 =cos π 5 =(α + α 4 = α +α 5 + α 8 = α ++α = α + α 4, = α + α + α 4 + α + α + α +=(α + α +(α + α 4 +=( + + + =0 = ± 4 ( = ± 5 =cos π 5 > 0 cos π 5 = + 5 cos 4π 5 =4cosπ 5 = 5 5
cos π 5 =cos8π 5 = + 5 4, cos 4π 5 =cosπ 5 = 5 4 cos π 5 = cos 4π 5 = + 5 4 9.. ; aaaaaaaa ; ; ; ; ; ; ; ; ; ; ; ; ; - - - - - - - - - - - - - - - - - - - - - ```````````````````` ( ( ( ( ( ( ( ( ( ( ( ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff d N ```````````````````` ( ( ( ( ( ( ( ( ( ( ( d l d l d l d l
9.. d iiiiiiiiiiiiiiiiiii sin(π/5 i π/5 U UUUUUUUUUUUUUUUUUUU sin(π/5 iiiiiiiiiiiiiiiiiii i sin π 5 π/5 sin π 5 U UUUUUUUUUUUUUUUUUUU d =sin π 5, l =sinπ 5 9..4 d/l l/d π d sin l = 5 sin π 5 = sin π 5 cos π 5 sin π 5 =cos π 5+ 5 =, l d = 5+ = 5 9..5 s N(0, 0, s ssss s ssss s ssss s ss s ssss s ssss s ssss s N(0, 0, 7
: = d : l :s = l : s = l 5 d =, s = d 5+ l = 9.. U UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii v v i U v vv v vvv v vvv UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i v v v v v v v v v v 8
0 0. O, c c P OP Q OP OQ = c P Q = P A A B O D O D c A O c A OO c D, D, D OA OA =, O O = OA OBD, ABD, OA : O = OD : OB OA : O = OA : = OB : OD O OA OBD O A O c B O c A D 9
0. c O c c 0. c O c c 0.4 0.5 O, S S P OP Q 0
OP OQ = c... x θ θ 0 0 0 θ θ 0
0 / x / x 40 / x 70 / x / x 0 90 00 / x.. x 80 π π π 80 0 π / π/ π/ 0
/ x π / x π / x 7π / x π / x π / x 5π.. θ x cos θ, sin θ cos θ, sinθ
θ O sin θ θ cos θ / x cos θ θ sin θ θ O O π / x???????? π? 4?????????????? / x..4 θ θ 4
O cos θ θ sin θ / x θ cos θ sin θ θ 0 <θ<π/ =, =..5 θ cos(θ + π = cos θ, sin(θ + π = sin θ 5
80 O sin θ θ cos(θ + π θ + π cos θ sin(θ + π / x 90 ( cos θ + π ( = sin θ, sin θ + π =cosθ θ = θ + π/ O sin θ θ θ = θ + π sin θ cos θ cos θ / x
0 0 cos 0 = sin 0 = 0 π 90 cos π = 0 sin π = π 80 cos π = sin π = 0 π 70 cos π = 0 sin 5π = π 0 cos π = sin π = 0 π 4 π 4 5π 4 7π 4 45 cos π 4 = 5 cos π 4 = 5 cos 5π 4 = 5 cos 7π 4 = sin π 4 = sin π 4 = sin 5π 4 = sin 7π 4 =?????????????????????????????? π π 4π 5π 0 cos π = 0 cos π = 40 cos 4π = 00 cos 5π = sin π = sin π = sin 4π = sin 5π = π 0 cos π = sin π = 5π 50 cos 5π = sin 5π = 7π 0 cos π = sin π = π 0 cos π = sin π = 7
... cos(α + β =cosα cos β sin α sin β, sin(α + β =sinα cos β +cosα sin β 0 <α,β,α+ β<π/ α β cos(α + β sin(α + β α β β cos α cos α cos β sin α sin β sin α β sin α cos β cos α sin β cos(θ + π/ = sin θ, sin(θ + π/ = cos θ.. α = β cos α =cos α sin α, sin α =sinα cos α cos α sin α (sin α (cos α cos α = sin α sin α = cos α cos α =cos α, cos α = sin α 8
β =α cos(α +α=cosα cos α sin α sin α =cosα ( cos α sin α sinα cos α sin(α +α=sinα cos α +cosα sin α =sinα ( sin α+cosα sinα cos α cos α =4cos α cosα, sin α =sinα 4sin α.. P(x, α Q(X, Y P(x, Q(X, Y α P(x, Q(X, Y = x + β P(x, α Q(X, Y 9
O P(x, Q(X, Y β x = cos β = sin β P(x, Q(X, Y β α Y = sin(α + β X = cos(α + β X = cos(α + β = cos α cos β sin α sin β = x cos α sin α, Y = sin(α + β = sin α cos β + cos α sin β = x sin α + cos α (X, Y =(cosα x sin α, sin α x +cosα... x, z = x + i a + bi, c + di a + bi = c + di a = c b = d (a + bi+(c + di =(a + c+(b + di (a + bi(c + di =(ac bd+(ad + bci i = a + bi 40
.. ImO =Im(z z = x + i x =Re(z / Re Gauss (eal axis Re (imagina axis Im z = x + i x z Re(z z Im(z.. x + i =0 x x +0i x x + i (x + i (x + i =( +0i(x + i =(x 0+( +0xi = x + i 0 z = x + i 0+z = z +0=z, z = z =z..4 x + i =(a + bi (c + di (x + i+(c + di =a + bi 4
(a + bi (c + di =(a c+(b di z = x + i 0 z z..5 x + i = a + bi (c + di(x + i =a + bi c + di a + bi =(cx d+(dx + ci x, { cx d = a dx + c = b c + d 0 x = ac + bd c + d, ad + bc = c + d c + di 0 (a + bi/(c + di z 0 /z z.. z = x + i z = x + i = x + a + bi c + di = (a + bi(c di (c + di(c di = (ac + bd+( ad + bci c + d ac + bd ad + bc = + c + d c + d i z = x + i z 0 z = z z z = ( x x + x + + x + i z = x + z/ z 0 z 4
.4.4. z = x + i x + = z z ImO z/ z = x x + + x i + / Re z 0 z/ z θ z/ z =cosθ +sinθi θ z 0 θ<π θ z 0 θ<π θ z = (cos θ +sinθi z 0 θ cos θ +sinθi e θi =cosθ +sinθi.4. e αi =cosα +sinαi, e βi =cosβ +sinβi e αi e βi =(cosα +sinαi(cos β +sinβi =(cosα cos β sin α sin β+(sinα cos β +cosα sin β i 4
e (α+βi e (α+βi =cos(α + β+sin(α + β i e αi e βi = e (α+βi.4. e θi =cosθ +sinθi x + i (cos θ +sinθi(x + i =(cosθx sin θ+(sinθx+cosθi x + i cos θ +sinθi x + i θ 0 a + bi = a + b θ a + bi = (cos θ +sinθi (a + bi(x + i =(cos θ +sinθi(x + i = (cos θx sin θ+(sin θx+cosθi x + i a + bi x + i θ = a + b.4.4 cos(π/ sin(π/ ω = e (π/ i =cos π +sinπ i (cos(π/, sin(π/ < cos(π/ < 0 0 < sin(π/ < ω =cosπ +sinπi= 44
ω O ω sin π cos π ω ω = / x ω =0 (ω (ω + ω +=0 ω ω ω + ω +=0 ω = ± 4 = ± = ± i 0 < sin(π/ < ω = + i cos π =, sin π = cos(π/5 ζ = e (π/5 i =cos(π/5 + sin(π/5 i (cos(π/5, sin(π/5 0 < cos(π/5 < 0 0 < sin(π/5 < ζ 5 =cosπ +sinπi= 45
ζ O ζ ζ v v v vvv cos π 5 v vvv v vvv v vvv ζ ζ 4 ζ 5 = / x = ζ + ζ 4 ζ 5 = ζ 4 = ζ =cos( π/5 + sin( π/5 i =cos(π/5 sin(π/5 i = ζ + ζ 4 =cos π 5 ζ 5 = ζ 5 = 0 (ζ (ζ 4 + ζ + ζ + ζ +=0 ζ ω ζ 4 + ζ + ζ + ζ +=0 = ζ + ζ 4 ζ 5 = ζ 4 + ζ + ζ + ζ +=0 = ζ +ζ 5 + ζ 8 = ζ ++ζ + =0 = ± 4 ( = ± 5 ζ ζ 4 = ζ >0 = + 5 cos π 5 = = + 5 4 4
.5.5. x, x = +, = + ( x + = + + ( = ( +( = + + 4 = + ( + ( + (, 0 (x, x =cosθ, =sinθ θ θ an θ co θ.5. θ cos θ 0 an θ = sin θ cos θ, sin θ 0 co θ = cos θ sin θ an θ θ co θ θ θ an θ sin θ O θ cos θ an θ / x θ 0 <θ<π/ an θ 47
.5. (x, x =cosθ x = + =sinθ = + π <θ<π ϕ = θ/ O O sin θ an ϕ ϕ θ cos θ / x ϕ cos θ an ϕ sin θ θ / x :+cosθ = :sinθ θ =ϕ = sin ϕ +cosϕ = = sin θ +cosθ sinϕcos ϕ +cos ϕ = sin ϕ cos ϕ =anϕ an ϕ = + +anϕ = cos ϕ sin ϕ cos ϕ +sin ϕ =cos ϕ sin ϕ =cosϕ =cosθ + = anϕ +anϕ = sinϕcos ϕ cos ϕ +sin =sinϕcos ϕ =sinϕ =sinθ ϕ x =cosθ, =sinθ =an(θ/ x =( /( + =/( + x =( /( + 48
=/( + =an(θ/ θ x =cosθ =sinθ (, 0 (cos θ, sin θ (, 0 (cos θ, sin θ x =( /( + =/( +... u =(a, b, c, v =(x,, z u v = ax + b + cz ( u v u v 0 u v u, v u v 0 u v 0 do u, v u v θ u v = u v cos θ u v π θ.. u =(a, b, c v =(x,, z u v u v =(bz c, cx az, a bx 49
R f coss u, v u v 0 (bz c, cx az, a bx =(0, 0, 0 bz = c cx = az a = bx (a, b, c (0, 0, 0 a 0 λ = x/a x = λa a = bx = bx/a = λb cx = az z = cx/a = λc (a, b, c =λ(x,, z u v a =0 b 0 c 0 u v u, v u v θ u v = u v sin θ n n u v u v π θ n u v u 0 <θ<π θ v n u v % %% % %% % %% n R % %% % % %% %% % % %% %% % % %% %% % % %% %% θ - u k kkkkkkkkkkkkkkkkkkkkk 5 v u =(, 0, 0, v =(0,, 0 x u π/ v =(0, 0, u v (0, 0, u v =(0 0 0, 0 0 0, 0 0 = (0, 0, u v sin θ n = sin(π/ (0, 0, = (0, 0, 50
... v u = u v. v u = u v. u ( v =( u v ( u v u =(a, b, c, v =(p,,, =(x,, z u ( v =(a, b, c (z, x pz, p x = ( b(p x c(x pz,c(z a(p x,a(x pz b(z = ( bp bx cx + cpz, cz c ap + ax, ax apz bz + b ( u v ( u v =(ax + b + cz(p,, (ap + b + c(x,, z = ( (b + czp (b + cx, (ax + cz (ap + c, (ax + b (ap + bz = ( bp + cpz bx cx, ax + cz ap c, ax + b apz bz. u u l u v l θ O l u O E v J θ O l O E v J θ / i iiiiiiiiii 4 θ 5
u v =cosϕ v +( cos ϕ( u v u +sinϕ u v.. l v ( u v u v v ( u v u l l E ( u v u O v u O O u O O E v / v ( u v u.. v ( u v u, u v v ( u v u θ cos θ ( v ( u v u+sinθ u v 5
90 u v O / v ( u v u O u v < cos θ ( v ( u v u+sinθ u v.. ( u v u v =cosθ ( v ( u v u+sinθ u v +( u v u =cosθ v +( cos θ( u v u +sinθ u v O l O ( u v u E v J θ / v ( u v u i iiiiiiiiii 4 cos θ ( v ( u v u+sinθ u v θ u v =cosϕ v +( cos ϕ( u v u +sinϕ u v u v ϕ 5
..., x,, z = + xi + j + zk s + ai + bj + ck, s + a i + b j + c k (s + ai + bj + ck+(s + a i + b j + c k=(s + s +(a + a i +(b + b j +(c + c k (s + ai + bj + ck(s + a i + b j + c k =(ss aa bb cc +(sa + as + bc cb i +(sb + bs + ca ac j +(sc + cs + ab ba k ij = k, ji = k, jk = i, kj = i, ki = j, ik = j.. + xi + j + zk x = = z =0 + xi + j + zk + xi + j + zk ( + xi + j + zk ( + xi + j + zk =( +0i +0j +0k( + xi + j + zk =( 0x+(x +0z 0i +( +0x 0zj +(z +0 0xk = + xi + j + zk 0 = + xi + j + zk 0+ = +0=, = =.. x + i + zj + k =(s + a i + b j + c k (s + ai + bj + ck (x + i + zj + k+(s + ai + bj + ck =s + a i + b j + c k 54
(s + a i + b j + c k (s + ai + bj + ck =(s s+(a ai +(b bj +(c ck 0 p p =0 p + p =0 (0 p+p =0 ( p+p =0 p +( p =0..4 p = s + a i + b j + c k p = s + ai + bj + ck = + xi + j + zk p = p p p + xi + j + zk =(s + a i + b j + c k(s + ai + bj + ck ( + xi + j + zk(s + ai + bj + ck =s + a i + b j + c k, x,, z (s xa b zc+(a + xs + c zbi +(b + s + za xcj +(c + zs + xb ak, x,, z s ax b cz = s a + sx + c bz = a b cx + s + az = b c + bx a + sz = c = ss + aa + bb + cc, s + a + b + c x = (a s as +(bc cb, s + a + b + c = (b s bs +(ca ac, s + a + b + c z = (c s cs +(ab ba s + a + b + c s+ai+bj+ck 0 (s +a i+b j+c k(s+ai+bj+ck, x,, z + xi + j + zk 55
p = s + a i + b j + c k p = s + ai + bj + ck = + xi + j + zk p = p p p + xi + j + zk =(s + ai + bj + ck (s + a i + b j + c k (s + ai + bj + ck( + xi + j + zk =s + a i + b j + c k = ss + aa + bb + cc s + a + b + c x = (a s as +(b c bc s + a + b + c = (b s bs +(c a ca s + a + b + c z = (c s cs +(a b ab s + a + b + c s+ai+bj+ck 0 (s+ai+bj+ck (s +a i+b j+c k, x,, z + xi + j + zk p a = b = c =0 p p p p..5 = + xi + j + zk xi + j + zk 0 = + xi + j + zk = + xi + j + zk = + x + + z = + xi + j + zk = xi j zk =( + xi + j + zk( xi j zk = + x + + z = + xi + j + zk = = =, = p = s + ai + bj + ck 0 = ( + xi + j + zk = + x + + z xi j zk + x + + z x + x + + z i 5 + x + + z j z + x + + z k
p p p p.4, x,, z x z x z Q(, x,, z= z x z x Q(s, a, b, cq(s,a,b,c, x,, z = ss aa bb cc x = sa + as + bc cb = sb + bs + ca ac z = sc + cs + ab ba Q(, x,, z =Q(s, a, b, cq(s,a,b,c, x,, z + xi + j + zk =(s + ai + bj + ck(s + a i + b j + c k 0 0 0 0 0 0 0 0 0 E = 0 0 0,I= 0 0 0 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J = 0 0 0,K= 0 0 0 0 0 0 0 0 0 0 0 0 EI = IE = I, EJ = JE = J, EK = KE = K, IJ = JI = K, JK = KJ = I, KI = IK = J, i, j, k I,J,K I,J,K I,J,K 57
.5.5. v =(x,, z ( v =xi + j + zk p = s + ai + bj + ck p = + xi + j + zk p = s + ( u, p = + ( v, u =(a, b, c, v =(x,, z p p =(s u v+( u+s( v+( u v.5. ϕ cos ϕ +sinϕ(u (u (cos ϕ +sinϕ(u(v(cos ϕ sin ϕ(u =(cosϕ(v sin ϕ (u v+sinϕ(u (v(cos ϕ sin ϕ(u =(cos ϕ(v sin ϕ cos ϕ (u v+sinϕ cos ϕ(u v +sinϕ cos ϕ (v u sin ϕ cos ϕ(v u +sin ϕ (u v(u sin ϕ((u v u =cos ϕ(v+sin ϕ (u v(u+sinϕ cos ϕ(u v sin ϕ((u v u (u v u = (u vu +(u uv (cos ϕ +sinϕ(u(v(cos ϕ sin ϕ(u =cos ϕ(v+sin ϕ (u v(u+sinϕ cos ϕ(u v +sin ϕ(u v(u sin ϕ(u u(v =cosϕ(v+( cos ϕ(u v(u+sinϕ(u v cos ϕv+( cos ϕ(u v u +sinϕu v u v ϕ 58