( ) ( )

20 21 2 8

1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1

1 2 2 1 2 1 0 2 0 3 0 4 0 2

2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t)) γ(t) γ(t) = (x(t) y(t)) γ(t) = (ẋ(t) ẏ(t)) (ẋ = dx dt ẏ = dy dt ) t 22 γ(t)(a t b) s(t) = t a γ(u) du γ(t) [a t] γ 0 ds = γ(t) 0 dt [a b] γ(t) l s(t) [a b] [0 l] t = t(s) : [0 l] [a b] t(s) s γ(s) := γ(t(s)) (0 s l) s s 3

s ( ) γ(s) = dγ ds = dγ dt dt ds = γ(t) γ(t) 1 t 1 γ(s) = (x(s) y(s)) e(s) := γ(s) = ( x(s) ý(s)) γ(s) n(s) := ( ý(s) x(s)) e(s) e(s) γ(s) 23 γ(s) = (x(s) y(s)) 1 γ(s) = γ(s) 2 = 1 ( ) s 2γ (s) γ (s) = 0 γ (s) γ (s) γ (s) n(s) κ(s) γ(s) γ (s) = κ(s)n(s) κ(s) γ(s) 4

24 3 2 x(u v) y(u v) z(u v) (u v) 3 (x y z) (u v) uv 2 3 p(u v) = x(u v) y(u v) z(u v) p u = p u = (x u y u z u ) p v = p v = (x v y v z v ) p(u v) (u v) uv (u v) p u (u v) p v (u v) x(u v) y(u v) z(u v) p(u v) p u (u v) p v (u v) p(u v) p u (u v) p v (u v) p(u v) sp u (u v) tp v (u v) s t p u (u v) p v (u v) ν := p u(u v) p v (u v) p u (u v) p v (u v) ( ) ν = (a b c) (x o y o z o ) a(x x o ) b(y y o ) c(z z o ) = 0 25 p(u v) 2 p(u v) p(u u v v) s 2 u v : ( s) 2 = p(u u v v) p(u v) 2 5

p u (u v) p u (u v) 3 (u v) p u p u p u p v p v p v ds 2 = du 2 dudv dv 2 ( ) ( ) ( ) ds 2 du = du dv dv ( ) p(u v) p u p v 2 = (p u p u )(p v p v ) (p u p v ) 2 = 2 d 2 dudv 26 p ( u v) II = dp dν = (p u du p v dv) (ν u du ν v dv) ν = ν(u v) p(u v) II = ( p u ν u )du 2 ( p u ν v p v ν u )dudv ( p v ν v )dv 2 p u p v ν p u ν = 0 p v ν = 0 u v p uu ν = p u ν u p vv ν = p v ν v p uv ν = p u ν v = p v ν u 6

II = du 2 2 dudv dv 2 p u ν u = p uu ν p u ν v = p v ν u = p uv ν p v ν v = p vv ν 2 p(u 0 v 0 ) f (u 0 v 0 ) II p(u 0 v 0 ) f (u 0 v 0 ) II (u(s) v(s)) p(s) = p(u(s) v(s)) κ dp/ds = 1 p (s) p(s) p (s) p(s) p(u v) p (s) p(u v) p (s) = k g k n k g k n k n e k n = κ n e κ n κ n = κ n e e = (p k g ) e = p e = p e du = (p u ds p dv v ds ) (e du u ds e dv v ds ) = du du dv 2 du ds ds ds ds dv dv ds ds κ n (s) p(s) p (s) 1 p 0 = p(u 0 v 0 ) ω = ξp u (u 0 v 0 ) ηp v (u 0 v 0 ) 7

II(ω ω) = ξ 2 2 ξη η 2 κ n (s) = II(p (s) p (s)) II(ω ω) ω p 0 ω 2 = ξ 2 2 ξη η 2 = 1 II(ω ω) (ξ η) (0 0) η 2 η 2 λ = ξ2 2 ξη ξ 2 2 ξη ξ 2 2 ξη η 2 λ( ξ 2 2 ξη η 2 ) = 0 ξ η λ/ ξ = λ/ η = 0 ( λ )ξ ( λ )η = 0 ( λ )ξ ( λ )η = 0 (ξ η) 0 λ ( 2 )λ 2 ( 2 )λ 2 = 0 λ = κ 1 κ 2 κ 1 κ 2 = 2 2 1 2 (κ 1 κ 2 ) = 2 2( 2 ) κ 1 κ 2 = 1 2 (κ 1 κ 2 ) 27 0 0 8

28 ( ) x 2 a 2 y2 b 2 z2 c 2 = 1 x = a cos u cos v y = b cos u sin v z = c sin u p(u v) = (a cos u cos v b cos u sin v c sin u) p u = ( a sin u cos v b sin u sin v c cos u) p v = ( a cos u sin v b cos u cos v 0) p uu = ( a cos u cos v b cos u sin v c sin u) p uv = (a sin u sin v b sin u cos v 0) p vv = ( a cos u cos v b cos u sin v 0) e = 1 ( bc cos u cos v ca cos u sin v ab sin u) = b 2 c 2 cos 2 u cos 2 v c 2 a 2 cos 2 u sin 2 v a 2 b 2 sin 2 u a2 b 2 c 2 a 2 sin 2 u cos 2 v b 2 sin 2 u sin 2 v c 2 cos 2 u (a 2 b 2 ) sin u cos u sin v cos v a 2 cos 2 u sin 2 v b 2 cos 2 u cos 2 v abc 0 a2 b 2 c 2 cos 2 u abc[(a2 b 2 c 2 ) (a 2 cos 2 u cos 2 v b 2 cos 2 u sin 2 v c 2 sin 2 u)] 2 3 9

3 31 xz xz z z xz z (x z) = (f(u) g(u)) z (x y z) = (f(u) cos v f(u) sin v g(u)) 31 p(u v) = (f(u) cos v f(u) sin v g(u)) p u p u = (f (u) cos v f (u) sin v g (u)) (f (u) cos v f (u) sin v g (u)) = f (u) 2 g (u) 2 p u p v = (f (u) cos v f (u) sin v g (u)) ( f(u) sin v f (u) cos v 0) = 0 p v p v = ( f(u) sin v f (u) cos v 0) ( f(u) sin v f (u) cos v 0) = f(u) 2 10

e e = p u p v p u p v e = f(u)( g (u) cos v g (u)f(u) sin v f (u)) f(u) f (u) 2 g (u) 2 = ( g (u) cos v g (u)f(u) sin v f (u)) f (u) 2 g (u) 2 p uu e = (f (u) cos v f (u) sin v g (u)) ( g (u) cos v g (u)f(u) sin v f (u)) f (u) 2 g (u) 2 = f (u)g (u) f (u)g (u) f (u) 2 g (u) 2 p uv e = ( f (u) sin v f (u) cos v 0) ( g (u) cos v g (u)f(u) sin v f (u)) f (u) 2 g (u) 2 = 0 p vv e = ( f (u) cos v f (u) sin v 0) ( g (u) cos v g (u)f(u) sin v f (u)) f (u) 2 g (u) 2 = f(u)g (u) f (u) 2 g (u) 2 2 2 = g (u)(f (u)g (u) g (u)f (u)) f(u)(f (u) 2 g (u) 2 ) 2 2 2( 2 ) = 1 2 ( g (u) f(u)(f (u) 2 g (u) 2 ) 1/2 f (u)g (u) f (u)g (u) (f (u) 2 g (u) 2 ) 3/2 ) u f (u) 2 g (u) 2 = 1 11

f (u)f (u) g (u)g (u) = 0 (f (u)g (u) f (u)g (u))g (u) = f (u)g (u)g (u) f (u)g (u)g (u) f (u)f (u)f (u) f (u)(1 f (u)f (u)) = f (u) g (u)(f (u)g (u) g (u)f (u)) f(u)(f (u) 2 g (u) 2 ) 2 = f (u) f(u) 1 2 ( g (u) f(u)(f (u) 2 g (u) 2 ) 1/2 f (u)g (u) f (u)g (u) (f (u) 2 g (u) 2 ) 3/2 ) = g (u) 2f(u) f (u) 2g (u) 32 ( ) z x2 a 2 z2 = 1 a 0 b 0 b2 (x z) = (f(u) g(u)) = (a cos u b sin u)) p(u v) = (f(u) cos v f(u) sin v g(u)) p(u v) p(u v) = (a cos u cos v a cos u sin v b sin u) 12

= = g (u)(f (u)g (u) g (u)f (u)) f(u)(f (u) 2 g (u) 2 ) 2 b 2 (a 2 sin 2 u b 2 cos 2 u) 2 g (u) f(u)(f (u) 2 g (u) 2 ) f (u)g (u) f (u)g (u) 1/2 (f (u) 2 g (u) 2 ) 3/2 b ab a(a 2 sin 2 u b 2 cos 2 u) 1/2 (a 2 sin 2 u b 2 cos 2 u) 3/2 b c (b = a) (c = b) 4 0 0 41 0 41 0 0 = 0 u f (u) f(u) = 0 f (u) = 0 f (u) 2 g (u) 2 = 1 x = f(u) = au b z = g(u) = ±u 1 a 2 + c (a b c a 1) 13

g(u) = u 1 a 2 +c π/2 g(u) = u 1 a 2 +c g(u) = u 1 a 2 c g(u) = u 1 a 2 + c z = g(u) = u 1 a 2 a = 0 (x z) = (f(u) g(u)) = (b u) 0 a 1 (x z) = (f(u) g(u)) = (au + b u 1 a 2 ) a = 1 (x z) = (f(u) g(u)) = (u + b 0) 42 42 0 p(u v) c cp(u v) 1/c 2 1/c p(u v) = cp(u v) p u = cp u p v = cp v p p c 2 p p (u v) p uu = cp uu p p c 1 1 = 1 f = f x = f(u) = α cos u β sin u(α β ) = a cos(u δ)(a = α 2 β 2 δ = arctan( β α )) f(u) = a cos u a 0 (f ) 2 (g ) 2 = 1 z = g(u) = ± u 0 1 a 2 sin 2 tdt 14

0 z = g(u) = u 0 1 a 2 sin 2 tdt a = 1 = 1 f = f x = f(u) = ae u be u (a b ) (f ) 2 (g ) 2 = 1 z = g(u) = ± u 0 1 (ae u be u ) 2 dt 0 u z = 1 (ae u be u ) 2 dt 0 a = 0 b = 1 u x(u) = e u z(u) = 1 (e 2t )dt 0 y y z z x x K = 1 a < 1 K = 1 a = 1 4 1 4 2 43 0 0 = 0 u g (u) 2f(u) f (u) 2g (u) = 0 15

f(u)f (u) = g (u) 2 = 1 f (u) 2 (f(u)f (u)) = f (u) 2 f(u)f (u) = 1 1 2 (f(u)2 ) = f(u)f (u) = u c d f(u) 2 = u 2 2cu d f(u) = u 2 2cu d g (u) = ± 1 f (u) 2 d c = ± 2 u 2 2cu d c = 0 d = a 2 (a 0) z = g(u) = ± x = f(u) = u 2 a 2 u u 0 a t 2 a = ±a u 2 sinh 1 a x = a cosh z a ( ) z 0 0 0 y x z 16

44 0 35 4y 25 3 15 2 05 1 2 4 6 8 10 12 x xy (r θ) r = r(θ) = a (a 0 0 ɛ 1) 1 ɛ cos θ 1 ɛ γ(θ) := r(θ)(cos θ sin θ) = γ(θ) γ a sin θ (θ) = ( (1 + ɛ cos θ) 2 ɛ + a cos θ (1 + ɛ cos θ )2 ξ cos ξ = ɛ sin θ sin ξ = 1 + ɛ cos θ 1 + 2ɛ cos t + ɛ 2 1 + 2ɛ cos t + ɛ 2 x 1 θ x s(θ) γ(θ) s(θ) = θ 0 a 1 + 2ɛ cos t + ɛ 2 (1 + ɛ cos t) 2 dt (x y) = (x(θ) y(θ)) = (s(θ) + r(θ) cos ξ(θ) r(θ) sin ξ(θ)) (x(θ) y(θ)) x 1 2 ( x y x y x (x 2 y 2 ) 3/2 y(x 2 + y 2 ) ) 1/2 17

= 1 + 2ɛ cos θ + ɛ 2 = ɛ sin θ (cos ξ) = ɛ2 sin 2 θ 2 ɛ cos θ 3 x = a(1 + ɛ cos θ) 3 x = ( aɛ sin θ) 3 3a(1 + ɛ cos θ) 2 y = aɛ sin θ 3 6 y = (aɛ cos θ) 3 3(aɛ sin θ) 2 6 1 ɛ2 2a (ɛ 1) 5 6 1 1995 2 2002 3 2000 18