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1

2 Frenet Frenet-Serret Frenet-Serret Frenet-Serret Bouquet

3 1 R I E 2 f(t) (x(t), y(t)), (t I) (1) x(t), y(t) C (2) t f(t) df(t) ( dx(t), dy(t) ) 0 (3) 1 f(t) 2 3 f(t) f(t) R I E 3 f(t) (x(t), y(t), z(t)), (t I) (4) x(t), y(t), z(t) C (5) df(t) ( dx(t), dy(t), dz(t) ) 0 (6) f(t) 1 4 t 2 5 C C r C r 2

4 2 f(t) (x(t), y(t), z(t)) f(t 0 ) f(t) [t 0, t] : t 0 < t 1 < t 2 < < t n 1 < t n t (7) 1: d( ) max{t i t i 1 i 1, 2,, n} (8) f(t 0 ) f(t) n lim f(t i ) f(t i 1 ) (9) d( ) 0 i1 f(t i ) f(t i 1 ) (x(t i ) x(t i 1 ), y(t i ) y(t i 1 ), z(t i ) z(t i 1 )) (ẋ(ξ i )(t i t i 1 ), ẏ(η i )(t i t i 1 ), ż(ζ i )(t i t i 1 )) (t i 1 < ξ i, η i, ζ i < t i ) (ẋ(ξ i ), ẏ(η i ), ż(ζ i ))(t i t i 1 ) (10) n lim f(t i ) f(t i 1 ) d( ) 0 i1 n lim ẋ(ξ i ) 2 + ẏ(η i ) 2 + ż(ζ i ) 2 (t i t i 1 ) d( ) 0 i1 (11) 3

5 d( ) 0 (t i t i 1 ) 0 (ξ i η i ) 0 (η i ζ i ) 0 lim n d( ) 0 i1 n lim d( ) 0 i1 n lim d( ) 0 i1 t ẋ(ξ i ) 2 + ẏ(η i ) 2 + ż(ζ i ) 2 (t i t i 1 ) ẋ(ξ i ) 2 + ẏ(ξ i ) 2 + ż(ξ i ) 2 (t i t i 1 ) f(ξ i ) (t i t i 1 ) f(t) (12) t 0 f(t 0 ) f(t) s(t, t 0 ) s(t, t 0 ) t f(t) t 0 ẋ(t) 2 + ẏ(t) 2 + ż(t) 2 (13) t t 0 s(t, t 0 ) t df(t) ( df(t) ) 2 (14) df(t) 0 (15) 0 (16) 2.1 x(t), y(t), z(t) r(t), θ(t), φ(t) x(t) r(t)sinθ(t)cosφ(t) y(t) r(t)sinθ(t)sinφ(t) z(t) r(t)cosθ(t) (17) (0 < r(t) < 0 < θ(t) < π 0 < φ(t) < π) (18) 4

6 2: ẋ ṙsinθcosφ + r θcosθcosφ r φsinθsinφ ẏ ṙsinθsinφ + r θcosθsinφ + r φsinθcosφ ż ṙcosθ r θsinθ (19) ẋ 2 + ẏ 2 + ż 2 ( ṙ 2 sin 2 θcos 2 φ + r 2 θ2 cos 2 θcos 2 φ + r 2 φ2 sin 2 θsin 2 φ +2rṙ θsinθcosθcos 2 φ 2rṙ φsin 2 θsinφcosφ 2r 2 θ φsinθcosθsinφcosφ ) + ( ṙ 2 sin 2 θsin 2 φ + r 2 θ2 cos 2 θsin 2 φ + r 2 φ2 sin 2 θcos 2 φ +2rṙ θsinθcosθsin 2 φ + 2rṙ φsin 2 θsinφcosφ + 2r 2 θ φsinθcosθsinφcosφ ) + ( ṙ 2 cos 2 θ 2rṙ θsinθcosθ + r 2 θ2 sin 2 θ ) ṙ 2 + r 2 θ2 + r 2 φ2 sin 2 θ (20) (20) (13) s(t, t 0 ) t f(t) t 0 ẋ(t) 2 + ẏ(t) 2 + ż(t) 2 t t 0 t t 0 ṙ 2 + r 2 θ 2 + r 2 φ 2 sin 2 θ (21) 5

7 3 6 df(t) ( dx(t), dy(t), dz(t) ) (22) 3 f f(t) 3: e 1 (t) df(t) df(t) (23) 23 e 1 (t) df df df df (24) (25) 14 e 1 (s) df(s) s f (26) 6

8 k(s) de 1 ( ) d2 f 2 (27) 27 k(s) κ(s) k(s) 3.1 κ(t) f(t) (x(t), y(t)) (28) 27 e 1 (s) k de 1 κ(t) e 1 (t) df ( df dx, dy ) (29) (30) 13 z(t) 0 ż(t) 0 ẋ 2 + ẏ 2 (31) 1 ẋ2 + ẏ 2 (32) 7

9 e 1 (t) ( ) ẋ ẋ2 + ẏ, ẏ 2 ẋ2 + ẏ 2 (33) 27 k de 1 32 de 1 33 t ( de 1 ẍ ẋ2 + ẏ + ẋ d 2 ( ) 1 ẋ2, + ẏ 2 ÿ ẋ2 + ẏ + ẏ d ( 2 )) 1 ẋ2 + ẏ 2 (34) (35) ( ) d 1 ẋ2 + ẏ (ẋẍ + ẏÿ) 2 (ẋ 2 + ẏ 2 ) 3 2 ẋẍ + ẏÿ (ẋ 2 + ẏ 2 ) 3 2 (36) de 1 ẍ ẋ2 + ẏ ẋ ẋẍ + ẏÿ ÿ, 2 (ẋ 2 + ẏ 2 ) 3 ẋ2 2 + ẏ ẏ ẋẍ + ẏÿ 2 (ẋ 2 + ẏ 2 ) 3 2 ẍẏ ẋÿ ÿẋ ẍẏ ẏ, ẋ (37) (ẋ 2 + ẏ 2 ) 3 2 (ẋ 2 + ẏ 2 ) 3 2 k ( ) ẍẏ ẋÿ ÿẋ ẍẏ k ẏ (ẋ 2 + ẏ 2 2, ẋ ) (ẋ 2 + ẏ 2 ) 2 (38) κ k κ κ(t) ẋ(t)ÿ(t) ẍ(t)ẏ(t) (ẋ(t) 2 + ẏ(t) 2 ) 3/2 (39) x y y y(x) (40) 8

10 39 x t, y y(t) (41) κ(t) ÿ(t) (1 + ẏ(t) 2 ) 3/2 (42) 3.2 f(t) (r(t), φ(t)) (43) 18 θ π/2 z t { x(t) r(t)cosφ(t) y(t) r(t)sinφ(t) (44) (0 < r(t) < 0 < φ(t) < π) (45) { ẋ ṙcosφ r φsinφ ẏ ṙsinφ + r φcosφ (46) t { ẍ rcosφ 2ṙ φsinφ r φsinφ r φ2 cosφ ÿ rsinφ + 2ṙ φcosφ + r φcosφ r φ 2 sinφ (47) ẋ 2 + ẏ 2 ṙ 2 + r 2 φ2 (48) ẋÿ ẍẏ 2ṙ 2 φ + rṙ φ r r + r 2 φ3 (49) κ(t) 2ṙ2 φ + rṙ φ r r + r 2 φ3 (ṙ 2 + r 2 φ 2 ) 3 2 (50) 9

11 50 φ t φ 1 φ 0 r f(φ) f(t) (r(t), t) (51) κ(t) r2 + 2ṙ 2 r r (ṙ 2 + r 2 ) 3 2 (52) 3.3 κ(t) f(t) (x(t), y(t), z(t)) (53) e 1 (s) k de 1 κ(t) e 1 (t) df ( df dx, dy, dz ) (54) (55) 13 ẋ 2 + ẏ 2 + ż (56) 1 ẋ2 + ẏ 2 + ż 2 (57) 10

12 f(t) (x(t), y(t), z(t)) e 1 (t) 58 t de 1 1 ẋ2 + ẏ 2 + ż 2 (ẍ, ÿ, z) + d 1 ẋ2 (ẋ, ẏ, ż) (58) + ẏ 2 + ż2 ( 1 ẋ2 + ẏ 2 + ż 2 ) (ẋ, ẏ, ż) (59) ( ) d 1 ẋ2 + ẏ 2 + ż (ẋẍ + ẏÿ + ż z) 2 (ẋ 2 + ẏ 2 + ż 2 ) 3 2 ẋẍ + ẏÿ + ż z (ẋ 2 + ẏ 2 + ż 2 ) 3 2 (60) de 1 ẍ (ẏ2 + ż 2 ) ẋ (ẏÿ + ż z) (ẋ 2 + ẏ 2 + ż 2 ) 3 2, ÿ (ẋ2 + ż 2 ) ẏ (ẋẍ + ż z) (ẋ 2 + ẏ 2 + ż 2 ) 3 2 (57) (61) k(t), z (ẋ2 + ẏ 2 ) ż (ẋẍ + ẏÿ) (ẋ 2 + ẏ 2 + ż 2 ) 3 2 k(t) de 1 de 1 (ẍ (ẏ 2 + ż 2 ) ẋ (ẏÿ + ż z) (ẋ 2 + ẏ 2 + ż 2 ) 2, ÿ (ẋ2 + ż 2 ) ẏ (ẋẍ + ż z) (ẋ 2 + ẏ 2 + ż 2 ) 2, z (ẋ2 + ẏ 2 ) ) ż (ẋẍ + ẏÿ) (ẋ 2 + ẏ 2 + ż 2 ) 2 κ(t) κ(t) k(t) 1 (ẋ 2 + ẏ 2 + ż 2 ) 2 ( (ẍ (ẏ2 + ż 2) ẋ (ẏÿ + ż z) ) 2 (ÿ (ẋ2 + + ż 2) ẏ (ẋẍ + ż z) ) 2 + ( z ( ẋ 2 + ẏ 2) ż (ẋẍ + ẏÿ) ) 2 ) (63) (61) (62)

13 4 Frenet e 2 (s) k(s) k(s) k(s) κ(s) (64) f(t) (x(t), y(t)) ( ) ẏ e 2 (t) ẋ2 + ẏ, ẋ 2 ẋ2 (65) + ẏ 2 f(t) (x(t), y(t), z(t)) (66) e 1 s e 1 e 1 1 (67) de 1 e 1 0 (68) e 1 e 2 0 (69) e 1 e 2 e 3 e 1 e 2 (70) (e 1, e 2, e 3 ) Frenet e 2 e 3 12

14 4: Frenet 13

15 de 1 κ(s)e 2(s) (71) κ(s) de 2 e 3 τ(s) κ(s) de 1 e 2(s) (72) τ(s) de 2 e 3(s) (73) 14

16 6 Frenet-Serret 6.1 Frenet-Serret e 1 e 2 e 3 e 2 e 2 1 (74) e 3 e 3 1 (75) s e 1, e 2, e de 2 e 2 0 (76) de 3 e 3 0 (77) e 1 e 2 0 (78) e 2 e 3 0 (79) e 1 e 3 0 (80) e 1 de 2 de 1 e 2 (81) e 2 de 3 de 2 e 3 (82) e 1 de 3 de 1 e 3 (83) e 1 de 2 κ (84) de 2 κe 1 + τe 3 (85) e 2 de 3 τ (86) 15

17 de 1 e 3 0 (87) e 1 de 3 0 (88) de 3 τe 2 (89) de 1 κe 2 de 2 κe 1 + τe 3 (90) de 3 τe 2 Frenet-Serret Frenet-Serret 6.2 Frenet-Serret 26 e 1 (t) df (91) df e 1 (92) d 2 f d2 s 2 e d2 s 2 e 1 + ( ) de1 ( ) 2 de 1 (93) 16

18 e 1 e 1 0 (94) e 1 e 2 e 3 (95) df d2 f 2 ( ) 3 κe 3 (96) df f d2 2 3 κ (97) df e 1 df (98) κ df f d2 2 df 3 (99) f(t) f(x(t), y(t), z(t)) df(t) (ẋ, ẏ, ż) (100) d 2 f(t) 2 (ẍ, ÿ, z) (101) κ κ df f d2 (ẏ z żÿ, żẍ ẋ z, ẋÿ ẏẍ) (102) 2 (ẏ z żÿ) 2 + (żẍ ẋ z) 2 + (ẋÿ ẏẍ) 2 (ẋ 2 + ẏ 2 + ż 2 ) 3 2 (103) 17

19 (93) d 3 f 3 d3 s 3 e 1 + d2 s 2 de 1 d3 s 3 e 1 + (90) + 2 ( d 2 ( ) s 2 de ( ) 3 d 2 e 1 2 ) de1 + ( ) 3 d 2 e 1 2 (104) (90) d 2 e 1 2 de 1 κe 2 (105) dκ e 2 + κ de 2 (106) (105) (108) (104) de 2 κe 1 + τe 3 (107) d 2 e 1 2 dκ e 2 + κ ( κe 1 + τe 3 ) κ 2 e 1 + dκ e 2 + κτe 3 (108) d 3 ( f d3 s d 2 ( ) 3 e s ) ( ) 3 ( κe κ 2 e 1 + dκ ) e 2 + κτe 3 ( ) 3 ( ) d3 s κ 2 e d2 s κ + 2 ( ) 3 2 κ + dκ κe 2 ( ) 3 + κτe 3 (109) ( ) 3 κ d 3 f 3 d 3 f 3 e 3 ( ) 3 κe 3 ( ) 3 κτ (110) 18 ( ) 6 κ 2 τ (111)

20 (99) ( ) 6 κ 2 df 6 e 1 κ 2 df 6 κ 2 df f 2 d2 2 (112) (111) (96) (112) τ d 3 ( ) f df f 3 d2 2 df f 2 (113) d2 2 f(t) f(x(t), y(t), z(t)) τ (100) (102) (113) d 3 f 3 ( x (3), y (3), z (3)) (114) τ x(3) (ẏ z żÿ) + y (3) (żẍ ẋ z) + z (3) (ẋÿ ẏẍ) (ẏ z żÿ) 2 + (żẍ ẋ z) 2 + (ẋÿ ẏẍ) 2 (115) 19

21 7 Bouquet f(s) Maclaurin 26 f(s) f(0) + s df(0) + s2 d 2 f(0) + s3 d 3 f(0) + O(s 3 ) (116) 2! 2 3! 3 df(0) e 1 (0) (117) s 85 d 2 f(0) 2 κ(0)e 2 (0) (118) d 3 f(0) 3 κ(0) e 2(0) κ(0) 2 e 1 (0) + κ(0)τ(0)e 3 (0) (119) f(s) f(0) + se 1 (0) + s2 2! κ(0)e 2(0) ( ) + s3 κ(0) 3! e 2(0) κ(0) 2 e 1 (0) + κ(0)τ(0)e 3 (0) + O(s 3 ) ( ) ( f(0) + s κ(0) 2 s3 e 1 (0) + κ(0) s2 3! 2! + κ(0) s 3 ) e 2 (0) 3! ( ) + κ(0)τ(0) s3 e 3 (0) + O(s 3 ) (120) 3! Bouquet Bouquet 20

22 : a, b O (ellipse) x 2 a 2 + y2 b 2 1 (121) { x acost y bsint (122) a b (circle) 122 { ẋ a sin t ẏ b cos t (123) { ẍ a cos t ÿ b sin t (124) 39 21

23 κ (t) ( a sin t) ( b sin t) ( a cos t) (b cos t) ( a2 sin 2 t + b 2 cos 2 t ) 3/2 ab ( a2 sin 2 t + b 2 cos 2 t ) 3/2 (125) 22

24 : a (parabola) y ax 2 (126) dy dx 2ax (127) 42 k(x) d 2 y 2a (128) dx2 2a (1 + 4a 2 x 2 ) 3/2 (129) 23

25 : a, b O (hyperbola) x 2 a 2 y2 b 2 1 (130) { x ±acosht y bsinht (131) cosh(t) et + e t 2 sinh(t) et e t 2 (132) d (cosh(t)) sinh(t) d (133) (sinh(t)) cosh(t) { ẋ ±asinht ẏ bcosht (134) { ẍ ±acosht ÿ bsinht (135) 24

26 39 k(t) ±ab(sinh2 t cosh 2 t) (a 2 sinh 2 t + b 2 cosh 2 t) 3/2 (136) 136 sinh 2 t cosh 2 t 1 (137) k(t) ab (a 2 sinh 2 t + b 2 cosh 2 t) 3/2 (138) 25

27 : a (catenary) y a cosh x a (139) dy dx sinh x a (140) d 2 y dx 2 1 a cosh x a (141) 42 k(x) 1 a cosh x a ( ) 1 + sinh 2 x 3/2 (142) a 26

28 : r (cycloid) { x rθ rsinθ r(θ sinθ) y r rcosθ r(1 cosθ) (143) dx r(1 cosθ) dθ dy (144) dθ rsinθ d 2 x dθ 2 rsinθ d 2 y dθ 2 rcosθ (145) 39 k(θ) 1 2 3/2 r 1 cos θ (146) 27

29 : (involute) r CAB θ BC rθ α tan 1 ( rθ r ) tan 1 θ (147) AC R R (rθ) 2 + r 2 r 1 + θ 2 (148) { x Rsin(θ α) y Rcos(θ α) r (149) { x r 1 + θ2 sin(θ tan 1 θ) y r 1 + θ 2 cos(θ tan 1 θ) r (150) 150 A { R r 1 + θ 2 Θ θ tan 1 θ (151) 28

30 d ( tan 1 θ ) 1 (152) dθ 1 + θ 2 dr dθ rθ 1 + θ 2 dθ dθ 1 1 (153) 1 + θ 2 θ2 1 + θ 2 d 2 R dθ r 2 (1 + θ 2 ) 3/2 d 2 (154) Θ dθ 2θ 2 (1 + θ 2 ) k(θ) θ6 + 2θ 4 + θ 2 1 rθ 3 (θ 2 + 1) (155) 29

31 Z X Y 11: a (ordinary helix) a > 0 b > 0 x acost y asint z bt (156) ẋ asint ẏ acost ż b ẍ acost ÿ asint z 0 x (3) asint y (3) acost z (3) 0 (157) (158) (159) 30

32 κ (103) τ (115) κ τ a a 2 + b 2 (160) b a 2 + b 2 (161) 31

33 8.2.2 Z X Y 12: (conic helix) a > 0 b 0 x atcost y atsint z bt (162) ẋ acost atsint ẏ asint + atcost ż b (163) ẍ asint asint + atcost 2asint + atcost ÿ acost + acost atsint 2acost + atsint z 0 (164) 32

34 x (3) 2acost + acost atsint acost atsint y (3) 2asint + asint + atcost asint + atcost z (3) 0 (165) ẏ z żÿ 2abcost abtsint (166) żẍ ẋ z 2absint + abtcost (167) ẋÿ ẏẍ 2a 2 a 2 t 2 (168) (ẏ z żÿ) 2 + (żẍ ẋ z) 2 + (ẋÿ ẏẍ) 2 a 2 b 2 (4 + t 2 ) + a 4 (2 t 2 ) 2 (169) ẋ 2 + ẏ 2 + ż 2 a 2 (4 + t 2 ) + b 2 (170) x (3) (ẏ z żÿ) + y (3) (żẍ ẋ z) + z (3) (ẋÿ ẏẍ) a 2 b(1 + t 2 ) (171) κ (103) τ (115) κ a 2 b 2 (4 + t 2 ) + a 4 (2 t 2 ) 2 (a 2 (4 + t 2 ) + b 2 ) 3 2 (172) τ a 2 b(1 + t 2 ) a 2 (4 + t 2 ) + b 2 (173) 33

35 [1], 2001 [2], 2000 [3],, 1998 [4],,, 2001 [5],,, 1994 [6],,, 1985 [7],,,

36 2005/06/ /08/ /08/ /08/ /08/ /08/ /09/01 7 Frenet Frenet 2006/04/ /04/28 9 Frenet 2006/04/ /07/26 11 Frenet-Serret Frenet 2007/04/

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