Cej termin maye takozh inshi znachennya Dokladnishe u statti Skrut U diferencialnij geometriyi skrut krivoyi angl torsion of a curve ce kilkisna mira vidhilennya krivoyi vid stichnoyi ploshini Takim chinom skrut vkazuye naskilki kriva vidriznyayetsya vid formi ploskoyi krivoyi Dlya ploskoyi krivoyi skrut dorivnyuye nulyu Koli skrut krivoyi ye miroyu vidhilennya vid ploshini to krivina krivoyi ye miroyu vidhilennya vid pryamoyi Zmist 1 Viznachennya 2 Geometrichnij zmist absolyutnogo skrutu j znaka skrutu 2 1 Skrut krivoyi v dovilnij parametrizaciyi 2 2 Zauvazhennya 3 Priklad 4 Primitki 5 LiteraturaViznachennya Redaguvati nbsp Nehaj P displaystyle P nbsp dovilna tochka regulyarnoyi krivoyi g displaystyle gamma nbsp Q displaystyle Q nbsp tochka krivoyi sho blizka do P displaystyle P nbsp Poznachimo cherez D a displaystyle Delta alpha nbsp kut mizh stichnimi ploshinami krivoyi v tochkah P displaystyle P nbsp ta Q displaystyle Q nbsp a cherez D s displaystyle Delta s nbsp dovzhinu dugi P Q displaystyle PQ nbsp krivoyi Todi lim Q P D a D s displaystyle lim Q to P frac Delta alpha Delta s nbsp yaksho vin isnuye nazivayetsya absolyutnim skrutom krivoyi g displaystyle gamma nbsp v tochci P displaystyle P nbsp i poznachayetsya cherez k 2 displaystyle k 2 nbsp 1 Geometrichnij zmist absolyutnogo skrutu j znaka skrutu RedaguvatiAbsolyutnij skrut krivoyi v tochci P displaystyle P nbsp dorivnyuye kutovij shvidkosti obertannya binormali krivoyi navkolo tochki Q displaystyle Q nbsp tobto k 2 lim D s 0 D a D s displaystyle k 2 lim Delta s to 0 frac Delta alpha Delta s nbsp de D a displaystyle Delta alpha nbsp kut povorotu binormali sho vidpovidaye prirostu dovzhini dugi D s displaystyle Delta s nbsp Skrut bude dodatnim vid yemnim yaksho pri sposterezhenni z kincya vektora shvidkosti vektor binormali pri rusi tochki po krivij obertayetsya proti po godinnikovoyi strilki Teorema Nehaj g displaystyle gamma nbsp regulyarna kriva klasu C 3 displaystyle C 3 nbsp Todi v kozhnij tochci krivoyi v yakij krivina k 1 0 displaystyle k 1 neq 0 nbsp viznachenij absolyutnij skrut k 2 displaystyle k 2 nbsp Yaksho r r s displaystyle bar r bar r s nbsp naturalna parametrizaciya krivoyi to k 2 b s r s r s r s k 1 2 displaystyle k 2 bar beta s frac bar r s bar r s bar r s k 1 2 nbsp de b s displaystyle bar beta s nbsp vektor funkciya odinichnih binormalej krivoyi g displaystyle gamma nbsp Dovedennya Rozglyanemo vlastivosti vektora b displaystyle bar beta nbsp b b displaystyle bar beta bot bar beta nbsp bo b displaystyle bar beta nbsp odinichnij vektor otzhe b 2 c o n s t displaystyle bar beta 2 const nbsp 2 b b 0 displaystyle 2 bar beta bar beta 0 nbsp b t displaystyle bar beta bot bar tau nbsp oskilki b t n displaystyle bar beta bar tau bar nu nbsp z pershoyi formuli Frene d t d s k 1 n displaystyle frac d bar tau ds k 1 bar nu nbsp i d b d s d t d s n t d n d s t n displaystyle frac d bar beta ds frac d bar tau ds bar nu bar tau frac d bar nu ds bar tau bar nu nbsp Tut t n displaystyle bar tau bar nu nbsp poznadayut vidpovidno odinichni dotichnij i normalnij vektori k 1 displaystyle k 1 nbsp krivinu krivoyi u vidpovidnij tochci d b d s k 2 n displaystyle frac d bar beta ds k 2 bar nu nbsp tretya formula Frene Takim chinom k 2 d b d s displaystyle k 2 frac d bar beta ds nbsp Znajdemo teper d b d s displaystyle frac d bar beta ds nbsp b k 2 n displaystyle bar beta k 2 bar nu nbsp b n k 2 n 2 displaystyle bar beta cdot bar nu k 2 bar nu 2 nbsp abo k 2 b n 2 displaystyle k 2 bar beta cdot nu 2 nbsp Vrahovuyuchi vlastivist 2 ta pershu formulu Frene i rozglyadayuchi krivinu k displaystyle k nbsp yak funkciyu s displaystyle s nbsp mayemo k 2 t n n r 1 k 1 r 1 k 1 r 1 k 1 3 r r k 1 r k 1 r 1 k 1 3 r r k 1 r r r r k 1 r r r k 1 2 displaystyle k 2 bar tau bar nu cdot bar nu bar r frac 1 k 1 bar r cdot frac 1 k 1 bar r frac 1 k 1 3 bar r bar r k 1 bar r k 1 bar r frac 1 k 1 3 bar r bar r k 1 bar r bar r bar r bar r k 1 frac bar r bar r bar r k 1 2 nbsp Otzhe k 2 r s r s r s k 1 2 displaystyle k 2 frac bar r s bar r s bar r s k 1 2 nbsp Skrut krivoyi v dovilnij parametrizaciyi Redaguvati Nehaj r r t displaystyle bar r bar r t nbsp regulyarna parametrizaciya krivoyi g displaystyle gamma nbsp r C 3 displaystyle bar r in mathbb C 3 nbsp Todi k 2 r t r t r t r t r t 2 displaystyle k 2 frac bar r t bar r t bar r t bar r t bar r t 2 nbsp absolyutnij skrut v dovilnij parametrizaciyi Dlya F x y z 0 displaystyle F x y z 0 nbsp skrut krivoyi obchislyuyetsya za formuloyu k 2 x y z y z y x z x z z x y x y y z y z 2 x z x z 2 x y x y 2 displaystyle k 2 frac x y z y z y x z x z z x y x y y z y z 2 x z x z 2 x y x y 2 nbsp Zauvazhennya Redaguvati Yaksho skrut krivoyi dorivnyuye nulyu k 2 0 displaystyle k 2 0 nbsp to kriva ploska Priklad RedaguvatiObchislimo skrut gvintovoyi liniyi x t a cos t y t a sin t z t b t displaystyle begin cases x t a cdot cos t y t a cdot sin t z t b cdot t end cases nbsp Oskilki r t a sin t a cos t b displaystyle bar r t a cdot sin t a cdot cos t b nbsp r t a cos t a sin t 0 displaystyle bar r t a cdot cos t a cdot sin t 0 nbsp r t a sin t a cos t 0 displaystyle bar r t a cdot sin t a cdot cos t 0 nbsp to lt r r gt a 2 b 2 displaystyle lt bar r bar r gt a 2 b 2 nbsp lt r r gt a 2 displaystyle lt bar r bar r gt a 2 nbsp lt r r gt 0 displaystyle lt bar r bar r gt 0 nbsp r r 2 r 2 r 2 lt r r gt 2 a 2 b 2 a 2 displaystyle bar r bar r 2 bar r 2 bar r 2 lt bar r bar r gt 2 a 2 b 2 a 2 nbsp r r r b a 2 displaystyle bar r bar r bar r ba 2 nbsp Todi k 2 b a 2 b 2 displaystyle k 2 frac b a 2 b 2 nbsp Primitki Redaguvati Borisenko 36 Literatura RedaguvatiBorisenko O A Diferencialna geometriya i topologiya Navch posibnik dlya stud Harkiv Osnova 1995 S 41 46 ISBN 5 7768 0388 8 Arhivovano z dzherela 23 sichnya 2022 Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Otrimano z https uk wikipedia org w index php title Skrut krivoyi amp oldid 40618218
