U Vikipediyi ye statti pro inshi znachennya cogo termina Kriva znachennya Kriva liniya v evklidovomu prostori abo v mnogovidi Parabola odna z najprostishih krivihRivnyannya krivoyi mozhna zadavati v parametrichnij formi x i x i t displaystyle x i x i t de x i displaystyle x i koordinati tochok krivoyi v deyakij sistemi koordinat zadanij v evklidovomu prostori abo mnogovidi a t displaystyle t skalyarnij parametr jogo mozhna fizichno uyavlyati momentom chasu t time a samu krivu yak trayektoriyu ruhu tochki Rozglyanemo rivnyannya krivoyi v Dekartovij sistemi koordinat n displaystyle n vimirnogo evklidovogo prostoru Vvedemo poznachennya radius vektora tochki krivoyi r x 1 x 2 x n displaystyle mathbf r x 1 x 2 x n Zmist 1 Dotichnij vektor 2 Dovzhina krivoyi 3 Krivina krivoyi 4 Geometrichnij zmist krivini 5 Tipi krivih 5 1 Tipi tochok na krivij 6 Skrut 7 Formuli Frene Serre 8 Div takozhDotichnij vektor RedaguvatiPohidnu za parametrom poznachatimemo krapkoyu zverhu r d r d t displaystyle dot mathbf r d mathbf r over dt nbsp x i d x i d t displaystyle dot x i dx i over dt nbsp Ochevidno sho vektor v r displaystyle mathbf v dot mathbf r nbsp u fizichnij interpretaciyi shvidkist tochki ye dotichnim do krivoyi Dovzhina krivoyi RedaguvatiDokladnishe Dovzhina krivoyiKvadrat vidstani mizh dvoma neskinchenno blizkimi tochkami r displaystyle mathbf r nbsp i r d r displaystyle mathbf r d mathbf r nbsp dorivnyuye 1 d s 2 d r d r i d x i 2 i d x i 2 d t 2 displaystyle 1 qquad ds 2 d mathbf r cdot d mathbf r sum i dx i 2 sum i d dot x i 2 dt 2 nbsp Dovzhina vidrizka krivoyi koli parametr t displaystyle t nbsp probigaye znachennya vid t 1 displaystyle t 1 nbsp do t 2 displaystyle t 2 nbsp dayetsya integralom 2 s t 1 t 2 d s t 1 t 2 x i x i d t displaystyle 2 qquad s int t 1 t 2 ds int t 1 t 2 sqrt dot x i dot x i dt nbsp Yaksho v integrali 2 rozglyadati verhnyu mezhu yak zminnij parametr to mayemo funkciyu s s t displaystyle s s t nbsp viznachenu z tochnistyu do konstanti tochki vidliku abo nizhnoyi mezhi v integrali 2 Cya velichina s displaystyle s nbsp takozh parametrizuye tochki nashoyi krivoyi s displaystyle s nbsp nazivayetsya naturalnim parametrom krivoyi Yaksho vektor shvidkosti v r displaystyle mathbf v dot mathbf r nbsp nide ne peretvoryuyetsya v nul to pidintegralna funkciya v 2 dodatnya a otzhe funkciya s s t displaystyle s s t nbsp vsyudi monotonno zrostaye i maye obernenu funkciyu t t s displaystyle t t s nbsp Krivina krivoyi RedaguvatiIz rivnosti d s 2 d r d r displaystyle ds 2 d mathbf r cdot d mathbf r nbsp sliduye sho pohidna radius vektora za naturalnim parametrom krivoyi t d r d s displaystyle boldsymbol tau d mathbf r over ds nbsp ye dotichnim vektorom odinichnoyi dovzhini 3 t 2 t t 1 displaystyle 3 qquad boldsymbol tau 2 boldsymbol tau cdot boldsymbol tau 1 nbsp Diferenciyuyuchi 3 za naturalnim parametrom mayemo d d s t t 2 t d t d s 0 displaystyle d over ds boldsymbol tau cdot boldsymbol tau 2 boldsymbol tau cdot d boldsymbol tau over ds 0 nbsp Otzhe vektor k d t d s d 2 r d s 2 displaystyle mathbf k d boldsymbol tau over ds d 2 mathbf r over ds 2 nbsp ortogonalnij do krivoyi Cej vektor prijnyato rozkladati na dobutok odinichnogo vektora n displaystyle mathbf n nbsp normali do krivoyi ta skalyara k displaystyle k nbsp yakij nazivayetsya krivinoyu k k n displaystyle mathbf k k mathbf n nbsp Geometrichnij zmist krivini RedaguvatiPokazhemo navit dvoma sposobami sho krivina dorivnyuye obernenij velichini do radiusa R displaystyle R nbsp dotichnogo kola 4 k 1 R displaystyle 4 qquad k 1 over R nbsp Pershij sposib cherez kut mizh dotichnimi vektorami odinichnoyi dovzhini v susidnih tochkah krivoyi Nehaj v tochci z parametrom s displaystyle s nbsp mayemo dotichnij vektor t displaystyle mathbf tau nbsp a v tochci z parametrom s s D s displaystyle s s Delta s nbsp dotichnij vektor t t D t displaystyle mathbf tau mathbf tau Delta mathbf tau nbsp Ci dva vektora mayut odnakovu dovzhinu odinicyu i yaksho yihni pochatki zvesti v odnu tochku utvoryat rivnobedrenij trikutnik Yaksho kut mizh vektorami poznachiti D a displaystyle Delta alpha nbsp to dovzhina tretoyi storoni bude dorivnyuvati D t 2 sin D a 2 D a displaystyle Delta boldsymbol tau 2 sin Delta alpha over 2 approx Delta alpha nbsp Oskilki dlya kola radiusa R displaystyle R nbsp mayemo D s R D a displaystyle Delta s R Delta alpha nbsp to mayemo dlya krivini krivoyi k d t d s D t D s D a R D a 1 R displaystyle k d boldsymbol tau over ds approx Delta boldsymbol tau over Delta s Delta alpha over R Delta alpha 1 over R nbsp Drugij sposib cherez rivnyannya kola Dlya prostoti formul vizmemo pochatok koordinat evklidovogo prostoru v tochci krivoyi dlya yakoyi mi budemo shukati najblizhche kolo a takozh budemo vidrahovuvati naturalni parametri krivoyi i kola vid ciyeyi zh tochki Z tochnistyu do chleniv drugogo poryadku malosti mayemo dlya tochok krivoyi 5 r d r d s s 1 2 d 2 r d s 2 s 2 t s 1 2 k s 2 displaystyle 5 qquad mathbf r approx d mathbf r over ds s begin matrix frac 1 2 end matrix d 2 mathbf r over ds 2 s 2 mathbf tau s begin matrix frac 1 2 end matrix mathbf k s 2 nbsp Kolo radiusa R displaystyle R nbsp dotichne do vektora t displaystyle mathbf tau nbsp matime centr v ortogonalnij do t displaystyle mathbf tau nbsp giperploshini Zapishemo koordinati centra kola u viglyadi r c R n displaystyle mathbf r c R mathbf n nbsp de n displaystyle mathbf n nbsp ye dovilnim poki sho odinichnim vektorom sho lezhit u cij giperploshini Mayemo ortogonalnist n t 0 displaystyle mathbf n cdot boldsymbol tau 0 nbsp Rivnyannya tochki kola v parametrichnij formi parametrom ye centralnij kut 6 r R sin t t R 1 cos t n displaystyle 6 qquad mathbf r R sin t boldsymbol tau R 1 cos t mathbf n nbsp Vrahuyemo sho dovzhina dugi kola dorivnyuye s R t displaystyle s Rt nbsp i rozklademo ostannye rivnyannya v ryad z tochnistyu do dodankiv drugogo poryadku malosti 7 r R t t 1 2 R t 2 n t s 1 2 R n s 2 displaystyle 7 qquad mathbf r approx Rt boldsymbol tau begin matrix frac 1 2 end matrix Rt 2 mathbf n boldsymbol tau s 1 over 2R mathbf n s 2 nbsp Porivnyuyuchi rivnosti 5 i 7 mayemo sho kolo bude zbigatisya z krivoyu z tochnistyu do chleniv drugogo poryadku r r displaystyle mathbf r approx mathbf r nbsp yaksho 8 k 1 R n displaystyle 8 qquad mathbf k 1 over R mathbf n nbsp Tipi krivih RedaguvatiZamknuta kriva kriva u yakoyi pochatok zbigayetsya z kincem Ploska kriva kriva vsi tochki yakoyi lezhat v odnij ploshini Prosta kriva te same sho kriva Zhordana Shlyah neperervne vidobrazhennya vidrizka 0 1 displaystyle 0 1 nbsp v topologichnij prostir Transcendentna krivaTipi tochok na krivij Redaguvati Tochka zlamu Tochka pereginuSkrut RedaguvatiYaksho evklidiv prostir maye rozmirnist n 3 displaystyle n geqslant 3 nbsp to mozhna postaviti pitannya pro zminu oriyentaciyi dotichnoyi ploshini v yakij lezhat dotichnij vektor t displaystyle mathbf tau nbsp ta vektor normali n displaystyle mathbf n nbsp pri rusi vzdovzh krivoyi Rozglyanemo bivektor specialnu antisimetrichnu matricyu komponenti yakoyi virazheni cherez koordinati vektoriv t displaystyle mathbf tau nbsp i n displaystyle mathbf n nbsp s t n displaystyle mathbf sigma mathbf tau wedge mathbf n nbsp s i j t i n j t j n i displaystyle sigma ij tau i n j tau j n i nbsp Velichina cogo bivektora dorivnyuye odinici ploshi kvadrata pobudovanogo na vektorah t displaystyle mathbf tau nbsp i n displaystyle mathbf n nbsp i lt j s i j 2 1 2 i j t i n j t j n i 2 1 2 i j t i 2 n j 2 t j 2 n i 2 2 t i n i t j n j t t n n t n 2 1 displaystyle sum i lt j sigma ij 2 1 over 2 sum i j tau i n j tau j n i 2 1 over 2 sum i j tau i 2 n j 2 tau j 2 n i 2 2 tau i n i tau j n j boldsymbol tau cdot boldsymbol tau mathbf n cdot mathbf n boldsymbol tau cdot mathbf n 2 1 nbsp Pohidna bivektora za naturalnim parametrom dorivnyuye s t n t n k n n t n t n displaystyle dot boldsymbol sigma dot boldsymbol tau wedge mathbf n boldsymbol tau wedge dot mathbf n k mathbf n wedge mathbf n boldsymbol tau wedge dot mathbf n boldsymbol tau wedge dot mathbf n nbsp Zvidsi robimo visnovok sho dvi ploshini s displaystyle boldsymbol sigma nbsp i s s D s displaystyle boldsymbol sigma boldsymbol sigma Delta boldsymbol sigma nbsp peretinayutsya po pryamij dotichnij do krivoyi mistyat vektor t displaystyle boldsymbol tau nbsp s t n t n D s t n n D s displaystyle boldsymbol sigma boldsymbol tau wedge mathbf n boldsymbol tau wedge dot mathbf n Delta s boldsymbol tau wedge mathbf n dot mathbf n Delta s nbsp Otzhe dotichna ploshina pri rusi vzdovzh krivoyi obertayetsya dovkola dotichnoyi pryamoyi Povorot v trivimirnomu prostori maye ochevidnij zmist v prostorah bilshoyi rozmirnosti povorot oznachaye kut mizh normalyami do spilnoyi pryamoyi Pohidna kuta povorotu za naturalnim parametrom nazivayetsya skrutom ϰ d ϕ d s t n displaystyle varkappa d phi over ds boldsymbol tau wedge dot mathbf n nbsp Formuli Frene Serre RedaguvatiDokladnishe Trigrannik FreneRozglyanemo detalnishe vipadok krivoyi v trivimirnomu prostori Dva odinichni vektora t displaystyle boldsymbol tau nbsp i n displaystyle mathbf n nbsp mi mozhemo dopovniti tretim yih vektornim dobutkom f t n displaystyle mathbf f boldsymbol tau times mathbf n nbsp Ci tri vektori utvoryuyut reper zminnij bazis u trivimirnomu prostori i mi mozhemo postaviti pitannya yak pohidni za naturalnim parametrom vid vektoriv repera t displaystyle dot boldsymbol tau nbsp n displaystyle dot mathbf n nbsp i f displaystyle dot mathbf f nbsp rozkladayutsya po comu zh bazisu Mi vzhe znayemo sho t k n displaystyle dot boldsymbol tau k mathbf n nbsp Zalishayetsya znajti pohidni she dvoh odinichnih vektoriv Pochnemo z odinichnogo vektora normali n displaystyle mathbf n nbsp Iz postijnosti velichini cogo vektora znahodimo 0 d d s n n 2 n n displaystyle 0 d over ds mathbf n cdot mathbf n 2 mathbf n cdot dot mathbf n nbsp Tobto pohidna n displaystyle dot mathbf n nbsp ortogonalna do samogo vektora normali n displaystyle mathbf n nbsp a tomu rozkladayetsya po dvom inshim vektoram repera 9 n a t b f displaystyle 9 qquad dot mathbf n alpha boldsymbol tau beta mathbf f nbsp Koristuyuchis cim rozkladom mozhna znajti i pohidnu f displaystyle dot mathbf f nbsp f d d s t n t n t n k n n t a t b f b t f b n displaystyle dot mathbf f d over ds boldsymbol tau times mathbf n dot boldsymbol tau times mathbf n mathbf tau times dot mathbf n k mathbf n times mathbf n mathbf tau times alpha mathbf tau beta mathbf f beta boldsymbol tau times mathbf f beta mathbf n nbsp Znajdemo koeficiyenti rozkladu a displaystyle alpha nbsp i b displaystyle beta nbsp Z ostannoyi formuli vidno sho b displaystyle beta nbsp z tochnistyu do znaku ye shvidkistyu povorotu odinichnogo vektora f displaystyle mathbf f nbsp a otzhe i dotichnoyi do krivoyi ploshini f displaystyle mathbf f nbsp ye vektorom normali do ciyeyi ploshini Otzhe cej koeficiyent ye kruchennyam b ϰ displaystyle beta varkappa nbsp Koeficiyent a displaystyle alpha nbsp mozhna znajti skalyarno pomnozhivshi rivnist 9 na t displaystyle boldsymbol tau nbsp a t n d d s t n t n k n n k displaystyle alpha boldsymbol tau cdot dot mathbf n d over ds boldsymbol tau cdot mathbf n dot boldsymbol tau cdot mathbf n k mathbf n cdot mathbf n k nbsp U pidsumku oderzhuyemo sistemu troh rivnyan t k n displaystyle dot boldsymbol tau qquad k mathbf n nbsp n k t ϰ f displaystyle dot mathbf n k boldsymbol tau qquad varkappa mathbf f nbsp f ϰ n displaystyle qquad dot mathbf f qquad varkappa mathbf n nbsp Ci rivnyannya vidkrili dva francuzki matematiki Zhan Frederik Frene en 1852 i Zhozef Alfred Serre fr 1851 Koeficiyent ϰ displaystyle varkappa nbsp u formulah Frene Serre mozhe buti dodatnim abo vid yemnim v zalezhnosti vid togo pravoyu chi livoyu gvintovoyu liniyeyu aproksimuyetsya kriva v okoli danoyi tochki Div takozh RedaguvatiLiniya Algebrichna krivaCya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno zhovten 2023 Otrimano z https uk wikipedia org w index php title Kriva amp oldid 40694227
