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 x i d x i d t displaystyle dot x i dx i over dt Ochevidno sho vektor v r displaystyle mathbf v dot mathbf r u fizichnij interpretaciyi shvidkist tochki ye dotichnim do krivoyi Dovzhina krivoyi RedaguvatiDokladnishe Dovzhina krivoyiKvadrat vidstani mizh dvoma neskinchenno blizkimi tochkami r displaystyle mathbf r i r d r displaystyle mathbf r d mathbf r 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 Dovzhina vidrizka krivoyi koli parametr t displaystyle t probigaye znachennya vid t 1 displaystyle t 1 do t 2 displaystyle t 2 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 Yaksho v integrali 2 rozglyadati verhnyu mezhu yak zminnij parametr to mayemo funkciyu s s t displaystyle s s t viznachenu z tochnistyu do konstanti tochki vidliku abo nizhnoyi mezhi v integrali 2 Cya velichina s displaystyle s takozh parametrizuye tochki nashoyi krivoyi s displaystyle s nazivayetsya naturalnim parametrom krivoyi Yaksho vektor shvidkosti v r displaystyle mathbf v dot mathbf r nide ne peretvoryuyetsya v nul to pidintegralna funkciya v 2 dodatnya a otzhe funkciya s s t displaystyle s s t vsyudi monotonno zrostaye i maye obernenu funkciyu t t s displaystyle t t s Krivina krivoyi RedaguvatiIz rivnosti d s 2 d r d r displaystyle ds 2 d mathbf r cdot d mathbf r sliduye sho pohidna radius vektora za naturalnim parametrom krivoyi t d r d s displaystyle boldsymbol tau d mathbf r over ds ye dotichnim vektorom odinichnoyi dovzhini 3 t 2 t t 1 displaystyle 3 qquad boldsymbol tau 2 boldsymbol tau cdot boldsymbol tau 1 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 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 ortogonalnij do krivoyi Cej vektor prijnyato rozkladati na dobutok odinichnogo vektora n displaystyle mathbf n normali do krivoyi ta skalyara k displaystyle k yakij nazivayetsya krivinoyu k k n displaystyle mathbf k k mathbf n Geometrichnij zmist krivini RedaguvatiPokazhemo navit dvoma sposobami sho krivina dorivnyuye obernenij velichini do radiusa R displaystyle R dotichnogo kola 4 k 1 R displaystyle 4 qquad k 1 over R Pershij sposib cherez kut mizh dotichnimi vektorami odinichnoyi dovzhini v susidnih tochkah krivoyi Nehaj v tochci z parametrom s displaystyle s mayemo dotichnij vektor t displaystyle mathbf tau a v tochci z parametrom s s D s displaystyle s s Delta s dotichnij vektor t t D t displaystyle mathbf tau mathbf tau Delta mathbf tau 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 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 Oskilki dlya kola radiusa R displaystyle R mayemo D s R D a displaystyle Delta s R Delta alpha 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 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 Kolo radiusa R displaystyle R dotichne do vektora t displaystyle mathbf tau matime centr v ortogonalnij do t displaystyle mathbf tau giperploshini Zapishemo koordinati centra kola u viglyadi r c R n displaystyle mathbf r c R mathbf n de n displaystyle mathbf n ye dovilnim poki sho odinichnim vektorom sho lezhit u cij giperploshini Mayemo ortogonalnist n t 0 displaystyle mathbf n cdot boldsymbol tau 0 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 Vrahuyemo sho dovzhina dugi kola dorivnyuye s R t displaystyle s Rt 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 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 yaksho 8 k 1 R n displaystyle 8 qquad mathbf k 1 over R mathbf n 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 v topologichnij prostir Transcendentna krivaTipi tochok na krivij Redaguvati Tochka zlamu Tochka pereginuSkrut RedaguvatiYaksho evklidiv prostir maye rozmirnist n 3 displaystyle n geqslant 3 to mozhna postaviti pitannya pro zminu oriyentaciyi dotichnoyi ploshini v yakij lezhat dotichnij vektor t displaystyle mathbf tau ta vektor normali n displaystyle mathbf n pri rusi vzdovzh krivoyi Rozglyanemo bivektor specialnu antisimetrichnu matricyu komponenti yakoyi virazheni cherez koordinati vektoriv t displaystyle mathbf tau i n displaystyle mathbf n s t n displaystyle mathbf sigma mathbf tau wedge mathbf n s i j t i n j t j n i displaystyle sigma ij tau i n j tau j n i Velichina cogo bivektora dorivnyuye odinici ploshi kvadrata pobudovanogo na vektorah t displaystyle mathbf tau i n displaystyle mathbf n 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 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 Zvidsi robimo visnovok sho dvi ploshini s displaystyle boldsymbol sigma i s s D s displaystyle boldsymbol sigma boldsymbol sigma Delta boldsymbol sigma peretinayutsya po pryamij dotichnij do krivoyi mistyat vektor t displaystyle boldsymbol tau 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 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 Formuli Frene Serre RedaguvatiDokladnishe Trigrannik FreneRozglyanemo detalnishe vipadok krivoyi v trivimirnomu prostori Dva odinichni vektora t displaystyle boldsymbol tau i n displaystyle mathbf n mi mozhemo dopovniti tretim yih vektornim dobutkom f t n displaystyle mathbf f boldsymbol tau times mathbf n 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 n displaystyle dot mathbf n i f displaystyle dot mathbf f rozkladayutsya po comu zh bazisu Mi vzhe znayemo sho t k n displaystyle dot boldsymbol tau k mathbf n Zalishayetsya znajti pohidni she dvoh odinichnih vektoriv Pochnemo z odinichnogo vektora normali n displaystyle mathbf n 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 Tobto pohidna n displaystyle dot mathbf n ortogonalna do samogo vektora normali n displaystyle mathbf n 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 Koristuyuchis cim rozkladom mozhna znajti i pohidnu f displaystyle dot mathbf f 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 Znajdemo koeficiyenti rozkladu a displaystyle alpha i b displaystyle beta Z ostannoyi formuli vidno sho b displaystyle beta z tochnistyu do znaku ye shvidkistyu povorotu odinichnogo vektora f displaystyle mathbf f a otzhe i dotichnoyi do krivoyi ploshini f displaystyle mathbf f ye vektorom normali do ciyeyi ploshini Otzhe cej koeficiyent ye kruchennyam b ϰ displaystyle beta varkappa Koeficiyent a displaystyle alpha mozhna znajti skalyarno pomnozhivshi rivnist 9 na t displaystyle boldsymbol tau 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 U pidsumku oderzhuyemo sistemu troh rivnyan t k n displaystyle dot boldsymbol tau qquad k mathbf n n k t ϰ f displaystyle dot mathbf n k boldsymbol tau qquad varkappa mathbf f f ϰ n displaystyle qquad dot mathbf f qquad varkappa mathbf n Ci rivnyannya vidkrili dva francuzki matematiki Zhan Frederik Frene en 1852 i Zhozef Alfred Serre fr 1851 Koeficiyent ϰ displaystyle varkappa 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 nedostatno ilyustrovana Vi mozhete dopomogti proyektu dodavshi zobrazhennya do ciyeyi statti Otrimano z https uk wikipedia org w index php title Kriva amp oldid 36436680
