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Cursul – 11 Elemente de geometrie diferentiala a curbelor in spatiu

Cursul – 11 Elemente de geometrie diferentiala a curbelor in spatiu. Fie E 3 = ( E 3 , V 3 , ~ ) si R = ( O; i,j,k )- un reper ortonormat Definitia 1 . Numim arc regulat de curba in spatiu aplicatia bijectiva, continua si diferentiabila

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Cursul – 11 Elemente de geometrie diferentiala a curbelor in spatiu

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  1. Cursul – 11Elemente de geometrie diferentiala a curbelor in spatiu Fie E3 = ( E3 ,V3 ,~) si R = ( O;i,j,k)- un reper ortonormat Definitia 1. Numim arc regulat de curba in spatiu aplicatia bijectiva, continua si diferentiabila - puncte regulate (*), puncte singulare, curba in sens larg - reprezentari: Daca x,y,zCp(I) -  de clasa Cp(I) Definitia 2. Curbele (1) si (2) au in punctul comun Mo un contact de ordinul q p daca au q+1 puncte comune. q = 0 – curbe secante, q = 1 – curbe tangente, q = 2 – curbe osculatoare,.. supraosculatoare

  2. Teorema 1.Curbele (1) si (2) au in punctul Mo un contact de ordinul q daca si numai daca Teorema 2.Curbele (1)F(x,y) = 0 si ( 2) x = x(t), y=y(t) au in punctul Mo un cntact de ordinul q daca (t0) = ’(t0) =….= (q)(t0) = 0,  (q+1)( t0) = 0,unde (t0)= F(x(t) ,y(t) ). Aplicatii: 1o Tangenta la o curba: sau 2oCercul osculator: Fie ()x = x(t), y=y(t) si cercul

  3. Familii de curbe plane :(C) F(x,y,) = 0 Infasuratoarea unei familii de curbe plane; definitie Puncte singulare: Teorema 3.Multimea punctelor infasuratoarei familiei de curbe plane C satisfac sistemul: Aplicatie: evoluta unei curbe plane, evolventa Proprietati metrice Fie (  )

  4. Versorul tangent , respectiv dreapta tangenta intr-un punct regulat al curbei (  )Planul prin punct cu normala data de  se numesteplanul normal

  5. Curbura si torsiune Functia Teorema 3.Curba () este o dreapta daca si numai daca K(s) = 0. Teorema 4. Curba () este plana daca si numai daca T(s) = 0

  6. Formule de calcul pentru K si T

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