10.25.22LECTURE 26Definition For each t in the domain of r a vectorvalued function the unit tangentvectorto C tis thtTCH RICHHr t Ilwe also know that 11TH 11 1Definition if r is a smooth parameterization of CT is the unit tangent vector and T t 0the unit normal vector to C t noted Nct isN t T H this always points11T HII inthe inwards directionofconcavityexample Find unit tangent and unit normal vectorstothecurve r t Itt t2 1,0 Itangent r t 1 ZtHr CHIK TH TH TAK FEI EETnormal r t O 2Ur t 11 2 Nlt O 1 Observation if rls is an arc lengthparam ofC thenT s r SN s r SMr s 11Definition the binormal vector to C t isBA TH X NltBlt Nlt and T t make a mini moving coordsystemwhich is called the Ferret FrameTheorem B t r t x r tu r t X r t IlIf rls is the arc length parameterizationB s r S x r S11 r s Il14.4 Problems Lecture 26 Probs144Probleme3 TH YEA 44 thr t Zt 1 Fs Fs Not Yt 220 1.0711 r ro t tr prove that r r to ST tor ro t urdraw sftIlfull an turnt Fullr ro t s full because VedraI we know that TH Frithen substituting t to givesr r to ST to ALecture 26 Problems1 ret Att H THE 4 713 to101 1431 III2 ret tr Sint 2 cost 3 BltEffier t V3 cost sintr t I Sint cost r's cost sint it cost sin't Jf Boost s intiii n tsint cosI 654t 211 sin4t 3cost3 r t 2t et 372 tori o 0 1 07 Ir t 2 et 6 t 2 1 074 A
