The Frenet Serret equations | Differential Geometry 18 | NJ Wildberger

so although everyone Norman Wahlburgers we're  here at the University of New South Wales today   we're going to talk about the frente Saraya  equations which tell us about curves in space   and extend our notion of curvature to another  related concept called torsion so I'm going to   start with the planar case to motivate things  we were talking about this unit circle at the   end of the last lecture there wasn't a unit  circle was a circle of radius a and the curve   going around it parametrized by essentially  the angle which we're going to call T because   we like to think of a parameters being time and  here was the formula for the curve itself alpha   of T a Coastie and a sine T and here was its  derivative alpha Prime and we noted that the   size of the derivative was equal to a so this  is a constant speed curve but not unit speed   unless April's one now it's often nice for  us to try to repr ammeter eyes the curve so   we want to reaper ammeter eyes by introducing  or pred using use arc length s as a parameter and our arc length is defined by the formula that  is the integral from over the curve or whatever   part of the curve that we're considering of the  Alpha prime of T DT so in this case in the circle   case the arc length at a certain point which we  can denote by s there that represents the arc   length just the length of the curve from wherever  we started let's say we starting here to the point   where we're at so in this case s is just equal  to well the integral over the of a DT so that's a   times T so we can reap remet rise in this case  very simply just by replacing T with s over a all right so what would happen there so we  would get a slightly different curve not much   different but it's just slightly we normalize  and I'll generally tend to use beta to signify   that we're talking about a unit speed curve so  beta of s at least in the plane is going to be   a cosine s over a and a sine s over a just  replacing the T with s over a and then we   can calculate its derivative B prime of s will  be a will be minus sine s over a comma cosine   of s over a because the a is cancelled and then  we can check that in this case yes the length   of the derivative B prime of s is indeed 1 for  all s so this is indeed a unit speed curve its   velocity has is has length 1 at all times  and the acceleration beta double prime of   s one more derivative we're going to get minus  one over a cosine s over a and minus one or a   sine s over a and that's a vector pointing in  the opposite direction from where we started so maybe our tangent vector up a prime of  s and then beta double prime of s will be   in this direction okay and it's speed  is or its length is the length of beta   double prime of s is well I'm still just  the squared plus this square the coasts   squares and sine Squared's will still add  up to one and we'll just get square root of   one over a squared which is one over a which  is in this case the curvature of this circle what we usually call inkay so we see  that for a circular motion the if it's   unit speed the unit speed circular motion  then the length of the tangent vector is   1 because it's Union speed but the length of  the acceleration the second derivative will   be the curvature I should write curvature K so  this motivates us to think of the size of the acceleration vector as being how  we can define curvature in a more   general setting so we use this to  define curvature for more general curves in particular curves in space which  is what we're going to be talking about today   well before I do that let me just write down  one more formula formula for the normalized   acceleration vector what we get when we normalize  by dividing by the length so if we normalize and   write say n of s which is going to be beta  double prime of s over the length of beta   double prime of s then that's going to be well  we're just dividing by the what array so that's   going to be just minus cosine s over a minus  sine s over a that's going to be unit normal so on our picture well we don't know what a is you know where  where the one is in this picture but let's   say suppose that the one happened to be obvious  let's say say there's one for example then that   would maybe roughly be that's supposed to be our  tangent vector with length one sorry that case   then the normal vector which we've renormalized  beta double prime of s renormalize it so that   it has length one well it'll be something like  like this so there will be n of s pointing in   the same direction as beta double prime but just  of length one so now we have these two vectors   they are perpendicular in this case and they're  both of length 1 this is the normal vector this   is a tangent vector here and we're going to call  it T of s and this is the normal vector n of s all right so we now want to take this motivating  example and start thinking about curves in space   in particular we're interested in say a curve  something like this okay there's a curve in   space we're just analyzing it okay so as we move  along the curve we have to choose a direction so   let's say we choose this direction here so any  point we can talk about the tangent vector will   be in the direction of the curve clearly and of  course the the speed of the curve the velocity   will depend on how fast we're going but we want  to perhaps think about a unit speed curve for it   to make things sort of simple if we think about  a unit speed curve and say this is a unit vector   then our tangent vector is going to always be the  same size and it's just going to be exactly like   this kind of vector moving steadily along at the  same pace all the way around but it's moving and   what we're interested in doing when we're studying  the furnace or a equations is thinking about not   just the tangent vector but also a normal vector a  unit normal vector which is playing just the same   role as this one is which is in the direction  of the acceleration and then in addition is the   right-handed frame a third vector called the by  normal vector which is going to together with the   normal no deal with the tangent and the normal  vector the by normal vector is going to form a   right-handed system of three unit vectors like I  J and K okay and what's going to happen is that   as we Traverse whatever curve we have okay we all  go over here let's suppose that we were here okay   so I have to worry into myself so that the blue  is going in the direction of the of the tangent   and the edie one would be in the direction of  the normal and then the bye-bye normal would   just be perpendicular to it and we're going to  watch these three vectors move along as we go   along the curve so we're going to have is these  these three vectors are going to be moving in   space and they're going to be twisting as well  as their and as going along the curve and the   sprint a survey equations tell us how these three  vectors are changing give us very precise formulas   involving the curvature and this other quantity  called the torsion that allow us to complete   the control what's happening in this unit speed  case all right so that's what we're going to do and before I get into the real business  I just need a little bit of preliminaries   so just some differentiations of vector quantities okay so we're going to be thinking about  having vectors which depend on time so   I'll write R of T but most of the time  I would thinking of a vector R of T and   typically it has components X of T say Y  of T is n of T and if we have some other vector valued function s of T which is also  officially a vector maybe it'll have components   X prime of T X Prime's not good maybe X hat  of T Y hat of T is that hat of T then we can   perform two standard things with these two vector  valued functions we can first of all define the   dot product between them so R of T dot s of T is  just the obvious thing X of T X hat of T plus y   of T Y hat of T plus set of T zet hat of T and  that's actually not a number which is actually   a function of T because the dot product of two  vectors is a number so that's actually a single   valued function of T maybe we'll call it F of  T and another thing that we can do of course is   we take the the cross product between these two  vectors so R of T cross s of T equals okay well   it's a vector which is going to be Y of T Z hat  of t minus y hat of T zmt that'll be the first   component and then the other ones sort of pre  muted in the usual fashion okay maybe I won't   write it down so this is a house the function  function of that variable T this is a vector   valued function of variable T and what we're  interested in is what we get very useful for   us is how are we differentiate these two things  let me call this one say U of T which is really   also a vector all right so we have some useful  lemma so in this context here if I want to take   F prime of T ok so that's our dot s prime really  then the formula is that it's R prime of T dot   s of T plus R of T dot s prime of T where the  primes are always derivatives with respect to   T that's the first one and the related formula  for the cross product is if we take this vector   valued function U and we differentiate it now  that means differentiating in every component   okay then what we're going to get is so this  is like R cross s Prime that's what we're   doing then we're going to get our prime of T  cross s of T plus R of T cross s prime of T so it's just like the ordinary product  rule that we know and for functions of   one variable but extended to this  sort of vector case and the proof   is just simple computation where you just  expand in coordinates and use the usual the usual product rule so as a consequence of this  a corollary that's going to be very useful for   us is that if we have the situation where the  quadrants of a vector field is constant okay   by which I mean ie that are T dot R T equals  C which is constant for all T or equivalently   if you want to take the square root you can  just say it's the same thing as saying that   the length of R of T is is constant then then R  prime of T is perpendicular to R of T for all T and the proof is an immediate consequence of  this first formula we just take the equation   R of T dot R of T equals C and apply this  formula so we apply the lemma and we get   that R prime of T dot R of T plus R of T dot R  prime of T equals the derivative of C which is   zero because it's a constant and so since these  two things are actually the same because it's a   symmetric bilinear form therefore R prime  of T dot R of T equals zero and that tells   us that our prime of T is perpendicular to R of  T ok so it's a simple but useful little formula all right so we're going to  be considering space curves space curve perhaps alpha of T with  coordinates X of T Y of T and Z of T   and we make a definition and we say that  this curve is regular alpha is regular if alpha prime of T is not equal to 0 for all  T if the velocity doesn't actually ever become   zero the thing keeps moving ok that's going to  be a useful condition because we're going to be   dividing by by speeds occasionally so another  definition that if alpha is a regular then   we'll define T of T will be the derivative alpha  prime of T all over alpha prime of T so it's the   derivative divided by the length of the derivative  so this is a it's called the unit tangent vector okay now we've said that we often want to reap  Rama tries so if we do remote eyes using the arc   length so we have some curve that's doing  something in space alfe of tea and the tea   can be going faster and slower and doing very  whatever it wants to but then we want to reap   Rama tries it by just using the arc length so  that's one two three four just lengths starting   from wherever we're starting then that's going  that new parameterization is called arc length   okay and so we're going to reap Rama tries using  arc length we're going to usually call that s to   get a curve that will write beta of s so it looks  slightly different because we've renormalized it   and using s instead of T so this is now a unit  speed a unit speed curve so then well then the   length of its tangent vector is going to be one  and then this unit tangent vector is just beta   prime itself we don't have to divide by the  length because it's already of length of one all right so we've done that we've renormalized  we now have this unit tangent vector so it'll be   a tangent vector of length one at every point  there's a little vector of length one and at   some general point s that vector is going to  be called t of s okay so next thing we want   to do is we want to think about the derivative  so then from our corollary made a double prime   of s which is the same as T prime of s is  perpendicular to T of s alright so our lemma   says that if we have a vector function which  always has the same length in this case T of   s always has length 1 then it's derivative  is going to be perpendicular to itself ok so motivated by the case of the unit circle  where the length of this derivative was   the curvature we're going to motivate we  define the curvature define the curvature okay of s to be the length of this derivative of the tangent vector so K of s is  the length of t prime of s so in the case of a   circle of radius a that was 1 over a that was  really the curvature and this allows us to to define this more generally so let's have a  look in this situation here what are these   derivatives going to look like so over here the  curve is not bending very much so the derivative   of this T of s the rate of change of the TV  s is going to be in this direction but it's   going to be relatively small okay so here the  the vector I guess where I guess we're talking   about now we're now calling the current beta of  s so let's just stick with arc length so we have   T of s are the blue ones and then we have this  perp the derivative t prime of s okay which is   always going to be perpendicular to the tangent  vector but may have different lengths so here   for example the curve is curving more so we're  going to expect a bigger derivative t prime of s over here somewhere still have these unit  tangent vectors here there might be quite   quite a big t prime of s okay that's just the  acceleration of the the curve and do the size   of that acceleration is the curvature okay that's  good and now we're in a position to renormalize   to get a unit normal vector because remember we  want this sort of tripod of three unit vectors we   already have a unit tangent vector we now want  a unit normal vector so far we have a normal   vector but it's of different lengths depending on  the curvature so we're just going to renormalize so renormalize to define and of s to be this well beta double prime of s over its  length which the same thing as T prime of   s over its length and its length is the curvature okay do I have a little bit of red here yes  I have a little bit of red okay so now I can   renormalize yes we need like an extra position  like that a nationally regular as well for us   to be insured that unit won't exist yes so we  require that this curvature be nonzero because   we're dividing by this curvature so this  curvature becomes zero at some point well   then it's problematic okay absolutely right  all right so in this red here I'm going to   now show you the the normalized vectors and  they're all going to be the same unit length   we're saying that's roughly one so here they  are they're all going to be perpendicular to   the tangent vectors so these red these are the  new vectors that we're calling n of s yes yes   because what we can well we take given any  curve whatsoever we can just compute we can   well we can write down an integral that allows  us to calculate at least numerically the arc   length yeah and so basically finding a unit speed  parameterization is roughly equivalent to being   able to integrate the length of the of the curve  now you know in practice I'm a maybe I should say   most actual polynomial curves or occurs they  come up in space it's very hard to explicitly   give formulas for such a Reaper ammeter ization  exactly all the time okay but in principle we can yes also all our prep all our parameterizations  are continuous here we're all talking about nice   continuous curves like nice and smooth  curves we don't want them to have too   many jagged bumps because then they're not  when I speak and be able to differentiate alright great so we have these two vectors and at any point an MS and let say any   point on the curve the NMS points  towards the center of curvature so we could go a certain distance along there to  find the center of curvature that's the center of   the osculating circle it's the same concept as we  had in the planar case except it still is valid   in the in the spherical case so for example over  here here the curvature is big and so the radius   of convergent radius of curvature is going to be  small so it's going to be a small circle there   over here the the curvature was small and so the  osculating circle is going to be bigger okay but   the normal is still pointing towards the center  of the oscillating circle and the oscillating   circle can be found in the same way as we do  for a planar curve if we want to know what the   osculating circle at some point is we choose any  three points near there say one point there and   two two on each one on either side those three  points if they're not all collinear will form   a plane and in that plane there will be a unique  sneak circle and as the three points move together   towards a specific point that circle will take  a limiting value and that'll be the oscillating   circle that we're talking about here for this  reason the plane spanned by T of s and n of s   is called the oscillating plane the plane at  any point spanned by the unit tangent vector   and the unit normal vector is the oscillating  plane of the curve at that point so again we   if we have some thing like this then at any  point say that one there there is a tangent   vector and I could also by looking at the rate  of change of this unit tangent vector calculate   the normal vector and then the plane spanned  by those two is going to be a plane the are   the oscillating plane so as the point moves  along the curve there's this plane that moves   with it that all that contains the direction  of the tangent vector and also contains the   direction of the normal vector so this plane  that's moving along touching the curve the   oscillating plane and that's the plane that  contains within it the osculating circle all right so we're doing well we've got two out  of three but we don't want to stop here we want   a third vector to complete our our set maybe I'll  leave this diagram here all right so we complete   T this and s by defining by normal vector V of s  which is T of s cross n of s this is another unit another unit vector because T and n are  perpendicular and they're both with length   1 they're perpendicular both of length 1 so when  you take their cross product I remind you you have   to use right hand rule to do that all right that  means you have to use these three fingers one two   and three you put the thumb along the first one  which is would be T and then the second finger   along the second one which would be the N and  then the third finger well third finger there's   T there's N and there's that B okay so it's  another unit vector and they form a right-handed tripod just like the IJ and K of standard the  usual basis vectors but they can be sort of in   any position so there's a T in one position  an end in another position T and then the B   would be coming out might do it like this but  that's supposed to be perpendicular to both   of them okay they're all supposed to be unit  like okay so now so that therefore this this   set of vectors the ordered set T of s and M  s and B of s is called the the frenay frame   of the curve beta at s alright so we have to  augment our picture by we had say T's and n   so T and so there's going to be a be coming out  also perpendicular here is a T that way and in   this direction be coming out perpendicular  to both here we have a T here and an here   the beta is going to be going in the in the  direction of the board so at every point on   the curve we have this tripod these three  vectors and they're rotating around with   the curve and that's called a friend a frame and  the for any equations are all about what happens   to the derivatives of these three quantities  these three vectors so the frente equations tell us what the derivatives are and I might  just write write that down so here the frente   equations if you take T prime n prime and  B prime I'm going to all three of them at   once with a single equation and that's going  to be a certain matrix times t and b and the   matrix has mostly zeros but there's a K here  and there's a minus K there and there's a new   thing called tau here and minus tau here and  otherwise there's zeros okay so that's the   frente equations where K is the curvature that  we've already defined so it's really a function   of s everything here is a function of s yeah  I don't always put up s but everything is a   function of s and this tau is a new thing  tau is the torsion of the curve it's also   a function of s I have to tell you what that is  you have to tell you how you get these equations okay one of these equations we already have  okay one of these equations is right here   like the first one the first equation says  that T prime equals K times n T prime equals   K times n that's exactly this equation right  here T prime equals K times n the basically   be defined and so that that was going to be the  case alright so the first equation is T prime of   s equals K of s times n of s so I'll write  down the S dependency so we make sure it's   really everything is depending on s okay and so  that's really of already known from the definition okay so what what next well the next  easiest one is that actually the B   prime because it only involves  a single entry so let's have a   look at B prime and the claim is  that claim this is a multiple of N okay why is that well we'd have to we'd have to  do some differentiation so it's not immediately   obvious that this thing is a multiple of  n there's a sort of a bit of an argument   that's involved so the first first part of the  argument is that we first of all observe that   B dot B equals one okay I'm not going to write  the s everywhere some B of s dot B of s is one   this is a unit vector everywhere so it's dot  product with itself is constant and therefore   we know from the lemma that we started out with  that B prime is perpendicular to B right whatever   a vector had constant length then when we  differentiate it the derivative is going   to be perpendicular to the vector from our lemma  actually guess what's the corollary of our lemma okay so b-prime is perpendicular to be it but it  turns out that B prime is also perpendicular to   2t and how do we see that well also the second  fact that we need is that B dot T equals zero B   and T are perpendicular so if we differentiate  this equation we get B prime dot T plus B dot   T prime equals zero now we know that T prime  is a multiple of n T prime is a multiple of n   and n and B are perpendicular okay so this  is zero since well T prime is is K times n so therefore we conclude that B prime  dot T equals zero all right so what do   we know about this B Prime we know that it's  perpendicular to B and it's perpendicular to   T we have these three vectors T N and  B and they're changing okay and we know   that the rate of change of this one the  blue the green one that's by normal that   we're interested in we know that that rate of  change B prime whatever it is is perpendicular   to both the tangent and the normal the both  perpendicular to itself and to the tangent   vector but there's only one direction that's  perpendicular to B and the tangent vector   that's the normal vector so we conclude that  B prime is a multiple of the normal vector so b-prime is a multiple of the normal vector  so what we do is we define B prime to equal   well we're going to just make sure this  equation took minus tau Maybelle right s   so B prime of s equals minus tau s times n of  s so we define tau of s to be that multiple we   know B prime is a multiple of n of s let's call  them at the multiple towel but sort of the way   we introduced the curve which are actually  same kind of thing we saw that this had to   be a multiple that's when we introduced that so  we introduce now tau in the same kind of spirit   and we call this a torsion and that takes care  of our third equation so now we have only this   second equation to deal with so finally we  have to evaluate we need to evaluate n prime okay so remember we have this frame here is we  have T and then we have N and then we have B   this is a right handed system T and and then  B so we're interested in getting control of   n prime so we're going to write n as the  cross product of and we have to just make   sure that we get them in the right order  so here's T and here's n and here's B so   if I do B times T I'm going to get n so n is  equal to B times T cross product of B with T yeah like I times J equals K J times K equals  I K times I equals J it sort of cyclic like   this okay so that's an expression for  n we can differentiate that using the   fact that we know how to differentiate a  cross product so and the prime is going   to be the derivative of B times T plus B  times the derivative of T all right well   we know it B prime is we just already said  that's minus tau of s times minus tau of   s times n so we start to do it at minus tau  s times n times T might already need to ask   because I'm dropping all the assets so let me  just drive minus tau of an times n n times T   plus B times the derivative of t prime we can  read from the first one is K times K times n okay now we still have to do these cross  products so again the right hand rule T   and B all right now we're doing n times  T that's going to be well T times n is B   so n times T will be minus B okay so this  is going to be a Tau times B because the   minus sign from this one will cancel the  minus on that one and and B times n TN   and B so B times n will be minus T I think  yes so plus K times minus T for a total of minus K T plus tau B and that is our second  equation in the for any equations that tells   us that n prime is minus K times T  plus zero times n plus tau times B all right so those are the friend equations  and might take a little break and then I'll   come back and do an example just  to to have a see how how it goes

Channel: Insights into Mathematics

Views: 43,534

Keywords: mathematics, differential geometry, Frenet equations, Frenet Serret equations, curve, vector derivative, curvature, torsion, Wildberger, orthogonal frame, basis vectors, acceleration, tangent, normal, bi-normal

Id: 1HUpNAS81PY

Length: 50min 30sec (3030 seconds)

Published: Sun Nov 17 2013