Tangent conics and tangent quadrics | Differential Geometry 5 | NJ Wildberger

[Music] [Applause] [Music]  all right so hello everyone   i'm norman wahlberger we're here at  the university of new south wales   and this is a course in differential geometry and  in our last lecture we introduced or reintroduced   an old method of lagrange and euler to  basically initiate the differential calculus   and it's going to be a point of view that's  useful for us in our study of curves and surfaces   so let me illustrate what was going on  there with the example of a cubic polynomial so we'll look at a general cubic polynomial say p   equals a plus b x plus c x squared plus dx cubed  so it's a general one and let's take a point at the point r on the line and for us r is usually a rational number but if  you like you can think of it as a real number too   all right so we're going to expand p  around r and the way we do that is we   just evaluate p of x plus r so  that's a plus b x plus r plus c uh x plus r squared plus d x plus r cubed  [Music] and then we write that out in terms of   the various powers of x so there's the constant  term and then the stuff involving x will be b plus 2 c r plus 3 d r squared times x and the stuff involving x squared there  will be a c plus 3 d times r [Music] x squared and the x cubed term just d x cubed okay so then we rewrite this just by  replacing x with x minus r so we get   what we might call the taylor polynomial the taylor polynomial of p at r which is just p of x equals a  plus b r plus c r squared plus d   r cubed plus b plus two c r plus  three d r squared times x minus r plus c plus 3 d r times x minus  r squared plus d x minus r cubed so that's just actually a rewriting of the  original polynomial it's the same as the original   polynomial except now that it's sort of expressed  in powers of x minus r and we've only used   a very elementary high school algebra basically  we've just translated the polynomial that's all   okay so these various functions or polynomials  that appear here as the coefficients we uh we can give them some names so this  uh well let me just maybe write i'll write   this so this thing here is d0 of p at  evaluated at r this one here is the d1 of p at r which is just the same  as the ordinary derivative of p at r this one here is what we're going to call d2  of p of r and uh and this one is d3 p of r all right so these polynomials the coefficients are polynomials in r that are  derived from p by essentially   differentiation but they're not  exactly the usual derivatives   so where say d sub k of p is the same  as the kth derivative over k factorial so if we wanted to connect it with the usual  derivatives we're almost getting the derivatives   except that there's a k factorial  so here there'd be a two factorial   and a three factorial separating  what's happening here with the   the real second derivative so we  call these things uh sub derivatives they're almost like derivatives but they're  a little bit different they're a little bit   more convenient to work with in some  sense because the numbers involved   the coefficients involved are not as quite as  big if you take the fifth derivative of a of a   polynomial there will be a lot of 120s in all  the coefficients because of the five factorial   that will appear basically just extracting  that factorial out and dispensing with it   okay so we have the uh the various sub derivatives  and then what we're particularly interested in is   truncating this expression to various degrees  when we do that we get various approximations   to the to the polynomial that are valid near the  point r and have geometrical significance for us so by truncating this expression let's call it star this expression  star by truncating star we obtain the taylor polynomials the taylor polynomials of p at r   and let's write down what they are so we'll  use t for taylor so at r the zeroth one   in this case is just a plus b r plus  c r squared plus d r cubed you might   think of this now as actually a function as a  polynomial in x it's actually a polynomial n x   this one just happens to be a constant polynomial  and then the next one t r of 1 is what you   get when you take the first two terms okay a  plus b r plus c r squared plus d r cubed plus b plus 2 c r plus 3 d r squared times x minus r and the second one is well all the first  three terms so i'll just write e this and   then the same thing here and then finally  c plus three d r times x minus r squared and the last one tr3 of p of x well that's  going to be just the whole expression so   that'll just be in this case p of x because  we're dealing with a degree 3 polynomial so these taylor polynomials are going to  be for us geometrical simpler versions of   the original polynomial especially that are  especially valid near the point r so for example here is a polynomial something like that that  happens to be the polynomial   p of x equals 5 minus three x  minus four x squared plus x cubed okay so suppose we choose some  some point let's say r equals 2. then the value here this will be  the value t sub r of 0 of p   that's just that particular value there so the  first taylor polynomial will be the tangent line so that'll be t of well at two let's  say so i'll write two t two of one of p   and then also there'll be a tangent conic parabola maybe something like this these are all polymorphisms okay so let's have a look at first of  all the geometrical significance of these   things so we should think about this formula  as being valid star is useful [Music] when x minus r is very small okay it's very very small so if x minus r is  very very small so small that you cannot see it   then the only thing that you can see here  is this constant function here and that's   the situation where we basically have our  our our view very very close so close to   the function that we can only see that point in  question our face is almost right at the board   so now as we scale things up so that uh  we're looking at a very very very small   little neighborhood of of uh of r so x's would  be very very close in that case we only see that   point if we go a little bit further away move  a little bit further away maybe so that we're   out uh you know this far it's a little bit of  room then we see a little bit more of the curve in that case the x minus r  will be small but visible   if x minus r is small but divisible but still so  small that x minus r squared is negligible like   if this is one millionth maybe you can see one  millionth but you can't see one millionth squared   and you're certainly not going to be able to see  one millionth cubed so that means these two terms   would be invisible to you and you would only see  the linear part of the function in other words the   function would actually look to you like a line it  would look exactly like this tangent line it would   be in indistinguishable from the tangent line if  you were up if you had magnified it that close on the other hand if you zoomed a little bit  further away so you would go back a little   bit further so that now not only is x minus r  visible but x minus r squared is also visible   maybe x minus r is now a thousandth and  one thousand squares of millions maybe you   can see that but a billionth you still can't  see so if you've gone further away then this   term here will be invisible and you will see the  function as being the uh the parabola it will be   indistinguishable from the parabola so it's really  a question of scale of how much you're zooming   in or out near this point at r equals 2. okay so this thing here of course this  is just the function evaluated at at r   and if we have a look at this expression  and if we look at p of x minus p of r in   other words we take this thing to the left  hand side and then if we divide by x minus r then what's left on the right hand side will  be this term here so what we're calling d1p of r then the other terms there'll be a d2   p of r times now x minus r and a d3 p of r x  minus r squared since we've divided by x minus r and if we look now at the limiting value of both  sides when x approaches r well then these two   terms are going to disappear so if as x approaches  r this and this go to zero so we can say that the   limit we can see that the limit of p of x minus p  of r over x minus r is just this number d1 p of r so in other words this is the limit as x  approaches r so in other words we're we're seeing   that the standard definition of the derivative of  the polynomial at the point r isn't captured by   this uh this algebra break business it's just  a question of looking what happens when you   divide by x minus r let x go to r all right  so this really is the derivative of p at r in   the standard way so this is uh explains why this  really is no different from the usual story it's   completely equivalent to the usual framework but  we don't have to go via this limiting procedure okay something else i want to mention is the  the error okay so when we approximate the whole   polynomial by any one of these tangent polynomials  there's an error in fact we know exactly what the   error is in this kind of situation because it's  just the terms that we're dropping off okay so   if we take the linear approximation in other  words we use this then the error is just this   if we take the quadratic approximation  meaning those first three terms   well then the error is just this term here okay so in terms of the diagram if we're at  some point and the point doesn't have to be   close to r at all if we're at some point here and  we're interested in say the difference between   the functions value at x which is up here and  say the linear approximation of the tangent   line then it's given by the last the sum of  the last two terms in our expression star and   if we're interested in the difference between the  functions value and the quadratic approximation that difference there that  will just be exactly the last   term right here that will be d times x minus  r cubed so on this picture here this is the   tangent conic at r equals 2. this quantity  here it which is d times x minus 2 cubed   if r equals 2. that's just the error between  the curve itself and the tangent conic at   the point two so notice that this is a we have  complete control over this error complete control   over this error we know exactly how far we have  to go from the tangent conic to the curve itself okay this leads us to a very remarkable  i think and beautiful observation   which seems to have eluded researchers in calculus  for quite a long time it's only been discovered in   the last 10 years or so as far as i can see so  it's the theorem is that for a cubic polynomial p of x all the tangent conics the things that we're calling t sub r 2 of p of  x all the tangent conics are distinct or disjoint in other words none of them no two  of them [Music] meet each other that means if i went around graphing not just   that tangent conic but if i  graph the tangent conic here or maybe it would look like like this and i graphed it here  maybe it would look like this there's also an inflection point somewhere for  this curve right there's an inflection point   somewhere where the the tangent actually meets  the curve to order two it's actually the tangent   conic reduces to a line so that's also part  of the story so what we're saying is that all   these tangent conics not just the three that  i've shown you but all of them are disjoint okay so this seems to it seems  remarkable but i think it   was first only observed relatively  recently this is an observation that i believe is due to etienne geese a  french geometer and he actually described it   in a video of it's available on a video of a  seminar that he gave at some conference i believe and it's relatively recent so i also stumbled  across this thing and was also sort of amazed at   it it's really quite a beautiful thing and let's  see if we can see why it is the proof is actually almost a one line proof  really from what we've said well let's for example let's consider this  point here okay there's there's a point there   okay that happens to be a point on  uh on this particular tangent conic   let's ask what other tangent conics  could possibly go through that point okay so if there's some other tangent  conic that also goes through that point   it means that the error is going to be the same   the error between that other tangent conic which  would pass through there and the functions values   is also going to be the same area same error  as this one all right so if two tangent conics if two tangent conics say at r1 and r2 pass through the same path  pass through the same point then the errors are the same then the error is at that point are the same and we know exactly what the error  is the error is d times x minus r cubed so it   would have to be that d times x minus r one cubed  has to be equal to d times x minus r two cubed but d is non-zero because this is a cubic and so you can cancel the d's and then you have  this quantity cubed equals this quantity cubed   and so there's uh if two cubes are  the same then the numbers are the same and so therefore r1 has to equal r2 well if there is if if a tangent conic  passes through a point then it's the only   tangent conic that passes through that  point it's not necessarily the case that   every point has a tangent conic you're going  through it that's an interesting question so this naturally uh leads to some  some questions is this generalized [Music] and so problem think about first part generalize this to polynomials of degree five well and beyond if you want to and part two what happens or what can you say for quartic polynomials that was polynomials of degree four okay so um that's certainly an interesting story i think there's it shows that this this  whole subject of tangent conics is largely   unexplored right if it had been explored then this  would have been obvious to people long long time   ago so it probably suggests that there's probably  quite a lot of other interesting things that   you could you could say for a in exploring not  just tangent conics but also tangent cubics and   higher tangents we're going to see the  importance of tangent conics in this   course because differential geometry is all about  quadratic approximations to curves and surfaces   okay so that's in in some  sense that's a that's a broad framework for thinking about what differential  geometry is it's we're going to look at curves   and surfaces and we're going to look at them  so carefully that we don't just want to have   a linear approximation we want to look at them  quadratically so we're going to be interested   in in asking what does the tangent conic look like   and what do the properties of the  tangent conic or maybe some other conics   have to do with the properties of the curve or  the surface more generally so it's going to be   very important for us to to think in terms of  not just tangent lines but other higher degree   approximations and to really get drive this home i  want to now generalize the discussion by going to basically calculus of several variables and  i want to show you how lagrange's point of   view allows us to think about functions of two  variables in a very pleasant and natural way   all right so we're going to go up one step to  essentially second year calculus if you like   i want to talk about functions or polynomials of two variables and calculus  ideas in that that context and i'm going to do this in the context of  an example because i kind of like examples   but one could just as easily try to write down  general formulas but i'm going to have a look at   a particular example so i'm  interested in this following   function or polynomial of two variables  let me maybe i should call it uh   p also because it's still a polynomial it's  going to be x cubed plus y cubed plus three x y i hope that sounds a little bit  familiar i hope you remember that the if we set that equal to zero we get the  folium of descartes okay so that's what   you get when you set p of x y equal to zero  so what we're doing now is we're introducing   uh a function of two variables by setting this  thing equal to z and we're thinking of not having   a two dimensional picture but now rather having  a a three-dimensional picture where we have   an x and a y and a z and that that that volume  that we've just been talking about is a level it's in the plane z equals zero it's like a level  curve and this surface somehow is doing something   we'd like to sort of understand what's  it doing what does it look like what   does this surface look like because  this is an example of a cubic surface a cubic surface and these were very much studied  and loved by 19th century mathematicians so it   turns out that cubic surface is very rich subject  very visually rich all kinds of wonderful things   surprising things can happen with cubic surfaces  it's much richer than so just cubic curves in   the plane and we'll we'll get a sense of uh of  that with this example but this is just one of   many possible cubic surfaces all right there are  lots of other cubic surfaces that look completely   different from this one all right so we're  going to approach this exactly in the same way   that lagrange would suggest except we're going to   choose a point in the x y plane say with  coordinates r and s not just a single point okay and the idea is that we're  still going to do exactly the   same kind of thing and let's see  what happens if we do this so we we have to evaluate p of x plus r and y plus s so that's going to be x plus r q plus  y plus s cubed plus 3 x plus r y plus s   now we want to expand that in powers of x and y all right we have to use the binomial  theorem here so there's an x cubed plus   three x squared r plus three x r squared plus an  r cubed for that first term and then there'll be   a y cubed plus three y squared s plus three  y s squared plus s cubed for the second term and then expanding the last business we'll get uh   maybe down here three x y plus three  x s plus three r y plus three r s there's an s okay and what did we do in the  one variable case well we then just sort of   group this together in terms of increasing  powers of x so we'll do the same thing here   just write it out in terms of  increasing powers of x and y   so what do we get so the the constant  uh stuff is stuff that not involving any x's or y's so there's r cubed an s cubed and a 3rs   we'll put that in brackets so we got rid of those  three terms and now the the terms involving just x where is an x term so here there's  a 3r squared now we an x there's a   3s that looks like the x terms i better  underline them so i don't count them twice and then let's say the y terms so there's a  linear y term right here so plus 3 s squared y   and there's a 3rs there a 3ry and then there'll be the quadratic terms starting say with x squared there's a 3r x squared there   and there's no other x squared so 3rx squared  and 3sy squared any other quadratic terms there's 3xy that's also quadratic and now we only have the cubic terms left  and there's only x cubed plus y cubed okay so that's good so what was it that we did then well we  replaced the x the x with x minus r to get   the original polynomial back so we'll do the same  thing here we'll now take this expression replace   x with x minus r we'll replace y with y minus  s to get r cubed plus s cubed plus 2 3 rather   rs plus 3 r squared plus 3 s x minus r  plus 3 s squared plus 3 r times y minus s plus three r times x minus r  squared plus three s y minus r   squared plus three x minus r y  minus s sorry that's the y minus s plus the cubic terms x minus  r cubed plus y minus s cubed so we've ultimately just  rewritten the polynomial now   this is now a taylor polynomial about the point taylor polynomial of the  polynomial p at the point rs and the various coefficients we can give them  names we can call them various derivatives or   sub derivatives so we can define d zero of p to be  uh the coefficient well that's the constant term and then this this thing here what  should we call that well it's like a degree you know one we differentiated once   but we've differentiated in the in the x direction  and not in the y so we might put one zero there so d one zero of p will be 3 r  squared plus 3s while d01 of p is 3 s squared plus 3r this would this is just or this is the  polynomial p this is what would usually   be called the partial of p with respect to x  and this is the partial of p with respect to y and uh well then we can sort of keep going um   next would be the quadratic thing so  d two zero p would be three r d 0 2   of p would be 3s and then this thing here would  be kind of a mixed mixed partial that would be 3. and then the coefficients there would be three  zero p would be one i suppose and d zero three   p would be a one so you know these are sort of  analogs of partial derivatives but we can see   that we can get them just algebraically without  any notions of partial differentiation necessary   we're just doing the same kind of thing that we  did in one variable okay but what's really sort of   geometrically interesting is is not so much these  partial derivatives but rather the the truncations   that's what's geometrically interesting  and the truncations are going to be the analogs of tangent line  tangent conic and so on so the sum the the tangent at uh at rs of of degree k is going to be the sum of all terms of our the expression i just  wrote down of total degree less than or equal to k in x minus r y minus s so for example the the that when k is one we're talking about the tangent  plane the name we'll give that tangent plane it's the degree one term so exactly the first row  of that thing so r cubed plus s cubed plus 3 rs plus 3 r squared plus 3s times x minus r plus three s squared plus three r times y minus s so that will simplify a little bit you can  expand it out and hopefully you will get   r cubed plus s cubed minus three r s plus three  r squared plus s x plus 3 s squared plus r y and then the degree two or less terms well that's going to be the tangent conic or the   one degree one dimension up from a  tangent conic so the tangent quadratic well i'm not going to write it at all out it's  just it's the sum starting with r cubed plus   s cubed plus 3 rs and then you just keep  adding all the terms up to well up to here all the way up to three x minus r y minus s in this case the error is just these two   terms here so we could also write it we want  to be a little bit lazy we could write it as the whole polynomial which was  x cubed plus y cubed plus 3xy   minus the last two terms x minus r cubed minus  y minus s cubed it's actually a quadratic okay so can we visualize a little bit  what this surface looks like and you   know what what's going on here so  it's quite a pleasant situation so my next lecture i will get jojoba out and  we'll have a play and play around with some   figures it's very easy to illustrate these kinds  of things by drawing level curves so the original curve that we started with now the folium there so that's the  case that's sort of when p equals zero if you look at p equals minus  one you get something like this so that's p equals minus one p equals -2 and sort of so on here's one and one it turns  out that if you look at a different color if you look at the level curve when p equals one it appears to be a  straight line and then after that you get   things like that look like sort of like this  might be p equals two something like this   all right so we have this picture of the surface  first of if we're thinking of z coming out this   way right then the function is i'll remind you x  cubed plus y cubed plus 3xy that's our polynomial   and we're we're looking at level curves of  it so if we if we move in the positive x and   y direction well these cubic terms will  dominate and it will get it will get big   so it's sort of coming out at us if the z is like  this so so if we go in this direction it goes up   on the other hand if we go in this  direction it goes down it goes negatively okay and it just touches  the cuts the x y plane here   and it kind of goes down here and then inside here  i didn't label there's these um other ones like say right here there might be  one that sort of looks like this   that might be p equals a half and then  the p equals half also has a little   thing like this so there's a small  hill here there's a small hill okay so generally it's it's it's coming  down here and then there's a little hill   and then it goes down like this we can  see a little bit more by looking at this   axis of symmetry if the line x equals y which  restrict ourselves to that line x equals y   then we get p of x x equals  2 x cubed plus 3 x squared and if you graph that in just the ordinary if you  think of that as just an ordinary function of x it has a double zero at zero it gets a minus three halves it has a zero the derivative here f prime of x is six x squared  plus three x so that's equal to zero now plus six   x that's equal zero implies x equals  zero or x equals a minus one so there's a max like this so it looks like this because that's the cross section of  this surface above the line x equals y what's particularly interesting or one thing  that's quite interesting is this line the blue one   the line x plus y equals one you might  ask well is that uh is that really a level   curve is that really a line  that's lying in the cubic so if we look at the p of x y minus 1 so that's  x cubed plus y cubed plus 3 x y minus one   it's not entirely obvious but this thing factors   it factors as x plus y minus one times x squared  plus y squared minus x y plus x plus y plus one that implies that if x plus y minus one is zero   then this is zero which means that  that line really does lie on the cubic   cubic is very curvy but it does  have this line that lies on it in fact um there's a little bit more to   the this level curve p equals one can anybody tell  me what what i've missed from that level curve it's not entirely obvious but if  you if you look at this quadratic   you need to say well when could this be zero well this is a quadratic equation so you  could look at that quadratic equation and   and if you looked at it's discriminant  then you could convince yourself that   there's exactly one more point and that's the  point that's sitting right here at the maximum and that in fact we saw that right here that's   that's that point right there on that  diagonal part of the surface so this   level curve is a cubic this is an example of a  cubic that consists of well a line and a point so all the level curves are cubics and it's  a quite interesting thing and obviously it's   kind of an interesting question what do various  tangent planes look like let's choose a point on   here what does the tangent plane look like  what does the tangent quadratic look like so before we are able to well understand that  we're going to have to appreciate a little bit uh   more about uh what quadrants in general look  like all right so lagrange's point of view   it's a powerful point of view it allows us  to go to higher dimensions very pleasantly   and easily still only with high school  algebra in our next lecture we'll have a   look at uh some geogebra playing around with  with this example some others to illustrate well how three dimensional curves  might uh might look we'll see then [Music] you

Channel: Insights into Mathematics

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Keywords: differential calculus, differential geometry, Wildberger, Lagrange, tangent line, tangent conic, E. Ghys, cubics, Folium of Descartes, surfaces, tangent plane, tangent quadric, sub-derivatives, Taylor expansions, mathematics, education, Euler

Id: SfnsfVsFiFI

Length: 49min 16sec (2956 seconds)

Published: Thu Aug 22 2013