Classical curves | Differential Geometry 1 | NJ Wildberger

[Music] [Applause] [Music] [Music] so hello everyone i'm norman wahlberger  we're at the university of new south wales   and today we're going to start a  course in differential geometry   it's going to be an elementary or  beginners course in differential geometry   but we're also going to be looking  at some advanced things along the way   all right so the the topics that we're going  to cover include roughly the following although   there'll probably be some other things as  well we're going to look at classical curves   and surfaces we'll talk about frene surrey  frames associated to curves in space we'll   look at the idea of oscillating conics the idea  of curvature curves and surfaces we'll talk about   gauss's theorema agregium which is gauss's sort  of celebrated theorem on the curvature of surfaces   talk about the first and second fundamental  forms of a surface the so-called shape operator   we'll have a look at minimal surfaces and mean  curvature probably get some soap solutions and   make some soap bubbles we'll have a look at rural  surfaces and developable surfaces which are very   important in manufacturing applications we'll  look at geodesics and the gauss-bonnet theorem   and then there'll be also some material on  topology basic ideas of continuity topological   spaces indexes of curves classification of  surfaces and the oil are characteristic so   there's some natural connection between  differential geometry and algebraic topology okay so what's my orientation this is  going to be a classical course in in one way   and that the topics are rather classical but  it's going to be rather novel in another way   and that the approach that i'm going to take is  maybe innovative and rather modern to a number of   subjects so we're we're now just at this period  of time experiencing a whole bunch of new ideas   in geometry and perhaps also in calculus and  these new ideas are going to help us understand   differential geometry and perhaps some interesting  some new ways so my point of view roughly is oriented towards the  following ideas so i prefer rational or algebraic uh techniques or  objects to transcendental ones we're we're going to occasionally  not shy away from classical   geometry so we're going to  use some classical geometry in particular we're going to perhaps have  a little role for projective geometry   and especially uh talk about quadratics  that's going to be very important for us   and along the way we'll talk about  quadratic forms and sort of will adopt the   language of rational trigonometry in some  cases which helps us understand a few things i'm also going to encourage the use of  geogebra for visualization this is a   dynamical geometry package that allows us  to visualize planar situations and it's   quite helpful we're going to be using that we're  also going to be sticking to 2d and 3d mostly where we are in the realm of practical experience   the world in which we live in but we  are have a view nevertheless to physics so in particular about 100 years ago now  albert einstein introduced the special   theory of relativity and then the general  theory of relativity and this was really   remarkable watershed in the history of science  and differential geometry has a lot to do with   with his work so although we're not going to be  really getting to a full understanding at all of   of uh general relativity nevertheless  we're kind of oriented in that direction   and uh so we're we're interested in  understanding general relativity ultimately all right so that's the the basic  framework that we're going to adopt   today i'm going to have a  look at some classical curves going back to antiquity all right so the study  of curves took off in the 17th century with the   work of huygens newton then oiler monge then  taken up by koshi gauss and so on but earlier   back in ancient greece there were lots of interest  in classical curves so there's some classical   occurs the simplest one of course  i suppose is just the straight line   but then after that we get the conic sections and these are the curves that  we get when we take us a cone and we slice the cone with various planes we slice  the cone in this kind of way we get an ellipse   if we slice the cone with a plane  perhaps more up and down then we get a hyperbola and if we slice the cone with a plane which is   parallel to one of the generators  say parallel to this one then we get a parabola all right so we have three different classical conic sections the ellipse the hyperbola and sort of halfway in between them the parabola so these were named by apollonius and there's  some other sort of degenerate conic sections   that you get when for example if you slice the  cone with a plane that's exactly in the board   then you'll get two crossing lines or if you happen to slice the plane just through  that point there well then you can just pick up   one point or you can actually just get  a single line by slicing it like this all right so these uh these conic  sections are going to be important for us   and so of course i just remind  you that there's some alternate   points of view towards the conic sections  so we can also think of them metrically they also have metrical definitions and typically that's involving a point and a line   and a point x somewhere and then  we look at the relationship between the distance between the focus  f and the point x and the   vertical distance from the point to the line  but in fact i prefer to think rationally so   i'm going to use quadrants instead of distance  so quadrants is just the square of the distance all right and that allows me to say  this in a more algebraically simple way   so the point is that if we look at  the condition that the quadrants   of x and f equals the quadrants between x and  the line then that's the condition of a parabola so that would be something like like this perhaps so here the ratio between the quadrants  to the focus and the quadrants to the line   is some number epsilon less than  one in that case we get an ellipse and if we look at the case when the constant is bigger than one we get hyperbola all right and then we all know  that this is called the focus   of the conic and this is called the directrix and this was known back in antiquity  but what's not so clear what wasn't   really understood until the 19th century  was the relationship between these two definitions so if you have a  cone and we take some plane let's say that it's cutting out an ellipse then where is the focus actually there  are two foci for an ellipse so there's   a foci here but there's also another foci on  this side and another directrix on this side   so we can naturally ask where are the foci for  this ellipse if the ellipse is cut out from a cone   and that was answered only in the 19th  century by french mathematician danilon who   had the following solution he said take a little  sphere and place a little sphere in the cone okay so just sitting there in the cone like a  blob of ice cream and then expand that little   sphere so that it stays inside the cone make  it grow so it grows until eventually it just touches hits its head against  the plane of the ellipse   okay at some point it just touches the plane  and where it touches is one of the foci and to get the other foci what we do is we  start off with a very big sphere that's sitting   on the cone way above the uh the lips and we  shrink it and as we shrink it it comes down   down down down until eventually it hits the  plane of the ellipse from on top when it does so it will touch the uh the plane in the second  foci now you might also ask where are the   directrices in this story so it turns out that if  you take the equator of this sphere that's a plane that plane cuts the plane of the ellipse in a line that line is a directrix and similarly  if you take the equator of the sphere on   it has a plane and when you intersect that  plane again with the plane of the ellipse you   get another line and that's the other directrix  on the other side okay there are a few other incarnations there's quite a few other  incarnations of a of the conics for example   another way of getting an ellipse is to take a  circle and apply a affine transformation if we   stretch the plane in one direction more than the  other then [Music] we can get a certain ellipse there's another connection with complex function  theory so i remind you that the in complex in the   complex plane the cosh function in terms of  x plus i y is equal to cosh x cosh i y plus sinch x sinch i y and that can be rewritten as  cosh x times cosine y plus i times sinch x sine y and i also remind you that cosh z  can also be written as e to the z   plus e to the minus z over 2. so it turns out that the the  level curves of this function   give ellipses so if we're in the complex plane so maybe i should say if the real part of z  equals x is fixed let me say it this way the   real part of z is x equals fixed then y goes  to x plus i y which is cosh of x plus i y will   trace out x squared over cos squared x plus  y squared over sine squared x equals 1. so that's an ellipse with foci 1 0 and minus 1   0. so there's a natural connection  with a the cosh function as well all right so that's just a little bit of  an introduction to these conic sections   in the 17th century the geometry of the greeks was  perhaps somewhat replaced by the analytic geometry   of descartes and what happened there  is that it's a whole new dimension the analytic geometry of descartes allowed us to understand that conics have  an algebraic description in terms of well   first we should say that a line is something  of the form a x plus b y plus c equals zero   so a linear equation in x and y is of course  a line and a conic is then just a degree two curve so it's a curve that has the general form   say a x squared plus b x y plus c y squared  plus d x plus e y plus f equals zero so this algebraic form allowed  descartes to to of organize all the   conics into sort of one big family  just uh viewed algebraically okay so the greeks had other  ways of creating curves they had a method of creating derived  curves where we start with a certain curve   or sometimes a pair of curves and we  perform some operation to get a new curve   so an example of this was the  conscious construction of nicometes around 200 bc and the picture is we have a curve c let's say and then we choose a point o and then we choose some variable line  through o and it cuts the curve at some point   let's say q and then we go an equal  separation from q on either side so we go   maybe k in that direction to a point p1  and also k in this direction to a point p2 and then as q varies p1 and p2 are going to  trace in some sense some kind of parallel curves   to see from the point of view of o alright  so we're going to get these two new curves   sort of following c roughly but  maybe not exactly depending on the   position of the lines and these are  called controls of the original curve so the locus of p1 and p2 where  say the quadrants between p1 and q   equals the quadrants between p2 and q  equals say capital k is a controlled of the original curve c  with respect to the point o okay i hope we're familiar with the term locus  so locus means the path generated by something it's the path generated by something  so in this case the path generated   well we have actually two points  now creating a pair of curves well it's not exactly parallel because  it's parallel from the point of view of o   it's not that all these lines are exactly parallel  they're all going through the fixed point o   so by the time we get over here  for example we still have to go   k in this direction and k in this direction  so it would be something more like this maybe a similar kind of construction  is that of a systoid   i'm not sure if i'm pronouncing that right and  that was introduced by diocles also around 200   bc and he introduced it in fact for a very  famous reason there's a famous problem called   the doubling of the cube problem okay so  the the doubling of the cube problem is   you have a cube of side one and you  want to construct by ruler and compass a bigger cube which has exactly twice the volume  okay so we want to construct a bigger cube   which has exactly twice the  volume of the original one   so we know that the volume of a cube is  the is the cube of the the side length   so we really want this thing here to be cubed  root of two that would ensure that the volume   is actually two so the question is ultimately how  do you construct this number cubed root of two   by ruler encompassed constructions well  nowadays we know that that's not possible   okay but the ancient greeks  didn't know that and they tried to   find some ways of doing that and diocletes  did not find a ruler and construction   way but he introduced this other curve that  helped him to solve the problem in a slightly   different direction okay this so this systoid uh  construction here's the general one so it starts   off with two curves suppose we have a curve c1  and curve c2 they don't have to be parallel really and again we have some point o we draw again lines  through o meeting the two curves in two points   and then what we do is we take this  vector and we translate it back to o so we take this separation  and translate it back to hope   getting a point there [Music] and  as we do that for various lines through o we get some other curve and that's  called the sissoid of c1 and c2 now there's actually a famous  sissoid which is the following   so if we start in the xy  plane and we have a circle say going up to the point one and we have a line that's tangent  to that circle through there   the line y equals one this is a circle centered at  a half here then the systoid of these two curves   is particularly famous and it's  called the cysoide of diocles so what does that look like well so uh with  respect to which point well with respect to   this point o here there's the point o okay so  what we're doing is we're taking lines through   o seeing where they meet there are  two curves the circle and the line   and then taking this vector  and translating it down here and i hope you can see that as the  line approaches here right here the two   curves coincide so we're actually going  to be right here so this cyst side looks something like this this is sometimes called the cystoid of diocles all right so here's a problem for you find a rational parameterization of this curve of the sysoid of diocles and give its algebraic equation in terms of x and y i also want you to sketch it on geogebra all right there's some other famous constructions there's the evolute and the involute constructions and i'm not going to say them  say too much about that question okay ration yes rational parametrization  means uh so x of t let's say x of t let's   say the parameter is t so x of t and y i want  this expression to be a rational function in t   polynomial over polynomial no  cosines or transcendental uh business rational functions of t all right so for the evolute and involutes of  a curve it's a very important part of our story   but i hope that you will all  have a look at math history 16   which is a history short review of differential  geometry and in this video i talk a fair amount   about evolutes and envelopes okay so just  very quickly if you have some curve c then   an involute is what you get when you unravel a  string that's tied to it so if we imagine having   a string that's tied to the curve and then  we unravel it at some point keeping it tight   keeping it sort of tangent like this then the  the curve that we get here is an involute of c and we should say an involute because there are  lots of them if we started by unraveling at some   other point here we would get some other  involute these were introduced by huygens in the 17th century and then the somehow  dual construction to that is the evolute   where you have a curve and you take  tangents and then you take normals   to the tangents and then you see where  adjacent or very close normals meet so there's a point p and there's  a point q then the normal to q   and the normal to p will meet somewhere  and in the limit as q approaches p that   limiting position of the intersection of  the normals is a point that depends on   p so now if you change p and you do the same thing  somewhere else you're going to get some curve that   traces out the centers of curvature of  the original curve it's called the evolute of c let's see and it's really  the locus of centers of curvature okay so that's an important construction historically but  there are a few other ones too okay so another interesting way of getting  new curves is the idea of a petal curve all right so here we have some curve c and  we have some point o again and what we do is   we take some variable point on c and look at the  tangent to the curve through that variable point   and then we drop a perpendicular  from o onto the tangent giving us a new point let's choose another point let's  say the point happened to be there   its tangent would be here and then dropping  a perpendicular we would get this point here so the feet of all the perpendiculars from o   to various tangents of the curve that's called a  petal curve get some curve called the petal curve of c again with respect to the point all  this turned out to be quite an important   concept for newton in his study of  planetary motion and kepler's laws there's an interesting example  of this if you take a parabola there's a parabola and you take the point in  question to be the vertex   of the parabola then the petal curve of  the parabola with respect to the vertex is here's a point there's a tangent there would  be a perpendicular uh point my diagram's not that   good up here some other perpendicular  point okay what you're going to get is a curve like this it's exactly a sissoid of diocles so the pedal curve of a parabola with respect to its vertex is  the cysoid the cysoid of diocles so in the 17th century when newton introduced  the laws of motion and the calculus was being   rolled out people got very excited about the  possibility of using calculus to understand motion   of all kinds of things and so motion  and mechanics became important drivers   of of mathematics and also of curves so  there's a whole new family of constructions   which are generally under the  name of roulettes where one curve rolls on another without slipping and then any fixed point describes what's called a roulette this is a very general kind of story expose 1l so you have some some curve and on it you have  some other curve maybe it's in lips for example   and on this ellipse you have some  special point and as the ellipse rolls on you know your lips is just  rolling without slipping   on the other on the curve c then you trace what  happens to this to this point wherever it is and that's called uh that  curve is called a roulette a cycloid is an example right that's a very  famous example so for example a cycloid or a   relatively simple situation where you  just have a circle rolling on a line   and then you get some curve like this circles  rolling on the line and that's a cycloid   there are other examples if you have  a circle rolling on top of a circle   like this you get some kind of thing like  this it's sometimes called an epicycle or you could have a circle rolling inside a  circle when we were kids there was this toy   that children had called spirogram or something  like that with these little gears and you roll   these gears around these little wheels making all  kinds of pretty patterns exactly this kind of idea   where you have these little wheels attached to  each other with cogs and you can move them around so here this is called a hypo cycle and if the radii are suitably chosen and  you arrange so that you're going around   so it closes up after going three times that's an  interesting special case that's called a deltoid but we don't have to have circles rolling  on circles we can have any kind of curves   rolling on any other curves so another  interesting example is if you have a parabola   which rolls on a congruent parabola  okay so here are two congruent parabolas we imagine this one might be fixed  and this top one is allowed to roll   on the other one so after a  little while might be up like this something like that then if we take the locus say of this vertex  here and see it's not very clear what i'm   doing here but if you do this a little  bit more accurately then you find that   lo and behold you get once  again the cystoid of diocles that same curve that you're finding  the parametrization for in its equation   in that problem all right so a lot of these  curves are are what's called   transcendental curves they are curves  that involve non-algebraic aspects they're quite different from the  conics because the conics have an   algebraic equation in terms of x and y  you can write down the equation for them   these ones you need cosines and  sines or perhaps some other kinds of   functions to describe them naturally  mathematicians were also interested in well   what about the algebraic curves what happens when  you go beyond the conics the degree two curves that's a very natural question right once  you've once you've understood conics and   that's reasonably easy to do descartes thought  that he could prove everything that the ancient   greeks knew about comics just from his formulas  apollonius had hundreds of results about comics   descartes was confident that he  could prove all of them just by   maybe making algebraic manipulations instead of  any kind of geometry like apollonius was doing so   naturally after the conics was kind of exhausted  by the 17th century mathematicians they naturally   turned their attention to the next degree the  cubics okay so the cubics are a whole new world cubic curves so there's there's much to be  said about cubics and we are going to have   a look at some aspects of them in in this course  so there's uh first of all cubic functions this is   sort of a baby kind of version where you start  with y equals a function a general cubic okay okay we all know what the graph looks of these  they are going up and down and then up again   unless you the sign is changed in which case they  they go down and then they go up and then they go   down again okay but more or less any cubic  looks like this if it's a cubic of this form   but it's important to understand that  this is a very limited kind of a cubic   right the the general cubic is much more  complicated the general cubic has the equation   a x cubed plus b x squared y plus c x y  squared plus d y cubed those are all the   degree three terms then there'll be degree two  terms e x squared plus f x y plus g y squared   and then there will be linear terms h x  plus i y and then typically a constant term   j equals 0. so that's your  general cubic and it has   1 2 3 4 5 6 7 8 9 10 parameters in it so it's  a it's some kind of nine dimensional space   of cubics i say nine dimensional because you  can always multiply this equation by a scalar   and it's really the same cubic so there's one  degree of freedom there that's just that's scaled   all right so there's lots of cubics and the  natural question is what do they look like   it's much richer story than the cubic functions  so for example one kind of cubic that you can get   very simple cubic is just to take the product  of three lines that's an example of a cubic just take the equation of one line multiply by  the equation to the other line multiplied by   the equation of the third line you're going  to get a three degree three curve and its   set of points is going to look like that now  if you take that particular cube again you   modify the terms just a little bit okay  then it's going to perturb and it's no   longer going to be a product of three lines  typically what happens is something like this   you will get a blob inside here like this  and then these branches sort of become   curves that are sort of asymptotic roughly so  you may get something like that right so that's   that's a kind of a cubic curve but there are  many others and newton no less was naturally   interested in classifying cubic curves he  saw this as an important thing that one   should do one should classify cubics right and  in fact it's something that everyone who's doing   mathematics it should occur to you after you've  done conics you should ask well what about cubics   can we classify them is there anything like  ellipse parabola hyperbole that kind of thing   well it's much more complicated he classified  them into some 80 some different types   he had a few that were missing um the  story is is quite a long story we're   not going to go into it but these days  there's some sort of canonical forms   if we make something a little bit more general  y squared equals a x cubed plus b x plus c   so more or less with a suitable  transformation of variables and   actually we need a projective change of variables  you can cook up any cubic to be sort of of this   form and then there are various possibilities  so one possibility is something like this another possibility is something like this another possibility is something like like this well some of these are elliptic curves  so this is not this is an elliptic curve   now this kind of curve has a singularity there's  a point of non-differentiability there this curve   also has a non-differentiability so these curves  here are somewhat different from this one which is   non-singular okay so non-singular  cubics are called elliptic curves   so an important problem that that's related  to these cubics is how to parameterize them   how do you describe the points on a cubic like  this it was known since the ancient greeks that   you could parameterize conics any conic can be  parameterized in the sense you can you can find   a rational function to parameterize a conic it  turns out that not all cubics can be parametrized   so the story with cubics is considerably more  complicated than with conics so not all cubics can be parametrized okay so um that's probably a good  place for us to stop but the next   lecture we're going to delve a little bit  more into this issue of parameterizing   cubics we have a look a little bit more  at uh at cubics a whole new world there   and sometime in between then i'll i'll post  that video on on geogebra so i'll see you then [Music] you

Channel: Insights into Mathematics

Views: 219,893

Keywords: Differential Geometry, course, UNSW, N J Wildberger, curves, surfaces, topology, Einstein, general relativity, conchoid, cissoid, pedal curves, mathematics, education, evolute, involute, calculus, analytic geometry, robotics, physics

Id: _mvjOoTieTk

Length: 44min 10sec (2650 seconds)

Published: Fri Aug 02 2013