Good afternoon, I'm Norman Wildberger, and we're
here at the University of New South Wales. This is Lecture 16 of this course in the history of
mathematics. We're talking about differential geometry, a lovely combination of calculus
and analytic geometry applied to curves and surfaces. So that's what differential
geometry is, at least to begin with. And one of the key concepts that runs through the
history of the subject is the idea of curvature, first of curves and then of surfaces, and
then ultimately of higher dimensional things, although we won't be talking about that. But
in the 20th century, curvature turned out to play a very important role in modern physics due
to the theory of relativity of Einstein. Okay, so I guess our story can start with
the 17th century, and around 1673, we had some people interested in planar
curves and in particular Christiaan Huygens, who was a famous Dutch mathematician,
astronomer, inventor, started investigating the notion of an involute of a curve.
So he introduced involute of a curve. Christiaan Huygens also did other things. He
was also the inventor of the pendulum clock, which of course is, well, it's usually a long edge
box and it has some kind of pendulum that swings back and forth and then there are these various
gears and stuff up here that tick every time the thing swings back and forth. And the beauty about
the pendulum clock is that if this pendulum arm is rather long and if the amplitude is not too
big, then the time taken for every swing is pretty well independent of the amplitude. So a
pendulum clock lets the things swing over once and then it gives it another little kick and then
it swings over again and it gives another little kick and every time it swings once, that's
the unit of time that can be then measured by some gears. And that was Huygens' invention,
and it was a very, very important technical achievement because it meant that there was a big
improvement in timekeeping. So around that time, they had mechanical watches but those watches were
not very accurate and they typically lost around 15 minutes every day. Okay, so you know from
one day to another you could be not very sure up to about 15 minutes of your clock, but with
these ones here, that was dramatically reduced to down to about half a minute or 15 seconds.
The pendulum clocks were much better and they allowed people to make trigonometric large-scale
surveying measurements much more accurately. So that people could survey, for example France,
you need clocks to be able to figure out what longitude you're at. So large-scale surveying
requires clocks, and Huygens was prominent. He also was a notable mathematician around this time.
One of the French kings, Louis XIV, was persuaded by one of his advisers that he should set up a
royal academy. So he set up a French Academy of Science, where basically they got together lots of
prominent scientists and gave them lots of money and resources and said, "Go and do good things,
go and survey France, go and figure out what the shape of the earth is, various things like that."
So the French Academy was a prominent sort of directing force for various scientific initiatives
of the day, and Christiaan Huygens, even though he was a Dutchman, was invited to be a director
of this because of his prominence stature. Okay, so what did he do in terms of the involute of a
curve? Well, the idea is a pleasant one. You've got a planar curve, that means one that you can
draw on the plane. And then Huygens said, "All right, let's think about a piece of string that's
tight and wound around the curve." So imagine having a piece of string, maybe it's attached
there for the sake of argument, and it winds around the curve and it's kind of tight. And let's
stop it there. And then let's say, at this point, we're going to remove it. Maybe we should cut it,
maybe about here. And we're going to hold it and hold these edie ends of the string and move it in
this direction, but keeping the string taut all the time. So the next little position, it's like
this. And then it's tight there. And then a little bit further on, it's tight there. And then a
little bit further on, it's probably tight there. And so on. And then this gives a new curve. This
curve here is called an involute of the original curve. So the original curve is C, then this
blue curve, let's call it C prime, that's called an involute of the curve C. So it's obtained
by essentially unwrapping a string tightly. Now, because that endpoint is rather variable,
we could also have started with another endpoint here, in which case we would have gotten
another bunch of points and we would have gotten a different involute. So that would be
another involute, say C double prime. And you can do this in a number of ways. And Huygens proved
that all the involutes are disjoint. So the blue curves don't cross each other. And moreover,
they're always perpendicular to the tangents. So for example, if we look at one particular
position, say that one right there, at that very moment in time, we have a tangent
from the point of contact of the curve in the string. And what's happening is we're having a
rotation, it's been very small rotation really, essentially about this contact point. And
that means that very close to this point, this thing is actually going perpendicular
to the tangent. So this involute is always perpendicular to the tangent that you
can draw to the curve at that point. Well, it turned out that there was a dual kind
of notion to this which is called the evolute of a curve. And how does that go?
So, if you have a given curve, then at any given point on it, you can draw a
few interesting objects. The most obvious thing, I suppose, is the tangent line. So, it's the
line which approximates the curve at that point. Let's say the point is P; that's
the tangent line to the curve, same courtesy at P. But something else that
we can draw is the normal to that tangent. So, that is the normal line to the curve at P. So, what was realized is that if you take two
points which are very, very close to each other, let's say P and Q. Well, if you join those two
points, then you get something which is almost a tangent. That's well known in calculus. But
if you consider the two normals to P and Q, so if we consider the other normal here, then
those two normals will meet in some point, and as P and Q get closer and closer,
this point of intersection stabilizes and approaches a limiting value. This limiting
value is called the center of curvature. Center of curvature of the curve C at the point
P, that's assuming a trajectory approaching Q. And it has another description that
was discovered by Newton and Leibniz, which is that if you take three points
near P, so P, Q, and let's say R as well, so three points very close together. Well,
three points don't determine a line, but they do determine a circle. So, any three points,
unless they're collinear, they're not linear, they determine a circle. So, these three points,
PQ, and R, even though they're close together, they're not collinear. That means there's
a circle that we can draw through them. And that circle, well, it depends on
R and Q, but as R and Q approach P, that circle stabilizes just as the tangent
stabilizes. The circle stabilizes and as joins R and Q both approach P, this circle
is called the oscillating circle of the curve C at Point P, and the center of
curvature is the center of that circle. The place where adjacent normals meet is the
center of this oscillating circle to the conic at that point. This is the best circle that
approximates the curve near the point P, just as the tangent is the best line that approximates
the curve at the point P. This is the best circle that approximates the curve at Point P. So, how do
we get the evolute of the curve? What we do is we look at this point, and we see what happens to it
as we vary the point P. At this point, it depended on the point P. If we vary the point P, well,
then adjacent normals, it might meet in some other point, and over here, they might meet
in some other point. Now, we're going to get some curve over here. Who knows what it looks
like? So, the trajectory of the centers, that's called the evolute, the evolute of the curve C.
So, it is the locus of the centers of curvature. There's a few other associated concepts
which I might put on this board as well. So, this radius of this circle might call that
Rho. That's called the radius of curvature, of course, it depends on the point P. And then
one more definition: the curvature of the curve. The curvature at P is, by definition, 1 over the
radius of curvature. So, the bigger the radius is, the smaller is the curvature; the smaller
the radius is, the bigger is the curvature. That's the curvature of the curve at the point P,
and it's often given the letter K for curvature. So, these ideas were introduced by the
17th-century mathematicians and were studied. And then if you think about it for a little
while, you can convince yourself, as Huygens did, that the evolute and the involute are
kind of opposite constructions. So, the evolute of an involute is the original
curve. So, for example, if our original curve C, and then we had an involute like this, that
was what you get when you unwrap the string, and then if you just look at the strings
that we're unwrapping, so here it is. Okay, it's kind of obvious, because we said that
these were all going to be normal to the curve, it's almost obvious that where two adjacent
normals meet is going to be exactly a point on the curve. So, the evolute of the
involute is the original curve C. So, that's the involute C prime, and we're saying that
the evolute on C prime gives the original curve C. What happens if we have a function? Because after
all, that's where a lot of curves come from. We have some function f of X, and we're considering
its graph. A natural question is: suppose we have a point X here. We might be interested in what
is the curvature at that point. But not just be interested in tangent, might also be interested
in the osculating circle and its radius. Well, both Huygens and Newton gave a
formula for this radius of convergence, the radius of curvature, and it's this: it's
1 plus dy/dx all squared to the three-halves, divided by the second derivative
d squared y/dx squared. That's a formula for the radius of curvature
in terms of the derivative of the function at that point and also the second
derivative of the function at that point. "Why don't we learn that in
calculus?" you ask. Good question. Okay, so there are some nice examples of
involutes. We've already had a look at one of them when we were talking about the cycloid.
I remind you, the cycloid is what you get when you roll a ball on a circle or a lot of straight
line, and you look at the image of one point on the route circle as it rolls. So, that's what's a
cycloid. And if we invert the cycloid like this, and we have a piece of string that's
tied up there, and then we let it go, we saw that we got the cycloid pendulum,
and that thing there was also a cycloid. So, as Huygens realized with this pendulum business,
that the involute of a cycloid is another cycloid. Another important example is a catenary. So,
we had a catenary, which was the shape of a hanging chain. I remind you, that was basically a
hyperbolic function. And if you take its involute, so for example, you might stretch a
string from here around the curve, maybe to the midpoint there, and
then if you let that piece go, you get an important curve called tractrix.
So, that's the involute of a catenary. This is also the path that you get when
you take your pet rock for a walk. So, if you are here and you are walking in this
direction, and you have your pet rock which is sitting here and it's on a leash, and
this is a sort of a plainer representation, so this is the plane, and you're walking this
way, well, as you walk, once you're over here, the rock will have been dragged a certain amount.
It just drags behind you on its leash. And when you're over here, well, it'll be over here
like this. And when you're very far out, it'll be like this. So, the path that
that rock makes is also a tractrix. That's over for what we said here: the
tractrix is the involute of a catenary. So, the catenary is the evolute of the tractrix.
In other words, if we take normals to the tractrix and we look at adjacent normals or almost
adjacent normals, then they're going to meet on the catenary. Another interesting curve is the
parabola. So, if we take our favorite parabola y equals x squared, we can, of course, take its
involute or its evolute. So, if we take its involute, say we put a string from somewhere along
here and then cut it here and let it fall, well, we could do it on this side or on that side,
we would get something that looks like that. But perhaps more interesting is what happens
if we look at the evolute. So, if we look at normals to the parabola at every point, we draw a
normal, then what it turns out happening is that there is an interesting curve that results.
The evolute looks like something like this. The evolute is evolutive a parabola. Here's what's called a semi-cubical parabola, and
it has an equation that looks, well, in this case, it will have the equation y equals one-half plus
3/4 - x to the 2/3, and if you draw a diagram of this accurately, then what you see is that below
the evolute there is no crossing of the normals, but above the evolute, there's a rich
crisscrossing of normals. Down here, all the normals are sort of separate; above the
evolute, they're all forming this checkerboard kind of pattern. Alright, so those are some of the
investigations that were made with planar curves. There was also some important work done on
space curves or three-dimensional curves, and the person most responsible for this
was Alexis Claude Clairaut (1713-1765), and then Euler and Co. also contributed in big
ways. So, we have a three-dimensional situation now, and we have a curve in space,
and it's not necessarily in a plane; it's twisting all around. So, what can you do
with such a thing? Well, Clairaut said okay, one of the things we can do certainly is we
can look at a tangent line. So, at a point P, we can look at a tangent line as usual. Another
thing we can do is we can look at three adjacent points. Those three adjacent points are going to
form a plane; we're also going to form a circle, but that circle will lie in a plane
that's called the osculating plane. So, we have three adjacent points like P, Q,
and R, and when I say adjacent, I mean very, very close together, almost touching. Yes,
sorry, all three, yeah. If we take P and we take two other adjacent points R and Q,
then all three points together form a plane, and that's called the oscillating plane of the
curve. Let's call the curve C at the point P, and of course, that plane then has a
normal direction. So, there are, in fact, then sort of three special directions that you
can associate to the curve at the point P. You can specify its tangent direction, then you
can specify another direction that's in this osculating plane perpendicular to the tangent
direction, so usually called the binormal, I believe, or principle normal. That's the
principle normal, that's in this plane but perpendicular to the tangent. Here's the
tangent. And then there was a third one, which is perpendicular to both of those, which is
perpendicular to the oscillating plane, and that one's called the binormal. So, that's a basic
sort of theoretical framework that was realized was important. You have a space curve, then at
every point there are these three directions associated with the curve, and as you move along
the curve, these three directions change. So, they perform a role of a reference frame with respect
to the curve that plays an important role in the theoretical development. There's lots of formulas
associated with it which we won't write down. Alright, then the next study concentrated
on surfaces, a very natural extension. We're working in three-dimensional space; we have
some kind of two-dimensional surface. Okay, maybe try to make it some nice
surface, and we want to study this. So, I guess the main contributors here were Euler
in 1760 and then Moens a little later. Euler liked to be specific, so he often visualized the surface
in terms of a three-dimensional coordinate system. And if you have a three-dimensional coordinate
system, you can specify a curve by thinking of it in the form Z equals f of XY, they give us a
function lying over some region in the XY plane. But his considerations work for a general kind
of surface. So what did Euler do? What can you do in such a situation? Well, the thing to
do is to start with a point and say, "Well, let's study this surface near a point."
So if you have a point, let's call it P, and it's on a surface, then the natural thing
to do, which is sort of in the same spirit as what we're doing with the curves, is to
look at the tangent plane to the surface. The tangent plane, that's a plane that
just touches the surface at that point. Okay, so that's a tangent plane. A very good
surface to think about is your hand. The hand is a great surface. Okay, so if you look at
your hand, there's all kinds of points on it, but you could choose any point at all. Say, I
choose that point right there, then the tangent plane to that point is going to be something
like this. Well, if I choose the point up here, the tangent plane might be like this. Well,
if I choose a point there, the tangent plane would be like that. And of course, associated
to such a tangent plane is a normal direction because every plane has a normal direction.
So we have a tangent plane at every point, and we also have a direction that's perpendicular
to the tangent plane, which is a normal direction. And then Euler said, "Okay, that's alright, but
I know all about curvature of curves. So how can I talk about the curvature of a surface?" Well,
he said, "What I need to do is I need to slice this surface so that I get a curve." So suppose
I take a plane through P, and I won't just take any plane, I'll take a plane which contains the
normal direction. Okay, so Euler said, "All right, Euler considered planes through P containing
the normal." So in terms of one's hand, what would that mean? So, okay, let's use this hand
here. So there's a hand, there's a point there, okay, then the tangent plane is like this, and
the normal direction is coming out like this. So I want to look at planes which pass through the
point and contain the normal direction. So that would be such a plane. There's another one. All of
these planes as I rotate them around, they're all going through P and they're all going through the
normal vector, and each one of them is slicing my hand and would give you a cross-section of my hand
if you actually sliced. Okay, so imagine each one of these planes making a cross-section through the
surface. So if we take any one of those planes, let's sort of put it something like this. That's
supposed to be a plane that goes through P and also contains the normal, then that's going to
slice the surface in a curve, a planar curve. Right, so you can just actually see there will
be a line on my hand where that plane touches, cuts my hand. Question? Well, it's not, it doesn't
actually have to go through, you can just draw it right on the surface of the hand with a pen.
Right, so there's a point P, then that might be one such curve, there might be another such curve,
just the intersection of the surface with a plane containing the point and a normal to it. Okay, so
that's just one possible such curve, but we can take the curvature of that curve that's what Euler
did, and we can consider the planes through P containing the normal, and that gives us a curve,
gives us a planar, it's a cross-section curve. And that curve has a curvature, and maybe
we could think of the K, the curvature K, as depending on some angle, the
angle that this plane makes with some specified direction. So
as we rotate the plane around, we get different curvatures of all these
different cross-sections, one for every angle. And then Euler proved a theorem that in fact, as
we look at all these possible curvatures, as we vary the plane through P, look at all the possible
curvatures, the curvatures go up and down, but there's one maximum value and one minimum
value. There will be one direction in which it's curved the most and one direction which is
curved the least. Okay, so there, as we go around, there is then sort of two special, two critical
values. In the calculus language, critical values of the curvature, let's call them k1 and k2,
and these are sometimes called the principal curvatures of the surface at the point P.
Now, if you go back to our hand again, I have to say a little bit more. So, suppose
that I happen to take a point which is right in the crux right here. Again, that's a rather
interesting point because the tangent plane will be flat like this, and the normal will be straight
up. So, the planes that I'm considering are planes that are going right through my hand, up and down
like this. Then, the most extreme curves are the one that's going up like this and over, and the
other extreme one is one that's going like this. So, here, the circle of curvature is in this
direction, and here, the circle of curvature is down here. So, in this framework, what we have
to actually do is we have to consider signed curvatures. So, we have to give this curvature a
different sign from this one. So, this is positive something, then this is going to be negative
something. Okay, so this is a situation where the two different curvatures, the two principal
curvatures, have different signs. That's a rather different situation from a situation like this,
where the two curvatures are both positive. Okay, so the next person to really
take on this subject and really go a long way in it was the great German
mathematician Carl Friedrich Gauss. So, Carl Friedrich Gauss, preeminent
mathematician of the 19th century (1777 to 1855), sort of the Colossus of mathematics during his
time, spent most of his years in Gottingen, did very important work even as a young person.
As a teenager perhaps, he figured out that you could make a ruler and compass construction of
the regular 17-gon. He gave several proofs of the fundamental theorem of algebra. He wrote
the greatest work of number theory written, which is "Disquisitiones Arithmeticae". That was
in 1801 when he was still a relatively young man, but this just took number theory to a whole
new level and established him as one of the great mathematicians of the day. He also became
famous in a public way as a result of some work in astronomy. He predicted where a certain asteroid,
Ceres, would end up showing up. So, they had some asteroid that they were plotting and they lost
sight of it, and eventually he was able to calculate where they should look for it, and they
were right, he was right. And then later on in his life, he had some official positions involving
geodesy. He became involved in lots of field work. And this work probably contributed to his
interest in surfaces was actually going out there and making measurements and mountains
and rivers and so on. And we actually derive some of his work in the differential
geometry note as a result of this. So, he did a lot of things in this direction,
but there's one sort of main theorem that we can state in terms of these
principal curvatures of Euler, and this is a result that he liked it so much
that he called it his theorem "Egregium", which means something like excellent theorem.
And that is that if you take the two principal curvatures that we've been talking about, the ones
Euler and take the product, the product of the two principal curvatures is a new kind of curvature
which is determinable from the surface itself. In other words, if you were a bug and you were
living on some two-dimensional surface and you could only scoot around on this surface, but
let's say you had rulers you could drag around, you had protractors that you could
drag around and make measurements, you could figure out what the curvature of the
surface is at some point. You couldn't figure out what K1 and K2 were individually, but you could
figure out what the product was at the point P. Okay, one way of thinking about that is let's say
you're on a sphere. Suppose you were on a sphere and you suspect that you were on a sphere,
not a flat Earth, and you wanted to prove to someone you're on a sphere. How could you do it
by just making measurements on a sphere? Well, one way to do it would be to take a circle, a
circle of radius R, and to measure carefully the area of that circle. The area for a planar
circle would be πR², but if you didn't get πR² and you got something different, you might start
to suspect that maybe I'm not on a plane. In fact, you could tell by men if you took a smaller
sphere and you also went it the same distance R, you would find that the circle of radius R
on the big sphere and the circle of radius R on the little sphere would be different. And
by making simple measurements on the surface, you could tell whether you're on a big sphere
or a little sphere. So, in other words, you could tell the curvature just by making
measurements on the surface without having to look at it from outside. That's essentially the
content of Gauss's theorem, oh, hey, egregium. And this constant K is now called the
Gaussian curvature, Gaussian curvature of the surface at a point, at a point.
Of course, for a sphere, the Gaussian curvature is not too hard to figure out because
at any point P, if the sphere has radius say ρ, then any curve that you get by taking what
these normal sections is a great circle, and that great circle has radius ρ. And so the
curvature K1 will be the same as the curvature K2, will be 1 over ρ, and so the Gaussian
curvature will be 1 over ρ squared. So, for a sphere, the curvature is constant,
is the same at every point, sort of obvious from the symmetry, but the formula also works.
So, that's what's called a curve or a surface of constant curvature. And rather remarkably, it
was discovered that there were some other surfaces of constant curvature. The sphere is kind of
an obvious one, but there's another surface called a pseudosphere, which you get by taking
the tractrix that we considered earlier. So, you take the tractrix, that's the curve that you
get by taking the involute of the catenary or dragging your rock for a walk, and you rotate
it around the axis that you would walk on, and you get some surface of rotation. So, the
tractrix revolved, that's called the pseudosphere. Now, that thing has some remarkable properties. One of its remarkable properties is that it's of
constant curvature, constant Gaussian curvature. That means that if you look at some point on it,
well, there'll be one of the principal curves or directions will be in a circle going this way,
and the other principal direction will be a circle going this way, and one of them has positive
curvature, whichever way you're measuring, the other one has negative curvature. So
it's like a point on the saddle of your hand, and the product of those two curvatures
happens to be the same no matter where you are on the surface. So this thing has constant
negative curvature, constant negative curvature. This sphere, the pseudosphere, ended up being an
important model for hyperbolic geometry later on. So the notion of curvature was quite
important. First curvature of planes, then curvature of surfaces, and then the
story goes in more complicated directions. I might just mention two more names. One is
the essentially a student of Gauss, Riemann, who was a remarkable genius who extended ideas
of differential geometry from three-dimensional space to n-dimensional space and outlined ideas
of curvature even for higher-dimensional surfaces, whatever they might be. And it was all rather
abstract and seemingly not good for anything too practical until Albert Einstein came along. After
his special theory of relativity in the 1910-1918 area, he was trying to extend his special theory
of relativity to general theory of relativity, and it turned out exactly what he needed was
Riemann Raymond's notions of differential geometry and higher dimensions. In particular, notions of
curvature ended up playing a very important role. Einstein basically realized that somehow mass
curves space-time, and the amount of curvature induced by a mass in space-time by a mass is
proportional to the mass. So this curvature played and it still plays a very important
role in modern physics. So next time we're going to have a look at topology, the beginnings
of topology in the more modern era. See you then.
