- [Voiceover] So, in the last video I talked
about curvature and the radius of curvature,
and I described it purely geometrically
where I'm saying, you imagine driving
along a certain road, your steering wheel locks,
and you're wondering what the radius of the circle
that you draw with your car, you know through
whatever surrounding fields there are on the road
as a result, and the special symbol
that we have for this, for this idea of curvature
is a little kappa, and that's gonna be one divided
by the radius, and the reason we do
that is basically you want a large kappa,
a large curvature to correspond with a sharp turn
so, sharp turn, small radius, large curvature.
But the question is how do we describe
this in a more mathematical way?
So, I'm gonna to go ahead and clear up, get rid of the
circle itself and all of that radius
and just be looking at the curve itself in it's own right.
The way you typically describe a curve like this
is para-metrically, so, you'll have
some kind of vector valued function s
that takes in a single parameter t,
and then it's gonna output the x and y coordinates,
each as functions of t.
So, this will be the x coordinate,
the y coordinate.
And then this specific case, I'll just tell you
the curve that I drew happens to be parameterized
by one minus the sign of t as the x component function,
actually no, it's t minus sign of t, and the bottom part
is one minus cosign of t.
That's the curve that I drew.
And the way that you're thinking about this
is for each value t you get a certain vector
that puts you at a point on the curve,
and as t changes the vector you get changes,
but the tip of that vector kind of traces
out the curve as a whole, right?
And you can imagine the vector drawing the curve
as t varies,
and the thought behind making curvature mathematical,
here I'll kind of clear up some room for myself,
is that you take the tangent vectors here
so,
so, you might image a unit tangent vector
at every given point,
and you're wondering
how quickly those guys change direction,
right?
So with the little schematic that I have drawn here
I'm just gonna call this guy t one
for like the first tangent vector,
t2 , t3, and I haven't specified
where they started or anything
I just what give a feel for you've got
various different tangent vectors
and I'm just gonna say that all of them
each one of them t sub somethings has a unit length.
And the idea of curvature is to look at how quickly
that unit tangent vector changes directions.
So, you know you might imagine a completely different space
so, rather than rooting each vector on the curve,
let's see what it would look like if you
just kind of write each vector
in its own right off in some other spot.
So, this guy here would be t one,
and then t two points a little bit down,
and then t three points kind of much more down
So, all of these, this would be t one, that guy is t two
and these are the same vectors I'm just kind of drawing them
all rooted at the same spot so it's a little easier
to see how they turn, and you wanna say,
okay how much do you change as you go from t one
to t two, is that a large angle change?
and as you go from t two to t three,
is that a large change as well?
And you can kind of see how if you have a curve,
and let's say it's,
if you have a curve that curves quite a bit,
you know it's doing something like that
then the unit vector, the unit tangent vector
at this point changes quite rapidly
over a short distance to be something
almost 90 degrees different.
Whereas, if you take the unit vector here
and see how much does it change as you go from
this point over to this point
it doesn't really change that much.
So, the thought behind curvatures,
we gonna take the rate of change of
that unit tangent vector,
so, the rate of change of t, and I'm gonna
let capital T be a function
that tells you whatever the unit tangent vector
at each point is,
and I'm not gonna take the rate of change
in terms of, you know the parameter little t,
which is what we use to parameterize the curve,
because it shouldn't matter how you parameterize the curve,
maybe you're driving along it really quickly
or really slowly.
Instead, what you wanna take
is the rate of change with respect to arc length.
Arc length, and I'm using the variable s here
to denote arc length, and what I mean by
that is that if take just a tiny little step here
the distance of that step, the actual distance
in the x y plane, you consider to be the arc length.
And if you imagine it being really, really small
you're considering that a d s,
a tiny change in the arc length.
So this is the quantity that we care about.
How much does that unit tangent vector change
with respect to a tiny change in the arc length?
You know, as we travel along, let's say,
it was a distance of like 0.5, right,
you wanna know, did this unit vector change a lot
or a little bit.
But I should add something here,
this tiny change in the vector
that would tell you,
you know what the vector connecting
their two tips is,
so, this would be a vector valued quantity,
and curvature itself should just be a number.
So, what we really care about is the size of this guy.
So, what we really care about, the size of this,
which is a vector valued quantity,
and that'll be an indication of just
how much the curve curves.
But if I'm at a sharper turned curve
you go over that same distance
and then suddenly the change
in the tangent vectors goes by quite a bit
that would be telling you it's a high curvature.
And in the next video I'm gonna talk through
what that looks like, how you think about
this tangent vector function,
this unit tangent vector function,
and what it looks like to take the derivative of that
with respect to arc length.
It can get a little convoluted
in terms of the symbols involved,
and the constant picture you should have
in the back of your mind is that circle,
that circle that's hugging the curve very closely
at a certain point,
and this means of taking the magnitude of the rate of change
of the unit tangent vector with respect to arc length
is all just trying to capture that idea.