- [Voiceover] In the last video I started to talk about
the formula for curvature.
Just to remind everyone of where we are
you imagine that you have some kind of curve in let's say
two dimensional space just for the sake of being simple.
Let's say this curve is parameterized by a function S of T.
So every number T corresponds to some point on the curve.
For the curvature you start thinking about
unit tangent vectors.
At every given point what does the unit
tangent vector look like?
The curvature itself which is denoted by this sort of
Greek letter Kappa
is gonna be the rate of change of those unit vectors
kind of how quick they're turning in direction
not with respect to the parameter t
but with respect to arc length d s.
What I mean by arc length here is
just a tiny step you can think
the size of a tiny step along the curve would be d s.
You're wondering, as you take a tiny step like that
does the unit tangent vector turn a a lot or
does it turn a little bit?
The little schematic that I said you might have in mind
is just a completely separate space
where for each one of these unit tangent vectors
you go ahead and put them in that space
saying okay so this one would look something like this
this one is pointed down and to the right so it would
look something like this.
This one is pointed very much down.
You're wondering basically as you take tiny little steps
of size d s what is this
change to the unit tangent vector
and that change is gonna be some kind of vector.
Because the curvature's really just a value
a number that we want all we care about is the size
of that vector.
The size of the change to the tangent vector
as you take a tiny step in d s.
Now this is pretty abstract right?
I've got these two completely separate things that
are not the original functions that you have to think about.
You have to think about this unit tangent vector function
and then you also have to think about this
notion of arc length.
The reason, by the way, that I'm using an s here
as well as here for the parameterized of the curve
is because they're actually quite related.
I'll get to that a little bit below.
To make it clear what this means I'm gonna go ahead
and go through an example here where let's say our
parameterized with respect to t is
a cosine sine pair.
So we've got cosine of t as the x component
and then sine of t as the y component
sine of t.
Just to make it so that it's not completely boring
let's multiply both of these components by constant r.
What this means, you might recognize this,
cosine sine pair
what this means
is that in the x y plane
you're actually drawing a circle with radius r.
This would be some kind of circle with the radius r.
While I go through this example I also want to make a note
of what things would look like a little bit more abstractly.
If we just had s of t equals not specific functions
that I laid down but just any general function
for the x component and for the y component.
The reason I want to do this is because the concrete
version is gonna be helpful and simple and something
we can deal with but almost so simple as to
not be indicative of just how complicated the normal
circumstances but the more general circumstances
so complicated I think it will actually confuse
things a little bit too much.
It'll be good to kind of go through
both of them in parallel.
The first step is to figure out what is this
unit tangent vector?
What is that function that at every given point
gives you a unit tangent vector to the curve?
The first thing for that
is to realize that
we already have a notion of what should give the
tangent vector
the derivative of our vector valued function
as a function of t
the direction in which in points is
in the tangent direction.
If I go over here and if I compute this derivative
and I say s prime of t
which involves just taking the
derivative of both components so the
derivative of cosine is negative sine
of t multiplied by r and the derivative
of sine is cosine
of t multiplied by r.
More abstractly, this is just gonna be
anytime you have two different component functions
you just take the derivative of each one.
Hopefully you've seen this, if not
maybe take a look of videos on
taking the derivative of a position vector valued function.
This we can interpret as that tangent vector
but it might not be a unit vector right?
We want a unit tangent vector and this
only promises us the direction.
What we do to normalize it
and get a unit tangent vector function
which I'll call capital t of lowercase t
that's kind of confusing right?
Capital t is for tangent vector
lowercase t is the parameter.
I'll try to keep that straight.
It's sort of standard notation but
there is the potential to confuse with this.
What that's gonna be is your vector value derivative
but normalized. So we have to divide by
whatever it's magnitude is
as a function of t.
In this case, in a specific example
that magnitude, if we take the magnitude of
negative sine of t r
multiply by r
and then cosine of t
multiplied by r
so we're taking the magnitude of this whole vector
what we get
make myself even more room here
is the square root
of sine squared
negative sine squared is just gonna be sine squared
so sine squared of t multiplied by r squared
and then over here cosine square times r squared
cosine squared of t times r squared
we can bring that r squared outside of the radical
to sort of factor it out
turning it into an r
and on the inside we have sine squared plus cosine squared
I'm being too lazy to write down the t's right now
because no matter what the t is that whole value
just equals one.
This entire thing is just gonna equal r.
What that means is that our unit tangent vector up here
is gonna be the original function but divided by r.
It happens to be a constant usually it's not
but it happens to be a constant in this case.
What that looks like
given that our original function is
negative sine of t times r and cosine of t times r
we're dividing out by an r
the ultimate function that we get
is just negative sine of t
and then cosine of t.
For fear of running a little bit long
I think I'll call it an end to this video and
continue on with the same argument in the next video.