Application of the fundamental laws (setup)

WEBVTT
Kind: captions
Language: en

00:00:00.372 --> 00:00:01.205
- [Voiceover] Alright, now we're ready

00:00:01.205 --> 00:00:04.252
to learn how to do circuit analysis.

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This is what we've been shooting for

00:00:05.965 --> 00:00:08.322
as we've learned our Fundamental Laws.

00:00:08.322 --> 00:00:11.655
And the fundamental laws are, Ohm's law,

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and Kirchhoff's laws,

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which we learned was
Kirchhoff's Current Law

00:00:19.590 --> 00:00:22.554
and Kirchhoff's Voltage Law.

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This says that the sum of all
the currents going into a node

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adds up to zero.

00:00:27.912 --> 00:00:31.081
And the Voltage Law says
the sum of the voltages

00:00:31.081 --> 00:00:34.085
going around a loop adds up to zero.

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And the other thing we learned
was the sign convention.

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So, we had a passive sign convention,

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which was, if I label
the voltage like that,

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then I label the current
flowing into the positive end,

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the positive end and
the current go together.

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So, these are the tools that we have

00:00:54.347 --> 00:00:58.445
for analyzing a circuit and
I've drawn a circuit here.

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So, what we see is, we
have a voltage source,

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we'll call that VS, and we'll
give it a value of 15 volts.

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We have a resistor, R1,

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that has a value, we'll give
that a value of four K ohms.

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And another resistor, R2
is sitting right here,

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and we'll give it a value
of 2 K ohms, 2,000 ohms.

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And finally, there's a current
source sitting over here.

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And this has a value of three milliamps

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flowing downwards.

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So, the point of circuit analysis
means what we wanna do is,

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we wanna find all the
currents and all the voltages

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in this circuit using
our Fundamental Laws.

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So first, let's identify
the nodes in the circuit.

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We'll give them letter names.

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We'll call this node A,

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and that's the junction between R1 and Vs.

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We'll give this one a name,

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this is a node that's distributed,

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all three of these components
joined right about here,

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we'll call that node B.

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And that's one of the nodes.

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And down at the bottom here,
there's the third node,

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which is this distributed node here.

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It connects the voltage
source, the resistor,

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and the current source, and this is all,

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we'll call that node C.

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Now, we could also label
some voltages on here.

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We'll label the voltages,
we'll do that in orange.

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We'll call this V1 and we'll
give it a plus sign here.

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And the minus sign here,
that's the voltage across R1.

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We'll give R2 a name,
we'll get the voltage here,

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we'll call this plus, minus V2.

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That's the voltage across R2,

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it's also the voltage
across the current source.

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B, it's the voltage
between node B and node C.

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And now let's label the currents.

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Let's label the currents in this circuit.

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We have a current

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coming through R1, we'll call that i1.

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There's another current going through R2

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and we'll call that i2.

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There's a current going here through Is

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and we actually already know
that, we'll just call that Is.

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And that's all the currents,

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that's all the unknown
currents in our circuit.

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If we stare at this, we're
not gonna be able to tell,

00:03:42.532 --> 00:03:45.945
on just inspecting this, what's going on,

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we have to come up with
some solution technique.

00:03:48.586 --> 00:03:50.489
And we're going to solve
this by, what's called,

00:03:50.489 --> 00:03:53.616
Application of the Fundamental Laws.

00:03:53.616 --> 00:03:55.362
And here's our laws over here.

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And the whole technique is
based on solving simultaneous

00:04:04.915 --> 00:04:05.748
equations.

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That's the skill of circuit analysis,

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circuit analysis means setting up

00:04:14.285 --> 00:04:17.739
and solving simultaneous equations.

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So, let's write down some things
we know about the circuit.

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And first thing we're gonna apply is,

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we'll apply Ohm's law
to the two resistors.

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So, we can say

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V1 equals i1 times R1.

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And we can say that V2 equals i2 times R2.

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Alright, that's not a bad
start, we have two equations,

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how many unknowns do we have?

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I2 and V2 are unknown.

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I1 and R1 are unknown.

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So, we have four unknowns
and we have two equations,

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so, we're halfway there.

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And now, we need to come up
with some more equations.

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And I'm just gonna stare at the circuit

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and what I'm gonna do is,

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I'm gonna apply KVL around this loop here,

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that's gonna be my KVL loop.

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And after we get that equation,
what we're gonna do is,

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we're gonna write KCL,

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we're gonna write the
Current Law at node B.

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That's the two equations
we're gonna develop here.

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So, let's do KVL.

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Let's start at this node, at
this corner of the circuit

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and go around clockwise around.

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And what we'll see
first is a voltage rise,

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plus Vs.

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Next, I see a voltage
drop, so it's minus V1,

00:06:08.860 --> 00:06:11.610
going around this way, minus V1.

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I'm at this point, now
I'm gonna go through R2

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and I see a voltage drop of
V2, minus V2 equals zero.

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That's Kirchhoff's Voltage
Law around this loop.

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One rise, two drops, one rise, two drops.

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Now, we're gonna move up here,

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and do Kirchhoff's Current
Law at this node right here.

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So let me draw a few more arrows.

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This is i1, it comes right
through that resistor.

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And this is Is.

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So Kirchhoff's Current Law,

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let's count the currents
going into node B,

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current going into node B is i1.

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Currents go out, we'll set that equal

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to the sum of the currents
going out of node B.

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So, that's equal to i2 plus Is.

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So look, now I have my four equations,

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here's my four simultaneous equations.

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And this is what we have to solve.

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So let me move the screen up
and we'll start doing that.

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Let's do a double check,

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we have four unknowns, V1, i1,

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V2, i2.

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Here's V1 and V2, and here's i1 and i2.

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And I have four equations, so
we oughta be able to do this.


Curvature formula, part 1

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So, in
the last video I talked

00:00:01.765 --> 00:00:04.614
about curvature and the
radius of curvature,

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and I described it purely geometrically

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where I'm saying, you imagine driving

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along a certain road,
your steering wheel locks,

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and you're wondering what
the radius of the circle

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that you draw with your
car, you know through

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whatever surrounding fields
there are on the road

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as a result, and the special symbol

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that we have for this,
for this idea of curvature

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is a little kappa, and
that's gonna be one divided

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by the radius, and the reason we do

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that is basically you want a large kappa,

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a large curvature to
correspond with a sharp turn

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so, sharp turn, small
radius, large curvature.

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But the question is how do we describe

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this in a more mathematical way?

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So, I'm gonna to go ahead
and clear up, get rid of the

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circle itself and all of that radius

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and just be looking at the
curve itself in it's own right.

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The way you typically
describe a curve like this

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is para-metrically, so, you'll have

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some kind of vector valued function s

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that takes in a single parameter t,

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and then it's gonna output
the x and y coordinates,

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each as functions of t.

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So, this will be the x coordinate,

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the y coordinate.

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And then this specific
case, I'll just tell you

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the curve that I drew
happens to be parameterized

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by one minus the sign of t
as the x component function,

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actually no, it's t minus
sign of t, and the bottom part

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is one minus cosign of t.

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That's the curve that I drew.

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And the way that you're
thinking about this

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is for each value t you
get a certain vector

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that puts you at a point on the curve,

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and as t changes the
vector you get changes,

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but the tip of that vector kind of traces

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out the curve as a whole, right?

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And you can imagine the
vector drawing the curve

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as t varies,

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and the thought behind making
curvature mathematical,

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here I'll kind of clear
up some room for myself,

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is that you take the tangent vectors here

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so,

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so, you might image a unit tangent vector

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at every given point,

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and you're wondering

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how quickly those guys change direction,

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right?

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So with the little schematic
that I have drawn here

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I'm just gonna call this guy t one

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for like the first tangent vector,

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t2 , t3, and I haven't specified

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where they started or anything

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I just what give a feel for you've got

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various different tangent vectors

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and I'm just gonna say that all of them

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each one of them t sub
somethings has a unit length.

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And the idea of curvature
is to look at how quickly

00:02:38.619 --> 00:02:41.346
that unit tangent vector
changes directions.

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So, you know you might imagine
a completely different space

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so, rather than rooting
each vector on the curve,

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let's see what it would look like if you

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just kind of write each vector

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in its own right off in some other spot.

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So, this guy here would be t one,

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and then t two points a little bit down,

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and then t three points
kind of much more down

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So, all of these, this would
be t one, that guy is t two

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and these are the same vectors
I'm just kind of drawing them

00:03:09.779 --> 00:03:12.234
all rooted at the same spot
so it's a little easier

00:03:12.234 --> 00:03:14.138
to see how they turn, and you wanna say,

00:03:14.138 --> 00:03:16.261
okay how much do you
change as you go from t one

00:03:16.261 --> 00:03:18.677
to t two, is that a large angle change?

00:03:18.677 --> 00:03:20.955
and as you go from t two to t three,

00:03:20.955 --> 00:03:22.655
is that a large change as well?

00:03:22.655 --> 00:03:25.712
And you can kind of see
how if you have a curve,

00:03:25.712 --> 00:03:27.779
and let's say it's,

00:03:27.779 --> 00:03:29.606
if you have a curve
that curves quite a bit,

00:03:29.606 --> 00:03:31.687
you know it's doing something like that

00:03:31.687 --> 00:03:34.354
then the unit vector,
the unit tangent vector

00:03:34.354 --> 00:03:37.006
at this point changes quite rapidly

00:03:37.006 --> 00:03:38.581
over a short distance to be something

00:03:38.581 --> 00:03:40.143
almost 90 degrees different.

00:03:40.143 --> 00:03:42.487
Whereas, if you take the unit vector here

00:03:42.487 --> 00:03:44.290
and see how much does
it change as you go from

00:03:44.290 --> 00:03:46.376
this point over to this point

00:03:46.376 --> 00:03:48.383
it doesn't really change that much.

00:03:48.383 --> 00:03:49.502
So, the thought behind curvatures,

00:03:49.502 --> 00:03:50.815
we gonna take the rate of change of

00:03:50.815 --> 00:03:52.690
that unit tangent vector,

00:03:52.690 --> 00:03:55.039
so, the rate of change of t, and I'm gonna

00:03:55.039 --> 00:03:57.144
let capital T be a function

00:03:57.144 --> 00:03:58.952
that tells you whatever
the unit tangent vector

00:03:58.952 --> 00:04:00.606
at each point is,

00:04:00.606 --> 00:04:01.930
and I'm not gonna take the rate of change

00:04:01.930 --> 00:04:05.465
in terms of, you know
the parameter little t,

00:04:05.465 --> 00:04:07.744
which is what we use to
parameterize the curve,

00:04:07.744 --> 00:04:09.886
because it shouldn't matter
how you parameterize the curve,

00:04:09.886 --> 00:04:12.143
maybe you're driving
along it really quickly

00:04:12.143 --> 00:04:13.299
or really slowly.

00:04:13.299 --> 00:04:14.879
Instead, what you wanna take

00:04:14.879 --> 00:04:18.481
is the rate of change with
respect to arc length.

00:04:18.481 --> 00:04:20.887
Arc length, and I'm
using the variable s here

00:04:20.887 --> 00:04:22.391
to denote arc length, and what I mean by

00:04:22.391 --> 00:04:26.767
that is that if take just
a tiny little step here

00:04:26.767 --> 00:04:29.787
the distance of that
step, the actual distance

00:04:29.787 --> 00:04:32.759
in the x y plane, you
consider to be the arc length.

00:04:32.759 --> 00:04:34.367
And if you imagine it
being really, really small

00:04:34.367 --> 00:04:36.069
you're considering that a d s,

00:04:36.069 --> 00:04:38.581
a tiny change in the arc length.

00:04:38.581 --> 00:04:40.391
So this is the quantity
that we care about.

00:04:40.391 --> 00:04:43.511
How much does that unit
tangent vector change

00:04:43.511 --> 00:04:45.736
with respect to a tiny
change in the arc length?

00:04:45.736 --> 00:04:47.631
You know, as we travel along, let's say,

00:04:47.631 --> 00:04:50.714
it was a distance of like 0.5, right,

00:04:51.719 --> 00:04:55.576
you wanna know, did this
unit vector change a lot

00:04:55.576 --> 00:04:56.580
or a little bit.

00:04:56.580 --> 00:04:57.583
But I should add something here,

00:04:57.583 --> 00:04:59.913
this tiny change in the vector

00:04:59.913 --> 00:05:01.341
that would tell you,

00:05:01.341 --> 00:05:02.999
you know what the vector connecting

00:05:02.999 --> 00:05:04.887
their two tips is,

00:05:04.887 --> 00:05:06.589
so, this would be a
vector valued quantity,

00:05:06.589 --> 00:05:08.675
and curvature itself
should just be a number.

00:05:08.675 --> 00:05:11.647
So, what we really care about
is the size of this guy.

00:05:11.647 --> 00:05:14.767
So, what we really care
about, the size of this,

00:05:14.767 --> 00:05:17.358
which is a vector valued quantity,

00:05:17.358 --> 00:05:18.516
and that'll be an indication of just

00:05:18.516 --> 00:05:20.367
how much the curve curves.

00:05:20.367 --> 00:05:22.044
But if I'm at a sharper turned curve

00:05:22.044 --> 00:05:23.761
you go over that same distance

00:05:23.761 --> 00:05:25.216
and then suddenly the change

00:05:25.216 --> 00:05:27.793
in the tangent vectors goes by quite a bit

00:05:27.793 --> 00:05:30.535
that would be telling you
it's a high curvature.

00:05:30.535 --> 00:05:32.110
And in the next video
I'm gonna talk through

00:05:32.110 --> 00:05:34.755
what that looks like, how you think about

00:05:34.755 --> 00:05:36.939
this tangent vector function,

00:05:36.939 --> 00:05:38.699
this unit tangent vector function,

00:05:38.699 --> 00:05:41.236
and what it looks like to
take the derivative of that

00:05:41.236 --> 00:05:43.315
with respect to arc length.

00:05:43.315 --> 00:05:44.795
It can get a little convoluted

00:05:44.795 --> 00:05:46.452
in terms of the symbols involved,

00:05:46.452 --> 00:05:47.715
and the constant picture you should have

00:05:47.715 --> 00:05:50.011
in the back of your mind is that circle,

00:05:50.011 --> 00:05:52.093
that circle that's hugging
the curve very closely

00:05:52.093 --> 00:05:53.867
at a certain point,

00:05:53.867 --> 00:05:57.356
and this means of taking the
magnitude of the rate of change

00:05:57.356 --> 00:06:00.987
of the unit tangent vector
with respect to arc length

00:06:00.987 --> 00:06:04.320
is all just trying to capture that idea.


Curvature formula, part 4

WEBVTT
Kind: captions
Language: en

00:00:01.354 --> 00:00:03.746
- [Voiceover] So we've been
talking about curvature.

00:00:03.746 --> 00:00:06.834
And this means you've got
some sort of parametric curve,

00:00:06.834 --> 00:00:08.390
you might think as parameterized by

00:00:08.390 --> 00:00:11.178
a vector valued function S of t.

00:00:11.178 --> 00:00:12.502
And curvature is supposed to measure

00:00:12.502 --> 00:00:14.740
just how much this curve actually curves.

00:00:14.740 --> 00:00:17.620
So here it's curving quite a
bit, curvature should be high.

00:00:17.620 --> 00:00:18.988
But here it's, you know. relatively

00:00:18.988 --> 00:00:21.264
straight so curvature should be low.

00:00:21.264 --> 00:00:23.284
And I brought up in the last video

00:00:23.284 --> 00:00:25.212
this pretty complicated formula.

00:00:25.212 --> 00:00:27.731
And to remind you of the circumstances,

00:00:27.731 --> 00:00:31.075
I was saying that curvature,
denoted with this little kappa,

00:00:31.075 --> 00:00:32.907
is typically calculated as the

00:00:32.907 --> 00:00:35.975
derivative of the unit
tangent vector function.

00:00:35.975 --> 00:00:38.527
So we'll think of some kind of function

00:00:38.527 --> 00:00:41.755
that gives unit tangent
vectors at every single point.

00:00:41.755 --> 00:00:43.335
So you know, just a vector with

00:00:43.335 --> 00:00:46.563
a length of one lying
tangent to the curve.

00:00:46.563 --> 00:00:47.535
And that's going to be some kind

00:00:47.535 --> 00:00:49.303
of function of the same parameter.

00:00:49.303 --> 00:00:51.879
So big T for unit tangent
vector, little t for the

00:00:51.879 --> 00:00:55.315
parameter, hopefully
that's not too confusing.

00:00:55.315 --> 00:00:57.915
Then we're taking the derivative
of this, not with respect

00:00:57.915 --> 00:01:01.979
to t, the parameter, but with
respect to arc length, s.

00:01:01.979 --> 00:01:04.347
And by arc length, I mean
if you take a tiny step

00:01:04.347 --> 00:01:08.271
along the curve, and consider
this to be of size ds,

00:01:08.271 --> 00:01:11.103
so s typically denotes
length along a curve,

00:01:11.103 --> 00:01:12.915
this will be a tiny change in length.

00:01:12.915 --> 00:01:16.607
You're wondering how much that
unit tangent vector changes.

00:01:16.607 --> 00:01:18.303
And specifically what I mean by that,

00:01:18.303 --> 00:01:20.439
if you kind of imagine
tangent vectors off sitting in

00:01:20.439 --> 00:01:23.295
their own space somewhere,
where each one of them has unit

00:01:23.295 --> 00:01:26.939
lengths so they're all just
kind of of the same length

00:01:26.939 --> 00:01:28.373
and lets say they're all
stemming from the same points.

00:01:28.373 --> 00:01:30.319
So rather than drawing them
stemming from the curve,

00:01:30.319 --> 00:01:31.619
to show that they're tangent, I just

00:01:31.619 --> 00:01:33.223
want to show them in their own space.

00:01:33.223 --> 00:01:36.567
The derivative, the change in
that tangent vector, would be

00:01:36.567 --> 00:01:40.303
some, some other vector that
tells you how you move from one

00:01:40.303 --> 00:01:42.907
to the other, kind of tells
you how much it's turning.

00:01:42.907 --> 00:01:46.295
So the curvature itself is
given not by that vector,

00:01:46.295 --> 00:01:48.271
because this would be a vector quantity,

00:01:48.271 --> 00:01:51.055
but by the absolute value of it.

00:01:51.055 --> 00:01:54.003
And I said in the last
video that it turns out

00:01:54.003 --> 00:01:56.607
that this quantity, when you
work it out for a vector-value

00:01:56.607 --> 00:02:00.107
function with components, with componenets

00:02:01.386 --> 00:02:03.803
x of t, x of t and y of t,

00:02:06.950 --> 00:02:10.918
that it happens to equal
this complicated formula.

00:02:10.918 --> 00:02:13.150
And what I'm gonna do here,
I going to break it down

00:02:13.150 --> 00:02:16.933
and show why this formula
isn't actually all that random.

00:02:16.933 --> 00:02:19.278
It's actually kind of
sensible as a description

00:02:19.278 --> 00:02:22.437
for how much the curve
really curves at a point.

00:02:22.437 --> 00:02:24.941
So lets start by looking
at the numerator here.

00:02:24.941 --> 00:02:28.033
This x-prime of t, first
derivative of the first component,

00:02:28.033 --> 00:02:31.025
times the second derivative of
the second component, minus,

00:02:31.025 --> 00:02:32.073
and then kind of symmetrically,

00:02:32.073 --> 00:02:35.345
y-prime times x double-prime.

00:02:35.345 --> 00:02:36.462
You might be able to recognize

00:02:36.462 --> 00:02:38.918
this as a certain cross product.

00:02:38.918 --> 00:02:42.692
So the cross product, if
we take x-prime, y-prime,

00:02:42.692 --> 00:02:46.609
and these are still
functions of t, as a vector

00:02:47.916 --> 00:02:50.980
crossed with the vector
containing the double primes.

00:02:50.980 --> 00:02:52.500
And if you aren't familiar
with the cross products

00:02:52.500 --> 00:02:53.844
or you're feeling a little shaky

00:02:53.844 --> 00:02:55.028
and you kind of want to review,

00:02:55.028 --> 00:02:56.680
now would be a good time pause the video,

00:02:56.680 --> 00:02:58.536
take a look at the cross product videos,

00:02:58.536 --> 00:03:00.232
and remind yourself both of how

00:03:00.232 --> 00:03:03.364
to compute it and what the
underlying intuition is.

00:03:03.364 --> 00:03:06.572
Because the way that we compute
a cross product like this is

00:03:06.572 --> 00:03:09.520
you take the components in the
diagonal here, you know that

00:03:09.520 --> 00:03:12.576
rightward diagonal, and
multiply them together.

00:03:12.576 --> 00:03:16.056
That's where you get your
x-prime, y double-prime.

00:03:16.056 --> 00:03:18.704
And then you subtract off
the components in the other

00:03:18.704 --> 00:03:22.232
diagonal, you know it kind
of feels like a determinant.

00:03:22.232 --> 00:03:24.732
X double-prime, times y-prime.

00:03:25.856 --> 00:03:28.132
But the way that you
interpret this vector,

00:03:28.132 --> 00:03:30.060
well I'll get to the
interpretation in just a moment.

00:03:30.060 --> 00:03:32.356
First, first let me kind of write out

00:03:32.356 --> 00:03:34.588
what this is in terms of our function S.

00:03:34.588 --> 00:03:38.068
This first vector is just
the first derivative of S,

00:03:38.068 --> 00:03:41.692
so that's S-prime of t,
the first derivative of S.

00:03:41.692 --> 00:03:43.364
And we're crossing that with this one,

00:03:43.364 --> 00:03:46.428
which is the second derivative,
vector value derivative,

00:03:46.428 --> 00:03:49.011
but the second derivative of S.

00:03:50.748 --> 00:03:54.116
So before we, before we
do anything else lets just

00:03:54.116 --> 00:03:57.132
start to think about, what do
both of these vectors mean?

00:03:57.132 --> 00:04:00.499
How do you interpret the
S-prime and the S double-prime?

00:04:00.499 --> 00:04:02.379
I'm going to go ahead and give

00:04:02.379 --> 00:04:04.123
ourselves a little bit of room here.

00:04:04.123 --> 00:04:05.771
So I'll draw the curve again.

00:04:05.771 --> 00:04:09.159
We're on our x, y plane, and
you have some kind of curve.

00:04:09.159 --> 00:04:12.667
The function S itself is giving vectors

00:04:12.667 --> 00:04:16.403
whose tips trace out this curve, right?

00:04:16.403 --> 00:04:20.467
As T changes, the tips of these
vectors trace out the curve.

00:04:20.467 --> 00:04:22.999
And now the first derivative,
that first derivative

00:04:22.999 --> 00:04:26.332
vector, S-prime of t, is telling you

00:04:26.332 --> 00:04:29.512
how that tip should move
to go along the curve.

00:04:29.512 --> 00:04:33.088
As you go from one S
vector to another S vector,

00:04:33.088 --> 00:04:35.340
what direction should that tip move.

00:04:35.340 --> 00:04:38.380
And what this means is
that at every given point,

00:04:38.380 --> 00:04:40.076
when you kind of do this
in a limiting fashion

00:04:40.076 --> 00:04:43.560
and you only look at infinitesimal
changes in the original

00:04:43.560 --> 00:04:47.344
vector, you always get some
kind of tangent vector, right?

00:04:47.344 --> 00:04:50.062
So all of these are tangent
vectors, not necessarily unit,

00:04:50.062 --> 00:04:53.188
you might have like a very
long tangent vector to indicate

00:04:53.188 --> 00:04:56.808
that you're traveling very
quickly across that space.

00:04:56.808 --> 00:04:58.487
And now, how would you think about

00:04:58.487 --> 00:05:02.810
the second derivative vector
here, S double-prime of t?

00:05:02.810 --> 00:05:04.994
Well the way to do
that, I like to think of

00:05:04.994 --> 00:05:06.714
all of the tangents
vectors then just kind of

00:05:06.714 --> 00:05:08.966
living in their own space, right?

00:05:08.966 --> 00:05:12.982
If this here is representing
S of t, I just want to

00:05:12.982 --> 00:05:17.149
look in isolation at what
S-prime of t looks like.

00:05:18.231 --> 00:05:19.725
So each one of those
vectors, you know maybe this

00:05:19.725 --> 00:05:22.465
first one is just is
very long, gargantuan,

00:05:22.465 --> 00:05:24.345
indicating you're going very quickly.

00:05:24.345 --> 00:05:27.109
And then after that you've
got something else here,

00:05:27.109 --> 00:05:28.665
maybe it's pointing a little bit down.

00:05:28.665 --> 00:05:29.685
And you're kind of thinking about how

00:05:29.685 --> 00:05:31.357
all of these derivative vectors change,

00:05:31.357 --> 00:05:33.701
but I want them all
rooted at the same point.

00:05:33.701 --> 00:05:36.141
Just see what happens when
they're all rooted at the origin,

00:05:36.141 --> 00:05:38.809
because that, that gives
us a way to think about

00:05:38.809 --> 00:05:40.993
what changing the vector should look like.

00:05:40.993 --> 00:05:42.597
So in particular, as you're moving from

00:05:42.597 --> 00:05:45.453
this vector to this one, the tip should be

00:05:45.453 --> 00:05:47.541
kind of moving in this direction, right?

00:05:47.541 --> 00:05:49.677
So this second derivative
value is going to

00:05:49.677 --> 00:05:52.533
tell us how the tip of the
first derivative should move.

00:05:52.533 --> 00:05:54.065
And then similarly over here it tells us

00:05:54.065 --> 00:05:56.041
how the tip of that should move.

00:05:56.041 --> 00:05:59.081
And just as an example and
kind of a hint as to why this

00:05:59.081 --> 00:06:01.573
has something to do with curvature,

00:06:01.573 --> 00:06:05.667
if you have a curve
that turns very sharply,

00:06:05.667 --> 00:06:09.158
turns very sharply, you would
have you know a tangent vector

00:06:09.158 --> 00:06:11.328
that starts by pointing up
and to the right and then very

00:06:11.328 --> 00:06:14.740
quickly is pointing down,
down and to the right.

00:06:14.740 --> 00:06:16.808
So if they were off, you know
we'd kind of draw them on

00:06:16.808 --> 00:06:19.500
their own, rooted in their own location,

00:06:19.500 --> 00:06:22.472
you can maybe see how the
second derivative vector

00:06:22.472 --> 00:06:25.376
is telling it to turn
somehow in that direction.

00:06:25.376 --> 00:06:28.092
And if you, you know
capture not just those two

00:06:28.092 --> 00:06:30.996
but infinitesimally what's
going on as you move from

00:06:30.996 --> 00:06:32.836
one to the next you're gonna get

00:06:32.836 --> 00:06:36.577
this kind of turning motion
for all of these guys.

00:06:36.577 --> 00:06:38.713
So the second derivative
vector would be pulling

00:06:38.713 --> 00:06:41.081
perpendicular, perpendicular to that

00:06:41.081 --> 00:06:44.261
first derivative vector as
a way of telling it to turn.

00:06:44.261 --> 00:06:46.281
So just to draw that on
its own, if you have a

00:06:46.281 --> 00:06:48.694
first derivative vector, and
then the second derivative

00:06:48.694 --> 00:06:50.646
vector is perpendicular to it,

00:06:50.646 --> 00:06:53.686
it's telling it how it
should turn in some way.

00:06:53.686 --> 00:06:55.682
But if it was, you know
lets say it was not purely

00:06:55.682 --> 00:06:57.934
perpendicular but it was
also pointing against,

00:06:57.934 --> 00:06:59.514
it would be telling that, that vector,

00:06:59.514 --> 00:07:02.370
that first derivative vector,
to shrink in some way.

00:07:02.370 --> 00:07:03.554
So that would be an indication that

00:07:03.554 --> 00:07:05.898
not only is it turning
but it's getting smaller,

00:07:05.898 --> 00:07:09.686
meaning the trajectory based
on S is probably slowing down.

00:07:09.686 --> 00:07:13.998
And if it was kind of turning
it but also pointing away,

00:07:13.998 --> 00:07:15.037
that would mean it's telling the

00:07:15.037 --> 00:07:16.917
first derivative vector to grow.

00:07:16.917 --> 00:07:19.845
So not only should you
be turning on your curve

00:07:19.845 --> 00:07:21.633
but you should also be speeding up.

00:07:21.633 --> 00:07:23.810
But the case we care about
most is we're just trying to

00:07:23.810 --> 00:07:27.060
measure how perpendicular it is, right?

00:07:28.295 --> 00:07:30.978
And this is where the
cross product comes in,

00:07:30.978 --> 00:07:32.173
because if you think
about how you interpret

00:07:32.173 --> 00:07:36.609
that cross product, how you
interpret the cross product,

00:07:36.609 --> 00:07:38.825
it's basically the area
if you kind of take these

00:07:38.825 --> 00:07:42.285
two vectors, tip-to-tail as
they are, and think about

00:07:42.285 --> 00:07:45.069
the parallelogram that they trace out.

00:07:45.069 --> 00:07:46.813
Lets see, so this blue dotted line

00:07:46.813 --> 00:07:49.261
should be parallel to the blue vector.

00:07:49.261 --> 00:07:52.521
The area of this, of this
traced out parallelogram

00:07:52.521 --> 00:07:55.121
for each of them, that's what tells you

00:07:55.121 --> 00:07:57.745
that's how you interpret
the cross product.

00:07:57.745 --> 00:08:02.273
The cross product between
S-prime and S double-prime.

00:08:02.273 --> 00:08:05.085
So this cross product, by giving
you that area, is kind of a

00:08:05.085 --> 00:08:08.405
measure of just how perpendicular
these vectors are, right?

00:08:08.405 --> 00:08:11.701
Because if one of them,
if they point very much in

00:08:11.701 --> 00:08:13.793
the same direction, right,
and they're only slightly

00:08:13.793 --> 00:08:17.089
perpendicular, that means that
the parallelogram they trace,

00:08:17.089 --> 00:08:20.341
they trace out is gonna
have a very small area.

00:08:20.341 --> 00:08:22.941
It's gonna be a smaller
area in comparison.

00:08:22.941 --> 00:08:27.005
So with that, I don't want
this video to run too long,

00:08:27.005 --> 00:08:29.001
so I'm gonna call it in
here, but then I'll just

00:08:29.001 --> 00:08:30.905
continue on in the next
one through the same

00:08:30.905 --> 00:08:33.459
line of reasoning to build
to our original formula.


Curvature formula, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.374 --> 00:00:02.013
- [Voiceover] In the last
video I started to talk about

00:00:02.013 --> 00:00:03.773
the formula for curvature.

00:00:03.773 --> 00:00:06.464
Just to remind everyone of where we are

00:00:06.464 --> 00:00:09.571
you imagine that you have some
kind of curve in let's say

00:00:09.571 --> 00:00:12.394
two dimensional space just
for the sake of being simple.

00:00:12.394 --> 00:00:17.265
Let's say this curve is
parameterized by a function S of T.

00:00:17.265 --> 00:00:20.048
So every number T corresponds
to some point on the curve.

00:00:20.048 --> 00:00:22.013
For the curvature you start thinking about

00:00:22.013 --> 00:00:23.812
unit tangent vectors.

00:00:23.812 --> 00:00:25.652
At every given point what does the unit

00:00:25.652 --> 00:00:27.082
tangent vector look like?

00:00:27.082 --> 00:00:30.768
The curvature itself which
is denoted by this sort of

00:00:30.768 --> 00:00:32.571
Greek letter Kappa

00:00:32.571 --> 00:00:35.477
is gonna be the rate of
change of those unit vectors

00:00:35.477 --> 00:00:38.264
kind of how quick they're
turning in direction

00:00:38.264 --> 00:00:40.352
not with respect to the parameter t

00:00:40.352 --> 00:00:43.216
but with respect to arc length d s.

00:00:43.216 --> 00:00:45.998
What I mean by arc length here is

00:00:45.998 --> 00:00:48.292
just a tiny step you can think

00:00:48.292 --> 00:00:52.185
the size of a tiny step
along the curve would be d s.

00:00:52.185 --> 00:00:54.969
You're wondering, as you
take a tiny step like that

00:00:54.969 --> 00:00:57.015
does the unit tangent
vector turn a a lot or

00:00:57.015 --> 00:00:58.405
does it turn a little bit?

00:00:58.405 --> 00:01:00.616
The little schematic that I
said you might have in mind

00:01:00.616 --> 00:01:02.705
is just a completely separate space

00:01:02.705 --> 00:01:05.612
where for each one of
these unit tangent vectors

00:01:05.612 --> 00:01:08.314
you go ahead and put them in that space

00:01:08.314 --> 00:01:12.202
saying okay so this one would
look something like this

00:01:12.202 --> 00:01:14.374
this one is pointed down
and to the right so it would

00:01:14.374 --> 00:01:15.837
look something like this.

00:01:15.837 --> 00:01:18.386
This one is pointed very much down.

00:01:18.386 --> 00:01:20.448
You're wondering basically
as you take tiny little steps

00:01:20.448 --> 00:01:22.448
of size d s what is this

00:01:23.396 --> 00:01:26.307
change to the unit tangent vector

00:01:26.307 --> 00:01:30.028
and that change is gonna
be some kind of vector.

00:01:30.028 --> 00:01:32.323
Because the curvature's
really just a value

00:01:32.323 --> 00:01:34.699
a number that we want all
we care about is the size

00:01:34.699 --> 00:01:36.296
of that vector.

00:01:36.296 --> 00:01:39.963
The size of the change
to the tangent vector

00:01:41.482 --> 00:01:44.063
as you take a tiny step in d s.

00:01:44.063 --> 00:01:45.495
Now this is pretty abstract right?

00:01:45.495 --> 00:01:47.786
I've got these two completely
separate things that

00:01:47.786 --> 00:01:51.095
are not the original functions
that you have to think about.

00:01:51.095 --> 00:01:54.225
You have to think about this
unit tangent vector function

00:01:54.225 --> 00:01:55.704
and then you also have to think about this

00:01:55.704 --> 00:01:57.343
notion of arc length.

00:01:57.343 --> 00:01:59.389
The reason, by the way,
that I'm using an s here

00:01:59.389 --> 00:02:03.398
as well as here for the
parameterized of the curve

00:02:03.398 --> 00:02:05.320
is because they're actually quite related.

00:02:05.320 --> 00:02:07.160
I'll get to that a little bit below.

00:02:07.160 --> 00:02:09.262
To make it clear what this
means I'm gonna go ahead

00:02:09.262 --> 00:02:11.658
and go through an example
here where let's say our

00:02:11.658 --> 00:02:14.491
parameterized with respect to t is

00:02:17.042 --> 00:02:18.557
a cosine sine pair.

00:02:18.557 --> 00:02:21.347
So we've got cosine of
t as the x component

00:02:21.347 --> 00:02:24.051
and then sine of t as the y component

00:02:24.051 --> 00:02:24.897
sine of t.

00:02:24.897 --> 00:02:27.312
Just to make it so that
it's not completely boring

00:02:27.312 --> 00:02:32.221
let's multiply both of these
components by constant r.

00:02:32.221 --> 00:02:33.983
What this means, you might recognize this,

00:02:33.983 --> 00:02:35.341
cosine sine pair

00:02:35.341 --> 00:02:36.591
what this means

00:02:39.081 --> 00:02:40.685
is that in the x y plane

00:02:40.685 --> 00:02:43.343
you're actually drawing
a circle with radius r.

00:02:43.343 --> 00:02:47.928
This would be some kind of
circle with the radius r.

00:02:47.928 --> 00:02:50.710
While I go through this example
I also want to make a note

00:02:50.710 --> 00:02:53.984
of what things would look like
a little bit more abstractly.

00:02:53.984 --> 00:02:58.200
If we just had s of t equals
not specific functions

00:02:58.200 --> 00:03:00.532
that I laid down but
just any general function

00:03:00.532 --> 00:03:03.356
for the x component and
for the y component.

00:03:03.356 --> 00:03:06.222
The reason I want to do
this is because the concrete

00:03:06.222 --> 00:03:09.129
version is gonna be helpful
and simple and something

00:03:09.129 --> 00:03:12.239
we can deal with but
almost so simple as to

00:03:12.239 --> 00:03:14.901
not be indicative of just
how complicated the normal

00:03:14.901 --> 00:03:17.890
circumstances but the
more general circumstances

00:03:17.890 --> 00:03:19.731
so complicated I think
it will actually confuse

00:03:19.731 --> 00:03:21.084
things a little bit too much.

00:03:21.084 --> 00:03:22.475
It'll be good to kind of go through

00:03:22.475 --> 00:03:24.558
both of them in parallel.

00:03:25.402 --> 00:03:27.140
The first step is to
figure out what is this

00:03:27.140 --> 00:03:28.737
unit tangent vector?

00:03:28.737 --> 00:03:31.060
What is that function
that at every given point

00:03:31.060 --> 00:03:34.061
gives you a unit tangent
vector to the curve?

00:03:34.061 --> 00:03:36.128
The first thing for that

00:03:36.128 --> 00:03:37.600
is to realize that

00:03:37.600 --> 00:03:39.604
we already have a notion
of what should give the

00:03:39.604 --> 00:03:40.791
tangent vector

00:03:40.791 --> 00:03:43.494
the derivative of our
vector valued function

00:03:43.494 --> 00:03:45.336
as a function of t

00:03:45.336 --> 00:03:47.792
the direction in which in points is

00:03:47.792 --> 00:03:49.470
in the tangent direction.

00:03:49.470 --> 00:03:51.620
If I go over here and if
I compute this derivative

00:03:51.620 --> 00:03:53.453
and I say s prime of t

00:03:55.154 --> 00:03:56.547
which involves just taking the

00:03:56.547 --> 00:03:57.939
derivative of both components so the

00:03:57.939 --> 00:04:01.022
derivative of cosine is negative sine

00:04:03.093 --> 00:04:06.123
of t multiplied by r and the derivative

00:04:06.123 --> 00:04:07.540
of sine is cosine

00:04:08.809 --> 00:04:10.559
of t multiplied by r.

00:04:13.700 --> 00:04:16.633
More abstractly, this is just gonna be

00:04:16.633 --> 00:04:20.143
anytime you have two
different component functions

00:04:20.143 --> 00:04:22.448
you just take the derivative of each one.

00:04:22.448 --> 00:04:24.209
Hopefully you've seen this, if not

00:04:24.209 --> 00:04:26.094
maybe take a look of videos on

00:04:26.094 --> 00:04:30.338
taking the derivative of a
position vector valued function.

00:04:30.338 --> 00:04:32.678
This we can interpret
as that tangent vector

00:04:32.678 --> 00:04:34.930
but it might not be a unit vector right?

00:04:34.930 --> 00:04:36.522
We want a unit tangent vector and this

00:04:36.522 --> 00:04:38.770
only promises us the direction.

00:04:38.770 --> 00:04:40.937
What we do to normalize it

00:04:43.714 --> 00:04:46.881
and get a unit tangent vector function

00:04:51.970 --> 00:04:54.590
which I'll call capital t of lowercase t

00:04:54.590 --> 00:04:55.819
that's kind of confusing right?

00:04:55.819 --> 00:04:57.375
Capital t is for tangent vector

00:04:57.375 --> 00:05:00.426
lowercase t is the parameter.

00:05:00.426 --> 00:05:01.667
I'll try to keep that straight.

00:05:01.667 --> 00:05:04.932
It's sort of standard notation but

00:05:04.932 --> 00:05:07.761
there is the potential
to confuse with this.

00:05:07.761 --> 00:05:11.928
What that's gonna be is
your vector value derivative

00:05:13.883 --> 00:05:16.969
but normalized. So we have to divide by

00:05:16.969 --> 00:05:19.095
whatever it's magnitude is

00:05:19.095 --> 00:05:21.261
as a function of t.

00:05:21.261 --> 00:05:23.594
In this case, in a specific example

00:05:23.594 --> 00:05:27.177
that magnitude, if we
take the magnitude of

00:05:28.743 --> 00:05:30.410
negative sine of t r

00:05:31.832 --> 00:05:33.330
multiply by r

00:05:33.330 --> 00:05:34.997
and then cosine of t

00:05:36.461 --> 00:05:37.711
multiplied by r

00:05:40.416 --> 00:05:42.628
so we're taking the magnitude
of this whole vector

00:05:42.628 --> 00:05:43.545
what we get

00:05:44.635 --> 00:05:47.564
make myself even more room here

00:05:47.564 --> 00:05:49.064
is the square root

00:05:50.319 --> 00:05:51.916
of sine squared

00:05:51.916 --> 00:05:54.824
negative sine squared is
just gonna be sine squared

00:05:54.824 --> 00:05:58.552
so sine squared of t
multiplied by r squared

00:05:58.552 --> 00:06:02.444
and then over here cosine
square times r squared

00:06:02.444 --> 00:06:05.361
cosine squared of t times r squared

00:06:06.910 --> 00:06:09.447
we can bring that r squared
outside of the radical

00:06:09.447 --> 00:06:11.045
to sort of factor it out

00:06:11.045 --> 00:06:13.539
turning it into an r

00:06:13.539 --> 00:06:17.672
and on the inside we have sine
squared plus cosine squared

00:06:17.672 --> 00:06:19.803
I'm being too lazy to write
down the t's right now

00:06:19.803 --> 00:06:21.891
because no matter what
the t is that whole value

00:06:21.891 --> 00:06:23.224
just equals one.

00:06:24.294 --> 00:06:27.283
This entire thing is just gonna equal r.

00:06:27.283 --> 00:06:30.927
What that means is that our
unit tangent vector up here

00:06:30.927 --> 00:06:34.652
is gonna be the original
function but divided by r.

00:06:34.652 --> 00:06:36.087
It happens to be a
constant usually it's not

00:06:36.087 --> 00:06:38.483
but it happens to be a
constant in this case.

00:06:38.483 --> 00:06:40.860
What that looks like

00:06:40.860 --> 00:06:42.254
given that our original function is

00:06:42.254 --> 00:06:45.203
negative sine of t times
r and cosine of t times r

00:06:45.203 --> 00:06:47.170
we're dividing out by an r

00:06:47.170 --> 00:06:49.505
the ultimate function that we get

00:06:49.505 --> 00:06:51.672
is just negative sine of t

00:06:53.512 --> 00:06:55.262
and then cosine of t.

00:06:58.627 --> 00:07:00.307
For fear of running a little bit long

00:07:00.307 --> 00:07:02.230
I think I'll call it an
end to this video and

00:07:02.230 --> 00:07:05.803
continue on with the same
argument in the next video.


Curvature formula, part 5

WEBVTT
Kind: captions
Language: en

00:00:00.124 --> 00:00:02.071
- [Voiceover] So, let's
sum up where we are so far.

00:00:02.071 --> 00:00:04.221
We're looking at this formula

00:00:04.221 --> 00:00:07.691
and trying to understand why
it corresponds to curvature.

00:00:07.691 --> 00:00:10.000
Why it tells you how much
a curve actually curves.

00:00:10.000 --> 00:00:13.094
And the first thing we did is
we noticed that this numerator

00:00:13.094 --> 00:00:16.176
corresponds to a certain cross-product.

00:00:16.176 --> 00:00:17.838
The cross-product between
the first derivative

00:00:17.838 --> 00:00:19.958
and the second derivative of the function

00:00:19.958 --> 00:00:21.559
parameterizing the curve.

00:00:21.559 --> 00:00:23.214
And the way we started to understand that

00:00:23.214 --> 00:00:26.402
is we say, well, the function
parameterizing the curve

00:00:26.402 --> 00:00:30.069
s of t produces vectors
whose tips trace out

00:00:30.993 --> 00:00:32.793
that curve itself.

00:00:32.793 --> 00:00:35.950
And now if you think about
how one tip moves to the next

00:00:35.950 --> 00:00:37.248
the direction that it needs to go

00:00:37.248 --> 00:00:40.135
for that tip to move to the next one,

00:00:40.135 --> 00:00:41.788
that's what the first
derivative tells you.

00:00:41.788 --> 00:00:44.191
And when you treat that
in an infinitesimal way

00:00:44.191 --> 00:00:47.282
this is why you get tangent
vectors along the curve

00:00:47.282 --> 00:00:49.270
and there's an entire
series of videos on that

00:00:49.270 --> 00:00:51.911
for the derivative of a
position vector valued function,

00:00:51.911 --> 00:00:53.896
and they explain why the first derivative

00:00:53.896 --> 00:00:55.640
of a parametric function,

00:00:55.640 --> 00:00:58.023
gives you tangent vectors to that curve,

00:00:58.023 --> 00:01:00.111
but if we draw all of
those tangent vectors

00:01:00.111 --> 00:01:01.297
just in their own space,

00:01:01.297 --> 00:01:03.503
their own little s prime of t space,

00:01:03.503 --> 00:01:05.175
you get all of these vectors,

00:01:05.175 --> 00:01:06.653
and we're rooting them at the same point

00:01:06.653 --> 00:01:08.116
to be able to relate them more easily.

00:01:08.116 --> 00:01:10.500
The way that you move from
the tip of one of those

00:01:10.500 --> 00:01:13.338
to the next one, is given
by the second derivative,

00:01:13.338 --> 00:01:16.739
it kind of plays the same
roll for the first derivative

00:01:16.739 --> 00:01:18.357
as the first derivative plays

00:01:18.357 --> 00:01:20.153
for the original parametric function.

00:01:20.153 --> 00:01:22.333
And specifically if
you have a circumstance

00:01:22.333 --> 00:01:24.622
where the tangent
vectors are just turning,

00:01:24.622 --> 00:01:27.459
the only thing they're doing
is purely turning around,

00:01:27.459 --> 00:01:29.665
which is when your curve
is actually curving,

00:01:29.665 --> 00:01:32.222
this corresponds to a case when the

00:01:32.222 --> 00:01:35.455
second derivative function
is pretty much perpendicular

00:01:35.455 --> 00:01:36.288
as vector.

00:01:36.288 --> 00:01:37.968
The vector it produces is perpendicular

00:01:37.968 --> 00:01:39.857
to the first derivative vector.

00:01:39.857 --> 00:01:42.089
So, this is loosely why the cross-product

00:01:42.089 --> 00:01:43.958
is kind of a good measurement of curving

00:01:43.958 --> 00:01:45.884
cause it tells you how
perpendicular these guys are.

00:01:45.884 --> 00:01:47.754
But, there is a bit of a catch.

00:01:47.754 --> 00:01:49.872
The original formula for curvature,

00:01:49.872 --> 00:01:52.847
the whole reason we're doing
it with respect to arc length

00:01:52.847 --> 00:01:54.617
and not with respect to the parameter t,

00:01:54.617 --> 00:01:57.277
is that curvature
doesn't really care about

00:01:57.277 --> 00:01:58.844
how you parameterize the function.

00:01:58.844 --> 00:02:00.705
If you imagine zipping
along it really quickly,

00:02:00.705 --> 00:02:04.205
so your first derivative
vectors are all super long,

00:02:04.205 --> 00:02:05.246
it shouldn't matter,

00:02:05.246 --> 00:02:07.029
compared to crawling
along it like a turtle.

00:02:07.029 --> 00:02:09.667
The curvature should
just always be the same.

00:02:09.667 --> 00:02:12.239
But, this is a problem
if you think back to

00:02:12.239 --> 00:02:14.278
the cross-product that
we're now looking at,

00:02:14.278 --> 00:02:15.687
where you're taking the
cross-product between

00:02:15.687 --> 00:02:18.701
the first derivative and
the second derivative,

00:02:18.701 --> 00:02:21.459
because if you were
traveling along this curve

00:02:21.459 --> 00:02:23.873
twice as quickly, what that would mean is

00:02:23.873 --> 00:02:25.425
your first derivative vector,

00:02:25.425 --> 00:02:26.604
so I'll kind of draw it again over here,

00:02:26.604 --> 00:02:28.372
would be twice as long,

00:02:28.372 --> 00:02:30.500
to indicate that your going twice as fast.

00:02:30.500 --> 00:02:32.331
And similarly your
second derivative vector,

00:02:32.331 --> 00:02:34.240
to kind of keep up with
that changing rate,

00:02:34.240 --> 00:02:35.788
would also be twice as long.

00:02:35.788 --> 00:02:39.112
And as a result, the
parallelogram that they trace out,

00:02:39.112 --> 00:02:42.851
I should actually, kinda
going off screen here.

00:02:42.851 --> 00:02:46.606
The parallelogram that they trace out

00:02:46.606 --> 00:02:48.610
would be actually four times as big.

00:02:48.610 --> 00:02:49.443
Right?

00:02:49.443 --> 00:02:51.822
Because both of the vectors get scaled up.

00:02:51.822 --> 00:02:54.294
So, the way that we
really want to be thinking

00:02:54.294 --> 00:02:57.602
about this is not the tangent
vector due to the derivative,

00:02:57.602 --> 00:02:59.002
but normalizing this.

00:02:59.002 --> 00:03:00.130
And this should kind of make sense,

00:03:00.130 --> 00:03:01.866
because we are thinking in
terms of unit tangent vectors

00:03:01.866 --> 00:03:03.514
for the curvature as a whole.

00:03:03.514 --> 00:03:06.464
So, if you imagine instead
kind of cutting of the vector

00:03:06.464 --> 00:03:09.308
to make sure that it's got a unit length,

00:03:09.308 --> 00:03:10.586
a length of one.

00:03:10.586 --> 00:03:11.419
And what that means is,

00:03:11.419 --> 00:03:12.568
you're taking the derivative vector

00:03:12.568 --> 00:03:15.130
and dividing it by it's own magnitude,

00:03:15.130 --> 00:03:17.450
by the magnitude of
that derivative vector,

00:03:17.450 --> 00:03:19.986
and then similarly we'll want
to scale everything else down,

00:03:19.986 --> 00:03:23.722
so you're taking this
and kinda scaling it down

00:03:23.722 --> 00:03:25.139
by what you need.

00:03:26.362 --> 00:03:28.686
The resulting parallelogram they trace out

00:03:28.686 --> 00:03:31.841
is a more pure measurement of
how perpendicular they are,

00:03:31.841 --> 00:03:34.478
without caring how long they are.

00:03:34.478 --> 00:03:36.223
And with this guy would
be, by the way then,

00:03:36.223 --> 00:03:37.838
this is the second derivative vector

00:03:37.838 --> 00:03:40.291
not normalized with respect to itself,

00:03:40.291 --> 00:03:41.566
but we're still dividing,

00:03:41.566 --> 00:03:43.297
you know the thing we're diving by

00:03:43.297 --> 00:03:44.419
as we scale everything down,

00:03:44.419 --> 00:03:47.241
is still just the size of
that first derivative vector.

00:03:47.241 --> 00:03:49.766
So, this cross-product, if
we take the cross-product

00:03:49.766 --> 00:03:53.349
between s prime
normalized, s double prime,

00:03:55.911 --> 00:03:58.227
oh no, no, no, sorry,

00:03:58.227 --> 00:03:59.367
s prime itself,

00:03:59.367 --> 00:04:01.094
and I should be saying
vectors for all of these,

00:04:01.094 --> 00:04:02.090
these are all vectors.

00:04:02.090 --> 00:04:03.946
If we take the cross-product between that

00:04:03.946 --> 00:04:06.171
and s double prime,

00:04:06.171 --> 00:04:09.223
scaled down by that same
value that's still s prime,

00:04:09.223 --> 00:04:12.813
so it's not normalized, this
is just scaled down by s prime.

00:04:12.813 --> 00:04:16.837
This here is a more pure
measurement of how perpendicular

00:04:16.837 --> 00:04:19.210
the second derivative vector is to first.

00:04:19.210 --> 00:04:20.750
And the reason that we
don't really care about

00:04:20.750 --> 00:04:22.577
the second derivative being normalized is,

00:04:22.577 --> 00:04:23.485
if it was the case,

00:04:23.485 --> 00:04:24.318
that you know,

00:04:24.318 --> 00:04:25.779
the second derivative
was really, really strong

00:04:25.779 --> 00:04:27.428
and wasn't necessarily a unit vector,

00:04:27.428 --> 00:04:29.436
that's fine that's just telling us that

00:04:29.436 --> 00:04:31.195
the tangent vector turns much more quickly

00:04:31.195 --> 00:04:32.947
and the curvature should be higher.

00:04:32.947 --> 00:04:36.730
And in fact, it turns out
that this whole expression is

00:04:36.730 --> 00:04:40.864
the derivative of the
unit tangent vector t,

00:04:40.864 --> 00:04:42.728
that unit tangent vector
that I've talked about a lot,

00:04:42.728 --> 00:04:44.905
with respect to the parameter t.

00:04:44.905 --> 00:04:48.421
So, whatever the parameter
of your original function is.

00:04:48.421 --> 00:04:51.120
And now, if you think back to the,

00:04:51.120 --> 00:04:51.953
I'm not sure if it was the last video

00:04:51.953 --> 00:04:53.075
or the one before that,

00:04:53.075 --> 00:04:54.047
but I talked about how,

00:04:54.047 --> 00:04:55.329
when you take,

00:04:55.329 --> 00:04:57.315
when we're looking for this derivative of

00:04:57.315 --> 00:04:59.321
the tangent vector, with
respect to arc length,

00:04:59.321 --> 00:05:01.800
the way that you compute this is to

00:05:01.800 --> 00:05:05.130
first take its derivative
with respect to the parameter,

00:05:05.130 --> 00:05:06.535
which is something we can actually do,

00:05:06.535 --> 00:05:08.574
cause everything is expressed
in terms of that parameter,

00:05:08.574 --> 00:05:10.907
and then dividing it by the,

00:05:11.769 --> 00:05:13.205
basically, the change in arc length

00:05:13.205 --> 00:05:14.637
with respect to that parameter,

00:05:14.637 --> 00:05:17.277
which is the size of that
first derivative function.

00:05:17.277 --> 00:05:19.045
So, if this whole thing is the derivative

00:05:19.045 --> 00:05:21.517
of the tangent vector with respect to t,

00:05:21.517 --> 00:05:23.221
what that means is when we take this,

00:05:23.221 --> 00:05:26.054
and we divide that whole thing by,

00:05:28.937 --> 00:05:30.753
the derivative, by s prime,

00:05:30.753 --> 00:05:32.594
that should give us curvature,

00:05:32.594 --> 00:05:34.913
and in fact that's just worth
writing in its own here.

00:05:34.913 --> 00:05:36.466
That's curvature.

00:05:36.466 --> 00:05:38.657
Curvature is equal to,

00:05:38.657 --> 00:05:40.525
and what I'm going to do
is, I'm going to take,

00:05:40.525 --> 00:05:41.571
since we three different times,

00:05:41.571 --> 00:05:44.412
we see the magnitude of s prime here,

00:05:44.412 --> 00:05:45.680
magnitude of s prime here,

00:05:45.680 --> 00:05:47.752
magnitude of s prime here.

00:05:47.752 --> 00:05:48.905
Since we see that three times,

00:05:48.905 --> 00:05:50.221
I'm going to go over here,

00:05:50.221 --> 00:05:51.627
and I'm going to put
that on the denominator

00:05:51.627 --> 00:05:53.293
but cubing it,

00:05:53.293 --> 00:05:55.800
so s prime, that derivative vector,

00:05:55.800 --> 00:05:58.591
the magnitude of that
derivative vector cubed.

00:05:58.591 --> 00:06:00.552
And then on the top,
we still have s prime,

00:06:00.552 --> 00:06:04.719
that vector s prime cross-product
with s double prime,

00:06:05.860 --> 00:06:07.492
that vector.

00:06:07.492 --> 00:06:09.216
And this, I mean, you could
think of this as yet another

00:06:09.216 --> 00:06:11.055
formula for curvature,

00:06:11.055 --> 00:06:12.741
I think I've given you
like four at this point.

00:06:12.741 --> 00:06:14.711
Or, you could think of it as
just kind of the same thing,

00:06:14.711 --> 00:06:19.077
and if we look back up
at our original one,

00:06:19.077 --> 00:06:20.523
that I was trying to justify.

00:06:20.523 --> 00:06:23.236
This is just the spelled
out version of it,

00:06:23.236 --> 00:06:25.351
because what is this
bottom component here?

00:06:25.351 --> 00:06:27.739
If we take x prime squared
plus y prime squared,

00:06:27.739 --> 00:06:29.493
and if we think of the
square root of that,

00:06:29.493 --> 00:06:31.196
so kinda taking it to the one half power,

00:06:31.196 --> 00:06:34.195
that would be the magnitude
of the derivative,

00:06:34.195 --> 00:06:36.812
and I kind of showed that in
some of the previous videos.

00:06:36.812 --> 00:06:38.286
And what we're doing is we're cubing that,

00:06:38.286 --> 00:06:41.142
so that whole formula
that's the very explicit,

00:06:41.142 --> 00:06:42.946
you know, in terms of x and y's,

00:06:42.946 --> 00:06:46.119
what's going on is really
just expressing this idea,

00:06:46.119 --> 00:06:48.129
you take the cross-product
between the first derivative

00:06:48.129 --> 00:06:49.149
and the second derivative,

00:06:49.149 --> 00:06:51.135
and then because you're normalizing it,

00:06:51.135 --> 00:06:53.204
you know normalizing with
respect to the first derivative,

00:06:53.204 --> 00:06:54.676
you want to scale down
the second derivative

00:06:54.676 --> 00:06:55.890
by that same amount,

00:06:55.890 --> 00:06:57.717
just so the parallelogram we're thinking,

00:06:57.717 --> 00:07:00.619
kinda shrinks and everything
stays in proportion.

00:07:00.619 --> 00:07:03.005
And then once again we're
dividing about that s prime,

00:07:03.005 --> 00:07:05.452
basically because
curvature is supposed to be

00:07:05.452 --> 00:07:07.364
with respect to s, and
not with respect to t.

00:07:07.364 --> 00:07:11.358
So, that's a way of, kinda
getting a correction factor

00:07:11.358 --> 00:07:14.494
for how wrong you're going
to be if you just think

00:07:14.494 --> 00:07:15.823
in terms of the parameter
t instead of steps

00:07:15.823 --> 00:07:18.089
in terms of the arc length.

00:07:18.089 --> 00:07:20.458
So, hopefully this makes
the original formula

00:07:20.458 --> 00:07:22.816
a little bit less random, seeming,

00:07:22.816 --> 00:07:24.175
you know, in this two dimensional case.

00:07:24.175 --> 00:07:26.192
And it also gives another
strong conceptual tool

00:07:26.192 --> 00:07:28.985
for understanding yet another
way that you can think about

00:07:28.985 --> 00:07:32.783
how much a curve itself actually curves.

00:07:32.783 --> 00:07:35.195
And I think I've probably
done enough videos here going

00:07:35.195 --> 00:07:37.076
through all the different formulas,

00:07:37.076 --> 00:07:38.724
for what curvatures should be.

00:07:38.724 --> 00:07:41.725
And in the next one or
two I'll go through some

00:07:41.725 --> 00:07:43.783
specific examples, just
to see what it looks like

00:07:43.783 --> 00:07:45.200
to compute that.


Curvature formula, part 3

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:02.731
- [Voiceover] So continuing
on with where we were

00:00:02.731 --> 00:00:04.031
in the last video, we're looking for this

00:00:04.031 --> 00:00:08.095
unit tangent vector function
given the parameterization.

00:00:08.095 --> 00:00:10.603
So the specific example
that I have is a function

00:00:10.603 --> 00:00:14.434
that parameterizes a circle
with radius capital R,

00:00:14.434 --> 00:00:16.384
but I also wanna show in parallel

00:00:16.384 --> 00:00:18.358
what this looks like more abstractly.

00:00:18.358 --> 00:00:21.121
So here, I'll just write
down in the abstract half

00:00:21.121 --> 00:00:24.627
what we did here, what we did
for the unit tangent vector.

00:00:24.627 --> 00:00:27.692
So we actually have the
same thing over here

00:00:27.692 --> 00:00:29.690
where the unit tangent vector

00:00:29.690 --> 00:00:31.607
should be

00:00:32.933 --> 00:00:35.582
the derivative function,

00:00:35.582 --> 00:00:38.043
which we know gives a tangent.

00:00:38.043 --> 00:00:39.633
It's just it might not be unit.

00:00:39.633 --> 00:00:40.847
But then we normalize it.

00:00:40.847 --> 00:00:45.769
We take the magnitude of
that tangent vector function.

00:00:45.769 --> 00:00:47.720
And in our specific case with the circle,

00:00:47.720 --> 00:00:50.878
once we did this and we kinda
took the x-components squared,

00:00:50.878 --> 00:00:52.991
y-components squared, and
simplified it all out,

00:00:52.991 --> 00:00:54.686
we got the function R.

00:00:54.686 --> 00:00:57.844
But in general case, we
might not be so lucky.

00:00:57.844 --> 00:01:01.977
Because the magnitude of this derivative

00:01:01.977 --> 00:01:04.229
is gonna be the square root

00:01:04.229 --> 00:01:06.229
of x-prime of t-squared.

00:01:07.085 --> 00:01:10.894
Right? This is the
x-component of the derivative.

00:01:10.894 --> 00:01:13.540
Plus y-prime of t-squared.

00:01:13.540 --> 00:01:15.653
Just taking the magnitude
of a vector here.

00:01:15.653 --> 00:01:19.392
So when we take the entire
function and divide it by that,

00:01:19.392 --> 00:01:21.505
what we get doesn't simplify as it did

00:01:21.505 --> 00:01:23.362
in the case of a circle.

00:01:23.362 --> 00:01:26.752
Instead, we have that x-prime
of t, which is the x-component

00:01:26.752 --> 00:01:28.749
of our s-prime of t, and we have to

00:01:28.749 --> 00:01:31.118
divide it by that entire magnitude,

00:01:31.118 --> 00:01:33.161
which was this whole expression, right?

00:01:33.161 --> 00:01:35.971
You have to divide it by that
whole square root expression,

00:01:35.971 --> 00:01:38.084
and I'm just gonna write
"..." with the understanding

00:01:38.084 --> 00:01:40.708
that this square root expression
is what goes up there.

00:01:40.708 --> 00:01:44.585
And similarly, over here, we
have y-prime of t, divided by

00:01:44.585 --> 00:01:47.767
that entire expression again, right?

00:01:47.767 --> 00:01:50.042
So simplification doesn't always happen.

00:01:50.042 --> 00:01:52.364
That was just kind of a lucky happenstance

00:01:52.364 --> 00:01:53.827
of our circle example.

00:01:53.827 --> 00:01:55.244
Now what we want,

00:01:56.404 --> 00:01:59.168
once we have the unit tangent vector

00:01:59.168 --> 00:02:01.490
as a function of that same parameter,

00:02:01.490 --> 00:02:03.974
what we're hoping to
find is the derivative

00:02:03.974 --> 00:02:07.689
of that unit tangent vector
with respect to arc length,

00:02:07.689 --> 00:02:10.870
the arc length s, and
to find its magnitude.

00:02:10.870 --> 00:02:12.867
That's gonna be what curvature is.

00:02:12.867 --> 00:02:14.700
But the way to do this

00:02:15.670 --> 00:02:17.753
is to take the derivative

00:02:19.532 --> 00:02:24.036
with respect to the parameter
t; so d big T, d little t,

00:02:24.036 --> 00:02:25.953
and then divided out by

00:02:29.168 --> 00:02:31.931
the derivative of our function s

00:02:31.931 --> 00:02:35.971
with respect to t, which we already found.

00:02:35.971 --> 00:02:38.270
And the reason I'm doing this,
you know, loosely if you're

00:02:38.270 --> 00:02:39.617
just thinking of the
notation, you might say,

00:02:39.617 --> 00:02:42.333
oh, you can kinda cancel
out the dt's from each one.

00:02:42.333 --> 00:02:44.632
But another way to think
about this is to say,

00:02:44.632 --> 00:02:48.138
when we have our tangent vector
function as a function of t,

00:02:48.138 --> 00:02:51.157
the parameter t, we're not
sure of what it changes

00:02:51.157 --> 00:02:53.317
with respect to s, right?

00:02:53.317 --> 00:02:55.012
That's something we don't know directly.

00:02:55.012 --> 00:02:57.752
But we do directly know
its change with respect

00:02:57.752 --> 00:02:59.888
to a tiny change in that parameter.

00:02:59.888 --> 00:03:02.187
So then if we just kind
of correct that by saying,

00:03:02.187 --> 00:03:04.764
hey, how much does the
length of the curve change?

00:03:04.764 --> 00:03:06.831
How far do you move along the curve

00:03:06.831 --> 00:03:08.595
as you change that parameter?

00:03:08.595 --> 00:03:11.544
And maybe if I go back
up to the picture here,

00:03:11.544 --> 00:03:16.425
this ds/dt is saying for a
tiny nudge in time, right,

00:03:16.425 --> 00:03:19.880
what is the ratio of the
size of the movement there

00:03:19.880 --> 00:03:21.761
with respect to that tiny time?

00:03:21.761 --> 00:03:23.997
So the reason that this comes
out to be a very large vector,

00:03:23.997 --> 00:03:26.850
it's not a tiny thing, is
because you're taking the ratio.

00:03:26.850 --> 00:03:29.842
Maybe this tiny change was just
an itty-bitty smidgen vector

00:03:29.842 --> 00:03:32.581
but your dividing it by 1/1,000,000

00:03:32.581 --> 00:03:35.089
or whatever the size of
dt that you're thinking.

00:03:35.089 --> 00:03:36.863
And in this specific case for our circle,

00:03:36.863 --> 00:03:39.176
we saw that the magnitude of this guy,

00:03:39.176 --> 00:03:40.878
if we took the magnitude of that guy,

00:03:40.878 --> 00:03:42.155
it's gonna be equal to R.

00:03:42.155 --> 00:03:43.455
Which is a little bit poetic, right?

00:03:43.455 --> 00:03:45.615
That the magnitude of the derivative

00:03:45.615 --> 00:03:49.307
is the same as the
distance from the center.

00:03:49.307 --> 00:03:52.604
And what this means in our specific case,

00:03:52.604 --> 00:03:55.042
if we want to apply this
to our circle example,

00:03:55.042 --> 00:03:58.075
we take dT, the tangent vector fuction,

00:03:58.075 --> 00:04:00.852
and i'll go ahead and write it here,

00:04:00.852 --> 00:04:02.872
we have the derivative
of our tangent vector

00:04:02.872 --> 00:04:05.449
with respect to the parameter,

00:04:05.449 --> 00:04:07.191
and we go up and look
here and we say okay,

00:04:07.191 --> 00:04:09.768
the unit tangent vector

00:04:09.768 --> 00:04:12.578
has the formula -sin(t) and cos(t).

00:04:12.578 --> 00:04:17.222
So the derivative of -sin(t) is -cos.

00:04:17.222 --> 00:04:21.389
So over here, this guy
should look like -cos(t).

00:04:24.815 --> 00:04:27.230
The other component, the y component,

00:04:27.230 --> 00:04:29.621
the derivative of cos(t),
as we're differentiating

00:04:29.621 --> 00:04:33.788
our unit tangent vector
function, is -sin, -sin(t).

00:04:39.278 --> 00:04:42.817
And what this implies
is that the magnitude

00:04:42.817 --> 00:04:46.409
of that derivative of the
tangent vector with respect to t,

00:04:46.409 --> 00:04:47.802
well what's the magnitude of this vector?

00:04:47.802 --> 00:04:49.219
you've got a cosine, you've got a sine.

00:04:49.219 --> 00:04:50.287
There's nothing else in there.

00:04:50.287 --> 00:04:53.120
You're going to end up with
cosine squared plus sin squared.

00:04:53.120 --> 00:04:55.906
So this magnitude just equals one.

00:04:55.906 --> 00:04:57.625
And when we do what we're
supposed to do over here

00:04:57.625 --> 00:04:59.180
and divide by the magnitude
of the derivative.

00:04:59.180 --> 00:05:03.013
We take this and we
divide it by the magnitude

00:05:04.428 --> 00:05:07.888
of the derivative ds/dt.

00:05:07.888 --> 00:05:10.697
Well we've already computed the
magnitude of the derivative.

00:05:10.697 --> 00:05:11.882
That was R.

00:05:11.882 --> 00:05:15.457
That's how we got this R, is
we took the derivative here

00:05:15.457 --> 00:05:18.624
and took it's magnitude and found it.

00:05:19.916 --> 00:05:22.098
So we find that in the
specific case of the circle

00:05:22.098 --> 00:05:23.979
the curvature function that we want

00:05:23.979 --> 00:05:27.346
is just constantly equal to 1/R.

00:05:27.346 --> 00:05:29.436
Which is good and
helpful, because I said in

00:05:29.436 --> 00:05:30.899
the original video on curvature

00:05:30.899 --> 00:05:35.101
that it's defined by one divided
by the radius of the circle

00:05:35.101 --> 00:05:37.687
that hugs the curve most closely.

00:05:37.687 --> 00:05:39.008
And if your curve is actually a circle,

00:05:39.008 --> 00:05:40.262
it's literally a circle.

00:05:40.262 --> 00:05:43.048
Then the curve that hugs it
most closely is its self.

00:05:43.048 --> 00:05:44.766
So I should hope that its curvature

00:05:44.766 --> 00:05:47.204
ends up being one divided by R.

00:05:47.204 --> 00:05:48.737
And in the more general case,

00:05:48.737 --> 00:05:52.731
if we take a look at what this ought to be

00:05:52.731 --> 00:05:56.121
you can maybe imagine just how
horrifying it's going to be

00:05:56.121 --> 00:05:57.166
to compute this.

00:05:57.166 --> 00:05:59.279
We've got our tangent vector function,

00:05:59.279 --> 00:06:02.878
which itself is almost too
long for me to write down.

00:06:02.878 --> 00:06:05.293
I just put these dot dot
dots where you're filling in

00:06:05.293 --> 00:06:07.174
x prime of t squared
plus y prime of t squared

00:06:07.174 --> 00:06:08.985
and you're gonna have to take this,

00:06:08.985 --> 00:06:11.992
take it's derivative with respect to t.

00:06:11.992 --> 00:06:13.197
It's not going to get any simpler

00:06:13.197 --> 00:06:14.417
when you take it's derivative.

00:06:14.417 --> 00:06:17.622
Take the magnitude of that
and divide all of that

00:06:17.622 --> 00:06:19.642
by the magnitude of the derivative

00:06:19.642 --> 00:06:21.894
of your original function.

00:06:21.894 --> 00:06:22.916
And I think what I'll do,

00:06:22.916 --> 00:06:24.820
I'm not going to do all of that here.

00:06:24.820 --> 00:06:26.213
It's a little bit much and
I'm not sure how helpful

00:06:26.213 --> 00:06:28.256
it is to walk through all those steps.

00:06:28.256 --> 00:06:30.486
But for the sake of having
it, for anyone whose curious,

00:06:30.486 --> 00:06:32.784
I think I'll put that into an article,

00:06:32.784 --> 00:06:34.921
and you can go through
the steps at your own pace

00:06:34.921 --> 00:06:37.336
and see what the formula comes out to be.

00:06:37.336 --> 00:06:39.007
And I'll just tell you right now,

00:06:39.007 --> 00:06:40.795
maybe kind of a spoiler alert,

00:06:40.795 --> 00:06:42.862
what that formula comes out to be

00:06:42.862 --> 00:06:46.716
is x prime, the derivative
of that first component,

00:06:46.716 --> 00:06:50.687
multiplied by y double prime,
the second derivative of

00:06:50.687 --> 00:06:53.659
that second component, minus y prime,

00:06:53.659 --> 00:06:56.376
the first derivative of
that second component,

00:06:56.376 --> 00:06:59.673
multiplied by x double prime.

00:06:59.673 --> 00:07:02.090
And all of that is divided by

00:07:03.848 --> 00:07:04.681
the

00:07:07.349 --> 00:07:08.600
the magnitude component,

00:07:08.600 --> 00:07:12.267
the x prime squared plus y prime squared.

00:07:14.162 --> 00:07:17.181
That whole thing to the three halves.

00:07:17.181 --> 00:07:19.247
And you can maybe see why you're going

00:07:19.247 --> 00:07:20.846
to get terms like this, right,

00:07:20.846 --> 00:07:24.482
cause when you're taking the derivative,

00:07:24.482 --> 00:07:26.573
when you're taking the derivative of

00:07:26.573 --> 00:07:27.978
the unit tangent vector function

00:07:27.978 --> 00:07:29.603
you have the square root term in it,

00:07:29.603 --> 00:07:31.600
the square root that has
x primes and y primes,

00:07:31.600 --> 00:07:32.761
so that's where you're going

00:07:32.761 --> 00:07:33.829
to get your x double
prime, y double prime.

00:07:33.829 --> 00:07:36.546
As the chain rule takes you down there.

00:07:36.546 --> 00:07:38.450
And you can maybe see why this whole

00:07:38.450 --> 00:07:40.168
x prime squared, y prime squared term

00:07:40.168 --> 00:07:42.258
is going to maintain
itself and it turns out

00:07:42.258 --> 00:07:45.881
it comes in here as a three halves power.

00:07:45.881 --> 00:07:48.272
And what I'm going to do in the next video

00:07:48.272 --> 00:07:51.430
I'm going to go ahead and
describe kind of an intuition

00:07:51.430 --> 00:07:53.450
for why this formula isn't random.

00:07:53.450 --> 00:07:55.711
Why if you break down what this is saying

00:07:55.711 --> 00:07:59.401
it really does give a
feeling for the curvature.

00:07:59.401 --> 00:08:01.445
The amount that the curve curves

00:08:01.445 --> 00:08:03.511
that we want to try to measure.

00:08:03.511 --> 00:08:05.160
So it's almost like this is a third way

00:08:05.160 --> 00:08:06.019
of thinking about it.

00:08:06.019 --> 00:08:07.830
The first one, I said you have

00:08:07.830 --> 00:08:10.036
whatever circle most
closely hugs your curve

00:08:10.036 --> 00:08:12.567
and you take 1 divided by its radius.

00:08:12.567 --> 00:08:15.237
And then the second way
you're thinking of this dt/ds,

00:08:15.237 --> 00:08:18.186
the change in the unit tangent vector

00:08:18.186 --> 00:08:20.880
with respect to arc length
and taking its magnitude.

00:08:20.880 --> 00:08:22.784
And of course all of these are the same,

00:08:22.784 --> 00:08:24.897
they're just kind of three different ways

00:08:24.897 --> 00:08:26.592
to think about it or things
that you might plug in

00:08:26.592 --> 00:08:28.055
when you come across a function.

00:08:28.055 --> 00:08:29.169
And I'll go through an example,

00:08:29.169 --> 00:08:31.120
I'll go through something
where we're really computing

00:08:31.120 --> 00:08:33.070
the curvature of something
that's not just a circle.

00:08:33.070 --> 00:08:36.403
But with that, I'll see you next video.


Curvature intuition

WEBVTT
Kind: captions
Language: en

00:00:00.412 --> 00:00:01.956
- [Voiceover] Hello everyone.

00:00:01.956 --> 00:00:06.123
So what I would like to do
here, is talk about curvature.

00:00:07.076 --> 00:00:10.536
And, I've drawn on the X Y
plane here, a certain curve.

00:00:10.536 --> 00:00:11.976
So this is our X-axis.

00:00:11.976 --> 00:00:13.276
This is our Y-axis.

00:00:13.276 --> 00:00:15.075
And this is a curve running through space.

00:00:15.075 --> 00:00:16.493
And I'd like you to imagine that this

00:00:16.493 --> 00:00:19.394
is a road of some kind,
and you are driving on it.

00:00:19.394 --> 00:00:20.718
And you are at a certain point,

00:00:20.718 --> 00:00:22.378
so let's say this point right here.

00:00:22.378 --> 00:00:25.710
And if you imagine, what it
feels like to drive along

00:00:25.710 --> 00:00:27.568
this road, and where you need to have your

00:00:27.568 --> 00:00:29.948
steering wheel, you're turning
it a little bit to the right.

00:00:29.948 --> 00:00:31.178
Not a lot, because it's kind of

00:00:31.178 --> 00:00:33.431
a gentle curve at this point,
you're not curving a lot.

00:00:33.431 --> 00:00:34.812
But the steering wheel isn't straight,

00:00:34.812 --> 00:00:36.333
you are still turning on the road.

00:00:36.333 --> 00:00:38.876
And now, imagine that your
steering wheel's stuck.

00:00:38.876 --> 00:00:40.768
That it's not gonna move.

00:00:40.768 --> 00:00:42.150
And however you're turning it,

00:00:42.150 --> 00:00:43.814
you're stuck in that situation.

00:00:43.814 --> 00:00:45.459
What's gonna happen, hopefully you're on

00:00:45.459 --> 00:00:46.968
an open field or something,
because your car is

00:00:46.968 --> 00:00:50.056
going to trace out some kind of circle.

00:00:50.056 --> 00:00:50.889
Right?

00:00:50.889 --> 00:00:53.655
You know your steering wheel
can't do anything different,

00:00:53.655 --> 00:00:56.396
you're just turning it a certain rate,

00:00:56.396 --> 00:01:00.018
and that's gonna have you
tracing out some giant circle.

00:01:00.018 --> 00:01:01.818
And this depends on where you are, right?

00:01:01.818 --> 00:01:04.069
If you had been at a
different point on the curve,

00:01:04.069 --> 00:01:05.428
where the curve was rotating a lot,

00:01:05.428 --> 00:01:08.853
let's say you were back a
little bit towards the start.

00:01:08.853 --> 00:01:12.208
At the start here, you have
to turn the steering wheel

00:01:12.208 --> 00:01:14.141
to the right, but you're
turning it much more

00:01:14.141 --> 00:01:17.851
sharply to stay on this part of the curve,

00:01:17.851 --> 00:01:21.415
than you were to be on this
relatively straight part.

00:01:21.415 --> 00:01:24.550
And the circle that you draw
as a result, is much smaller.

00:01:24.550 --> 00:01:26.210
And this turns out to be a pretty nice way

00:01:26.210 --> 00:01:28.357
to think about a measure, for just how

00:01:28.357 --> 00:01:30.365
much the curve actually curves.

00:01:30.365 --> 00:01:31.910
And one way you could do this,

00:01:31.910 --> 00:01:34.064
is you can think, "Okay,
what is the radius

00:01:34.064 --> 00:01:36.233
of that circle, the circle you would trace

00:01:36.233 --> 00:01:38.389
out if your steering wheel
locked at any given point.".

00:01:38.389 --> 00:01:41.698
And if you kind of follow the point,

00:01:41.698 --> 00:01:43.149
along different parts of the curve,

00:01:43.149 --> 00:01:44.832
and see, "Oh, what's the different circle

00:01:44.832 --> 00:01:46.539
that my car would trace out if

00:01:46.539 --> 00:01:48.176
it was stuck at that point.".

00:01:48.176 --> 00:01:50.533
You would get circles of
varying different radii.

00:01:50.533 --> 00:01:51.369
Right?

00:01:51.369 --> 00:01:55.620
And this radius actual
has a very special name.

00:01:55.620 --> 00:01:57.253
I will call this "R".

00:01:57.253 --> 00:02:00.713
This is called the "Radius of Curvature".

00:02:00.713 --> 00:02:02.329
And you can kind of see how this is a good

00:02:02.329 --> 00:02:05.829
way to describe, how much you are turning.

00:02:07.460 --> 00:02:09.398
You may have heard with a car,

00:02:09.398 --> 00:02:11.939
descriptions of the turning radius.

00:02:11.939 --> 00:02:13.020
You know, if you have a car with a very

00:02:13.020 --> 00:02:14.566
good turning radius, it's very small,

00:02:14.566 --> 00:02:16.363
'cause what that means if you turned

00:02:16.363 --> 00:02:17.803
it all the way you could trace out

00:02:17.803 --> 00:02:19.440
just a very small circle.

00:02:19.440 --> 00:02:21.437
But a car with a bad turning radius,

00:02:21.437 --> 00:02:23.202
you know, you don't turn very much at all,

00:02:23.202 --> 00:02:25.304
so you'd have to trace
out a much larger circle.

00:02:25.304 --> 00:02:28.102
And, curvature itself isn't this "R".

00:02:28.102 --> 00:02:30.984
It's not the radius of curvature.

00:02:30.984 --> 00:02:31.817
But what it is, is it's the reciprocal

00:02:31.817 --> 00:02:34.067
of that, it's one over "R".

00:02:35.764 --> 00:02:38.093
And there is a special symbol for it.

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It's kind of a "K", and I'm not sure in

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handwriting I am going to distinguish

00:02:41.645 --> 00:02:43.828
it from an actual "K", maybe
give it a little curly there.

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It's the Greek letter Kappa.

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And this is curvature.

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And I want you to think for a second,

00:02:48.704 --> 00:02:51.003
why we would take one over "R".

00:02:51.003 --> 00:02:52.733
"R" is a perfectly fine description

00:02:52.733 --> 00:02:55.706
of how much the road curves.

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But why is it that you would think

00:02:57.261 --> 00:03:01.360
one divided by "R", instead of "R" itself?

00:03:01.360 --> 00:03:03.507
And the reason basically, is you want

00:03:03.507 --> 00:03:05.759
curvature to be a measure
of, how much it curves

00:03:05.759 --> 00:03:07.943
in the sense that, more sharp turns

00:03:07.943 --> 00:03:09.521
should give you a higher number.

00:03:09.521 --> 00:03:12.459
So, if you're at a point,
where you are turning

00:03:12.459 --> 00:03:14.560
the steering wheel a lot, you want that

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to result in a much higher number.

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But, radius of curvature
will be really small,

00:03:18.519 --> 00:03:19.842
when you are turning a lot.

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But if you are at a point that's basically

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a straight road, you
know, there's some slight

00:03:23.093 --> 00:03:25.067
curve to it, but it's
basically a straight road,

00:03:25.067 --> 00:03:27.702
you want the curvature to
be a very small number.

00:03:27.702 --> 00:03:30.268
But in this case, the radius
of curvature is very large.

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So it's pretty helpful to just have

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one divided by "R", as the measure

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of how much the road is turning.

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And, in the next video,
I am going to go ahead

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and start describing a little bit more

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mathematically, how we capture this value.

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'Cause as a loose
description, if you are just

00:03:47.045 --> 00:03:49.506
kind of drawing pictures,
it's perfectly fine

00:03:49.506 --> 00:03:50.690
to say, "Oh yeah, you
imagine a circle that's

00:03:50.690 --> 00:03:53.071
kind of closely hugging the curve,

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it's what your steering wheel would do

00:03:54.672 --> 00:03:55.624
if you were locked.".

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But in math, we will describe
this curve parametrically.

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It'll be the output of a
certain vector valued function.

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And I wanna know how you
could capture this idea.

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This, one over "R" curvature
idea, in a certain formula.

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And that's what the next few
videos are going to cover.


Curvature of a cycloid

WEBVTT
Kind: captions
Language: en

00:00:01.020 --> 00:00:03.200
- [Voiceover] So let's do
another curvature example.

00:00:03.200 --> 00:00:06.380
This time, I'll just take
a two-dimensional curve,

00:00:06.380 --> 00:00:10.547
so it'll have two different
components, x of t and y of t

00:00:11.661 --> 00:00:14.994
and the specific components here will be

00:00:16.741 --> 00:00:18.574
t minus the sine of t,

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t minus sine of t,

00:00:21.684 --> 00:00:24.267
and then one minus cosine of t.

00:00:25.600 --> 00:00:27.000
One minus cosine of t.

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This is actually the curve,

00:00:28.780 --> 00:00:30.680
if you watched the very
first video that I did

00:00:30.680 --> 00:00:33.797
about curvature introducing
it, this is that curve.

00:00:33.797 --> 00:00:35.618
This is the curve that I said,

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"Imagine that it's a
road and you're driving

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"along it, and if your
steering wheel gets stuck,

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"you're thinking of the
circle that you trace out

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"as a result and it
various different points

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"you're gonna be turning it
various different amounts,"

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so the circle that your
car ends up tracing out

00:00:50.118 --> 00:00:51.758
would be of varying sizes.

00:00:51.758 --> 00:00:54.198
So if the curvature's high,
if you're steering a lot,

00:00:54.198 --> 00:00:56.708
radius of curvature is
low and things like that.

00:00:56.708 --> 00:00:59.657
So here, let's actually compute it.

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And in the last example I walked through

00:01:01.858 --> 00:01:04.926
thinking in terms of the derivative

00:01:04.926 --> 00:01:07.798
of the unit-tangent vector
with respect to arc length

00:01:07.798 --> 00:01:09.638
but in this case, instead of doing that,

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I just want to show what it looks like

00:01:11.538 --> 00:01:15.658
when we take the explicit
formula that looks like

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x prime times y double-prime,

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minus y prime times x double-prime,

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and then all of that divided by

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x prime squared plus y prime squared.

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And then I'm writing x
prime and y prime and such

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and all of these you should think of as

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taking in the variable t.

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I'm must being a little
too lazy to write it.

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And you take that to
the three-halves power.

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So, this was the formula

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and I'm not a huge fan
of memorizing formulas

00:01:44.099 --> 00:01:45.639
and then hoping to apply them later.

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I really do think the one
thing you should take away

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from curvature is the idea that

00:01:49.499 --> 00:01:51.606
it's the derivative of
the unit-tangent vector

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with respect to arc length

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and if you need to, you can just look up

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a formula like this but
it's worth pointing out

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that it makes some things easier

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to compute because
finding the tangent vector

00:02:00.320 --> 00:02:02.356
and everything can be
kind of like reinventing

00:02:02.356 --> 00:02:05.019
the wheel when you already
have the results here.

00:02:05.019 --> 00:02:07.012
So, first thing to do is just find

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x prime, y double-prime,

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y prime and x double prime.

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So let's go ahead and write those out.

00:02:13.719 --> 00:02:16.519
So the first derivative of x of t

00:02:16.519 --> 00:02:18.719
if we go up here that's t minus sine of t.

00:02:18.719 --> 00:02:22.302
So its derivative is
one minus cosine of t.

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And the derivative of the y component

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of one minus cosine t,

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y prime of t,

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is gonna be, derivative
of cosine is negative sine

00:02:36.900 --> 00:02:38.940
so negative derivative of that is sine,

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and that one goes to a constant,

00:02:42.339 --> 00:02:46.560
and then when we take the second
derivatives of those guys,

00:02:46.560 --> 00:02:49.240
so maybe change the color for
the second derivative here,

00:02:49.240 --> 00:02:51.073
x double prime of the,

00:02:52.659 --> 00:02:54.819
so now we're taking
the derivative of this,

00:02:54.819 --> 00:02:55.959
which actually we just did because

00:02:55.959 --> 00:02:57.979
by coincidence the first derivative of x

00:02:57.979 --> 00:03:00.097
is also the y component

00:03:00.097 --> 00:03:02.597
so that also equals sine of t.

00:03:04.639 --> 00:03:08.806
And then y double prime is just
the derivative of sine here

00:03:10.200 --> 00:03:13.867
so that's just gonna
be cosine, cosine of t.

00:03:15.320 --> 00:03:17.620
So now, when we just
plug those four values

00:03:17.620 --> 00:03:21.405
in for kappa, for our
curvature, what we get

00:03:21.405 --> 00:03:24.738
is x prime was one minus cosine of t,

00:03:28.940 --> 00:03:32.614
multiplied by y double
prime is cosine of t.

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Cosine of t.

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We subtract off from that y prime,

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which is sine of t,

00:03:39.249 --> 00:03:40.082
sine of t,

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multiplied by x double prime.

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So x double prime is also sine of t,

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so I could just say

00:03:47.900 --> 00:03:49.576
sine of t squared,

00:03:49.576 --> 00:03:53.743
and the whole thing is
divided by x prime squared

00:03:54.620 --> 00:03:57.953
so x prime was one minus cosine of t,

00:04:00.020 --> 00:04:01.700
minus cosine of t,

00:04:01.700 --> 00:04:04.117
squared, plus y prime squared

00:04:05.080 --> 00:04:06.300
so y prime was just sine,

00:04:06.300 --> 00:04:10.583
so that's just gonna be sine squared of t.

00:04:10.583 --> 00:04:14.500
And that whole thing to
the power three halves.

00:04:15.859 --> 00:04:17.580
And that's your answer, right?

00:04:17.580 --> 00:04:19.720
You apply the formula, you get the answer.

00:04:19.720 --> 00:04:22.833
So for example, when I
was drawing this curve

00:04:22.833 --> 00:04:25.040
and kind of telling the
computer to draw out

00:04:25.040 --> 00:04:26.620
the appropriate circle,
I didn't go through

00:04:26.620 --> 00:04:29.521
the entire "find the unit tangent vector,

00:04:29.521 --> 00:04:31.961
"differentiate it with respect
with arc length" process,

00:04:31.961 --> 00:04:33.433
even though that's decently easy to do

00:04:33.433 --> 00:04:35.687
in the case of things
like circles or helixes

00:04:35.687 --> 00:04:37.208
but instead I just went to that formula.

00:04:37.208 --> 00:04:39.037
I looked it up because I had forgotten

00:04:39.037 --> 00:04:41.979
and I found the radius
of curvature that way.