- [Voiceover] So I wanna give you guys just one more example

of a transformation before we move on

to the actual calculus of multi variable calculus.

In the video on parametric surfaces

I gave you guys this function here.

Its a very complicated looking function

its got a two dimensional input

and a three dimensional output

and I talked about how you can think about it

as drawing a surface in three dimensional space

and that one came out to be the surface

of a doughnut which we also call a torus.

So what I wanna do here is talk about

how you might think of this as a transformation

and first let me just get straight

what the input space here is.

So the input space you could think about it

as the entire T,S plane, right?

We might draw this as the entire T axis

and the S axis

and just everything here and see where it maps.

But you can actually go to just a small subset of that.

So if you limit yourself to T going between zero

so between zero and lets say 2 pi

and then similarly with S going from zero up to 2 pi

can you imagine what, you know

that would be sort of a square region.

Just limiting yourself to that

you're actually gonna get all of the points

that you need to draw the torus.

And the basic reason for that is that

as T ranges from zero to 2 pi

the cosine of T goes over its full range

before it starts becoming periodic.

Sine of T does the same and same deal with S.

If you let S range from zero to 2 pi

that covers a full period of cosine

a full period of sine

so you'll get no new information by going elsewhere.

So what we can do is think about this portion

of the T,S plane kind of as living inside

three dimensional space as a sort of cheating

but its a little bit easier to do this

than to imagine moving from

some separate area into the space.

At the very least for the animation efforts

its easier to just start it off in 3D.

So what I'm thinking about here

this square is representing that T,S plane

and for this function which is taking all

of the points in this square as its input

and outputs a point in three dimensional space

you can think about how those points

move to their corresponding output points.

So I'll show that again.

We start off with our T,S plane here

and then whatever your input point is

if you were to follow it, and you were to

follow it through this whole transformation

the place where it lands would be

the corresponding output of this function.

And one thing I should mention is

all of the interpolating values

as you go in between these don't really matter.

Their function is really a very static thing

there's just an input and there's an output.

And if I'm thinking in terms of

a transformation actually moving it

there's a little bit of magic sauce

that has to go into making an animation do this

and in this case I kind of put it

into two different phases to sort of

roll up one side and roll up the other

it doesn't really matter but the general idea

of starting with a square and somehow warping that

however you do choose to warp it

is actually a pretty powerful thought.

And as we get into multi variable calculus

and you start thinking a little more deeply about surfaces

I think it really helps if you think about

what a slight little movement over here

on your input space would look like

what happens to that tiny little movement

or that tiny little traversal

what it looks like if you do that same movement

somewhere on the output space.

And you'll get lots of chances to wrap your mind

about this and engage with the idea.

But here I just want to get your minds churning on this

pretty neat way of viewing what functions are doing.