- [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.