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### Course: Multivariable calculus > Unit 1

Lesson 5: Transformations# Transformations, part 3

Learn how you can think about a parametric surface as a certain kind of transformation. Created by Grant Sanderson.

## Want to join the conversation?

- Is there a way to normalize the audio across videos? It's annoying to constantly switch between volumes across videos.(15 votes)
- hold up what is this im trying to study for a vague test. where am i(12 votes)
- here, the output of the function is a 3d vector. But what we see is a doughnut. How can I relate to points on the doughnut with 3d vectors?(4 votes)
- Take a point on the surface of the donut, and call it (x, y, z). The 3D vector is simply the vector from the origin to that point.(9 votes)

- What does the checkerboard pattern on the torus represent? Or is this just a fancy, non-boring way of coloring it?(5 votes)
- If you don't have a pattern on it, you won't be able to imagine it's shape. It would look like a 2-D disc(6 votes)

- The Surface Area of this donuts is 4*pi*pi(5 votes)
- Just a quick check, the output of the function is actually a plane of vectors right?(1 vote)
- It's no longer a plane. Rather, the output is a surface. However, it's not a surface of vectors. It's a surface of points (A given input produces a vector in R3 which uniquely determines a point in space. Doing so with multiple vectors gives us the donut shape).(3 votes)

- I get the feeling that a similar math method is used in 3d texturing, like when using the UVW unwrap modifier in 3ds max (a 3d surface get sliced and put in a 2d plane then after coloring and painting it gets wrapped as 3d again). Is that correct?(2 votes)
- By transformation, we are constrained by the shortest route from point a that gets transformed to point b. Suppose that the actual real world transformation take a different route other than the shortest route. What additional mathematics do we require if we want to trace the entire exact route of the transformation and show it exactly as it happened?(1 vote)
- Grant starts to address this towards the end when he talks about the "interpolated" values. In fact, the function as written says nothing about which path is followed--it defines no path, only the starting points and ending points. Grant himself put in a rule to make the animation look cool. So he could tell you what math he personally chose to transform the input to the output. The equation as written says precisely nothing about the path between the two spaces.(2 votes)

- for the third part of the lesson ;the diagram of the torus, what is the difference between the input space and the output space? Is it that they are same? Will the output space in the graph be equal to the input space?(1 vote)
- Hi. Is the doughnut hollow or is it filled inside? I am not asking this as a joke, I am asking this to see if the points would create the husk of a doughnut or an actual, full, doughnut.(1 vote)

## Video transcript

- [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.