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Transformations, part 3

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

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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.