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(swords clanking) - Welcome to Pixar. I'm Tony Derose, one of the computer scientists who works on our film here. And conveniently enough behind me is Mark Andrews, director of Brave. (whoosh) - Good to see you. - So we're talking today about some of the ways that math was used to create the forest in Brave. And I was wondering what it's like as a director to work with the technical staff here. - Oh I love them. I mean, everything that you see on a screen, on a Pixar movie, we couldn't put it out there without the technical staff. Our movies are so complex. The movie like Brave, the organics, and the grass, the forest, her hair, I mean everything. That just ups the game when it comes to the numbers that you're crunching in a computer so we rely completely on mathematics to make these movies. - That makes my heart warm. (Andrew laughs) Thank you so much. - Absolutely - And we're gonna be talking about some of that complexity in the rest of this lesson. We saw in the previous video, how parabolas are used to model grass in Brave. A complete parabola is actually an infinite curve, but we just want a little piece. That's called a parabolic arc. And to create believable grass, we have to create other attributes such as how the width varies up the blade, its color, and how it moves in response to things like horse hooves and wind. And we'll get to all of that later in the lesson. But for now let's just focus on the basic shape. Come on inside, I'll show you more. So the question is, how are we going to represent parabolic arcs in a way that artists can deal with, but computers can too. Well, there are a variety of ways of representing parabolic arcs. You may have seen them for instance, as grass in quadratic functions. The problem with quadratic functions is they're not very intuitive for artists. A more artist friendly way, it to use three points. Let me show you. Okay, so I have three points, and as I move them around the parabola updates accordingly and in computer graphics, these three points like this are called the control polygon. So if I'm only gonna store the three points, I some how have to recover that parabolic arc. So the question is how do I go from these three points to recovering my parabola? The idea is pretty simple. The first thing we're gonna do is lay out some evenly spaced points, the same number on each leg. And then next, what I'm gonna do is start connecting dots. And as I continue to connect these dots, you'll see the curve start to emerge almost magically. Now you can do this same construction in real life. It's called the string art construction. You take a pice of paper, you draw some lines on it, you spread out some evenly spaced points, and then with needle and thread, you start making these connections like down here. So we'll call this the strong art construction for parabolic arcs. In the next exercise you'll have an opportunity to connect the dots yourself.