[Sound of swords clanging] 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." [Sound of arrow
being shot and hitting target] TONY: Good to see you. So we're talking today about some
of the ways that math was used to create the forest in "Brave."
MARK ANDREWS: Oh, yeah. TONY: And I was wondering
what it's like, as a director, to work with the technical staff here. MARK: Oh, I love them. I mean, everything
that you see on a screen in a Pixar movie, we couldn't
put out there without the technical staff. Our movies are so complex. A movie like "Brave," the organics in it –
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. TONY: That makes my heart warm.
MARK: [Laughs] TONY: Thank you so much.
MARK: Absolutely. TONY: And we're going to 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 graphs of quadratic functions. The problem with quadratic functions
is they're not very intuitive for artists. A more artist-friendly way
is to use 3 points. Let me show you. Okay. So I have 3 points,
and as I move them around, the parabola updates accordingly. And in computer graphics, these 3 points
like this are called a "control polygon." So if I'm only going
to store the 3 points, I somehow have to recover
that parabolic arc. So the question is "How do I go from these 3 points
to recovering my parabola?" The idea is pretty simple. The first thing we're going do
is lay out some evenly spaced points – the same number on each leg. And then next, what I'm going to 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
a "string-art construction." You take a piece 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,
as I've done here. So we'll call this the "string-art
construction for parabolic arcs." In the next exercise, you'll have
an opportunity to connect the dots yourself.