Shapr At Pixar, we're all about telling（ 告诉） stories. But one story that hasn't been told very much is the huge degree to which math is used in the production（生产） of our films. The math that you're learning（学习） in middle school and high school is used all the time at Pixar. So let's start with a very simple example. Anybody recognize this guy? So this is Woody（多树木的） from Toy Story.

  And let's ask Woody to, say, walk across the stage from, say, left to right. Just like that. So, believe it or not, you just saw a ton（吨） of mathematics. Where is it? Well, to explain that, it's important to understand that artists and designers think in terms of shape and images, but computers think in terms of numbers and equations（方程式）. So to bridge those two worlds, we use a mathematical（数学的） concept（概念） called coordinate（坐标） geometry, right?

  That is, we lay（躺） down a coordinate（坐标） system with x describing how far something is to the right and y describing how high something is. So with these coordinates, we can describe where Woody is at any instant in time. For instance（实例）, if we know the coordinates（坐标） of the lower left corner of that image, then we know where the rest of the image is. And in that little sliding（滑） animation（活泼） we saw a second ago, that motion（运动） we call translation（翻译）, the x coordinate started with a value（价值） of 1 and it ended with a value（价值） of about 5. So if we want to write that in mathematics, we see that the x at the end is four bigger than x at the start. So, in other words, the mathematics of translation is addition（加）, all right?

  How about scaling? That is, making something bigger or smaller. Any guesses as to what the mathematics of scaling might be? Dilation? Multiplication（[数] 乘法）, exactly. If you're going to make something twice as big, you need to multiply（乘） the x and the y coordinates（坐标） all by two.

  So this shows us that the mathematics of scaling is multiplication. OK? How about this one? How about rotation（旋转）? Spinning（纺） around. The mathematics of rotation is trigonometry.

  So here's an equation that expresses that. It looks a little scary at first. You'll probably get this in eighth or ninth grade（等级）. If you find yourself sitting in trigonometry class, wondering（想知道） when you're ever going to need this stuff（材料）, just remember that any time you see anything rotate（旋转） in one of our films, there's trigonometry at work underneath（在下面）. I first fell in love with mathematics in seventh grade. Any seventh graders?

  A few of you? Yeah. My seventh grade science teacher showed me how to use trigonometry to compute（计算） how high the rockets（火箭） that I was building was going.