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## Pixar in a Box

### Course: Pixar in a Box > Unit 10

Lesson 2: Mathematics of animation curves- Start here!
- 1. Mathematics of linear interpolation
- Linear interpolation
- 2. Repeated linear interpolation
- 3. De Casteljau's algorithm
- Constructing curves using repeated linear interpolation
- 4. What degree are these curves?
- Bonus: Equations from de Casteljau's algorithm

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# 3. De Casteljau's algorithm

We can use de Casteljau's algorithm to calculate curves using any number of points.

## Want to join the conversation?

- is it important to teach younger kids about technology?(9 votes)
- Yes! Definitely! Technology is really the future to this world.(10 votes)

- i got a big question how do you put a piece of cheese on a phone and put on ceiling?(4 votes)
- Tie the piece of cheese in a phone cord, if it's a wired phone, and glue the phone to the ceiling. If it's a cellphone, glue the cheese on then glue the phone to the ceiling. xD(4 votes)

- Isn't it called De Casteljau's algorithm, and not Castlejau's?(2 votes)
- is the main point of animation for entertainment purposes(3 votes)
- Partially. Some animations are for educational purposes, and others are sadly used for hate (such as those gacha cringe children who animate their videos badly just to hate on a good creator). Most animation we see today is for entertainment, but schools still use edutainment animations to teach topics such as politics, history, math, and more.(2 votes)

- Is this similar to subdivision surface algorithms for smoothing 3D modeled surfaces?(3 votes)
- Yes, that's right. Subdivision is covered in this lesson: https://www.khanacademy.org/partner-content/pixar/modeling-character

Subdividing points to an infinite degree is mathematically the same as creating a bezier curve.(2 votes)

- I want to join pixar. HOW DO I ?(2 votes)
- To join Pixar you have to go to their website click on careers select the job you want click see open jobs click which one you want scroll down a bit click apply for this position and sign up(5 votes)

- Why do animators construct Bezier curves using four points?(2 votes)
- It's so they can have more refined control over the curve. As you can see from the video, combined with the program, the cubic Bezier curves work very efficiently for animating.(3 votes)

- De Casteljau's algorithm is pretty confusing for me, I am a 4th grader in Vietnam and at here we do not learn about this(2 votes)
- this lesson was fun until math came along.(1 vote)
- how do i work at pixar productions(1 vote)

## Video transcript

- So, how'd it go? Did you figure out how to extend a Casteljau's algorithm to 4 points? It's not so easy, so don't
worry if you had some trouble. Here's what De Casteljau came up with. First, we use linear interpolation along with our parameter
t, to find a point on each of the 3 line segments. Now we have a 3-point polygon,
just like the grass blade. As before, we find a point
on each of the new segments using linear interpolation
and the same t value. Now we have a 2-point polygon, or a line. We find a point on our line using linear interpolation, one more time. As we vary the parameter
t, this final point traces out our smooth curve. These are the kind of
curves we typically use to control the motion of our
characters as we animate. We use something called a graph editor, which lets us manipulate
the control points of these curves to get
smooth motion between poses. This is the graph editor
that we use at Pixar. In between each one of my
key poses is a Bézier curve. We also tend to group
the adjacent segments so they maintain the slope of the
curve across the key frame. This prevents sudden jerks in the motion. You can use the next interactive excercise to get some experience with Bézier curves that have 3, 4, or even more points.