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## Pixar in a Box

### Course: Pixar in a Box > Unit 3

Lesson 2: Mathematics of subdivision- Start here!
- Weighted average of three points
- Weighted average intuition
- Weighted average of three points
- 2. Weighted subdivision
- Weighted subdivision
- 3. Fun with weights
- Interativo: subdivisão ponderada
- Subdivision weights
- Bonus: Equations for points in subdivision

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# 3. Fun with weights

Let's take a closer look at the weights used during subdivision. Do we have to be careful when selecting weights?

Click here to learn more about Pascal's Triangle & the Binomial theorem

Click here to learn about eigenvalues & eignenvectors.

Click here to learn more about Pascal's Triangle & the Binomial theorem

Click here to learn about eigenvalues & eignenvectors.

## Want to join the conversation?

- How many different ways is/are Pascal Triangle used?(2 votes)
- Pascal's triangle is primarily used in the binomial theorem. It is basically a pattern triangle, which is practically infinite. You can look up the triangle, because I am unable to display it here, obviously. It is also used to expand binomials, such as (x-5y)^6.(3 votes)

- I don't understand the mathematics within subdivision. Please may someone HELP me?(3 votes)
- 1:11can it be in any direction(3 votes)
- sugestion talk louder on the videos i can hardley hear(1 vote)
- Make your sound louder then.(2 votes)

- The video makes the claim that the coefficients 1, 2, 1 are pieces of degree 3 curves, like y = x^3. However, (x + y)^3 = x^3 + 3x^2*y + 3xy^2 + y^3, whose coefficients are 1, 3, 3, 1, not 1, 2, 1. Similarly, (x + y)^2 has coefficients, 1, 4, 6, 4, 1, not 1, 3, 3, 1. Is there a mistake here or am I not understanding something?(1 vote)
- does any one still play this game(1 vote)
- where are the Lobsters 😩(0 votes)

## Video transcript

- Congratulations you're in
the last step of this lesson. To finish our look at weighted averages recall that if I pick weights one one I get subdivision curves
that look like this. And we learn that these
shapes are pieces of degree two curves like Y equals X
squared stitched together. And if I pick the weights one-two-one, I get curves that look smoother. Just like this. These curves turn out to be
pieces of degree three curves like Y equals X cubed stitched together. In this example there are four
different degree three pieces I can even get smoother curves by picking the weights
one-three-three-one. Which are degree four curves
like Y equals X to the fourth. And it turns out that the
weights have to be chosen very carefully for the
curves to be smooth. So what exactly does carefully mean? These numbers one-one,
one-two-one, one-three-three-one all come from a pattern
called Pascal's Triangle. Let's see what happens if I pick weights that aren't from Pascal's Triangle. I don't even have to
pick positive weights, so supposed I pick one-negative two-three? Nothing new with split, it
always just adds mid points. But when I do the averaging stuff, whoa! As I keep subdividing the curve
gets less and less smooth. It's something called a fractal. We don't have all of
the mathematical tools that we require to show this, there's some more advanced
concepts from linear algebra and something called eigenanalysis. Eigenanalysis sounds kinda scary
but mathematicians are just really good at coming up with
names that are kinda scary. And just like we've done so far you just learn one step at a time. The main thing that I
want you to walk away with is that the surfaces
that you see in our films are actually pretty easy to create. The art department creates
sketches for the modelers to refer to when determining
control point positions. From there the software tools
do the splitting and averaging to produce the beautiful surfaces that you see in the theater. Of course when you're
watching one of our films we don't want you to
think about any of that. We just want you to sit back, relax, and let the movie take you away. So play around with the weights. See what weird results you can get. And when you're done you can
move onto the final exercise.