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congratulations you're in the last step of this lesson to finish our look at weighted averages recall that if I pick weights 1 1 I get subdivision curves that look like this and we learn that these shapes are pieces of degree 2 curves like y equals x squared stitched together and if I pick the weights 1 2 1 I get curves that looks smoother just like this these curves turn out to be pieces of degree 3 curves like y equals x cubed stitched together in this example there are 4 different degree 3 pieces I can even get smoother curves by picking the weights 1 3 3 1 which our degree 4 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 carefuly mean these numbers 1 1 1 2 1 1 3 3 1 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 suppose I pick 1 negative 2 3 nothing new would split it always just adds midpoints but when I do the averaging step whoa as I keep subdividing the curve gets less and less smooth it's something called fractal we don't have all of the mathematical tools that require to show this there's some more advanced concepts from linear algebra and something called eigen analysis eigen analysis sounds kind of scary but mathematicians are just really good by coming up with names that are kind of 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 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 theatre of course when you're watching one of our films we don't want you to think about any of that you 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 on to the final exercise