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so far we've seen how to use subdivisions to create the surfaces that define the shapes of our characters in this lesson we're going to delve deeper into the mathematics behind weighted averages and just like before we're going to start with a simpler version which is looking at the 2d curves before looking at 3d surfaces and we will get to see that subdivision can be very flexible that is that we'll be able to get a bunch of results by just playing with different amounts of weights to continue our study of weighted averages recall that in the environment modeling lesson we looked at the weighted averages of two points so let's start this lesson with some review we wrote the weighted average M of two points a and B as M equals one minus T times a plus T times B the parameter T controls the weight and therefore the position of where m is along a and B recall also that the weights in front of a and B have to add up to 1 to represent a proper average we can rewrite the expression for M in a way that is easier to add more points and that is M equals little a times a plus little B times B all of that divided by little a plus little B notice we have to divide all of that by little a plus little B for the expression to be a proper average the geometry of this more symmetric form says that the ratio of the lengths am 2 MB is little B to little a now let's generalize to the case of averages of three points M equals little a times a plus little B times B plus little C times C all of that divided by little a plus little B plus little see the geometry says that the sub triangle areas are in the ratio a to B to C here all of the weights are 1 so that M is the midpoint and all of the areas are equal suppose we want to weight be twice as heavily as a or C the algebra is M equals a plus 2b plus C and all of that divided by for the geometry says that the area of the triangle opposite of B is twice as big as the area's in the other two triangles increasing B's weight to two moves the M closer to the beat increasing it to three moves it even closer setting A's weight to zero means that it doesn't matter anymore so M is on the line somewhere in between B and C I just love this connection between algebra and geometry and that's because it's just so elegant and it's really useful too sometimes the problem is best solved by looking at the geometry and other times is by looking at the algebra that's why it's so important to be fluent in both this next exercise will test how well you understand this concept I certainly could do the entire thing in Spanish aqui estamos otra vez school bus the subs BBC own attractiveness