I don't understand why: ``` M is the weighted average of three points, A, B and C. What is the position of M of using weights of 0, 0 and 0, respectively? ``` results in "M is undefined" rather than M defaulting to the center of △ABC triangle (which is the case for all other situations where the values are equal, like: 1, 1, 1, or 5,5,5 etc.). If there is no weight, it should be in the center of the triangle, right?

Because, according to the mathematical formula, the average is `(aA + bB + cC)/(a + b + c) = (0A + 0B + 0C)/(0 + 0 + 0) = 0/0 = Indeterminate`.

At 1:29 he says that to make the expression a proper average you have to divide by a+b. Why? By doing that you change the value don't you?

When you're taking an average of several numbers, you add all the numbers up, then divide by the number of numbers. So the average of the set [1,7,3] is (1 + 7 + 3) / 3. When you're taking a _weighted_ average, you count one of your numbers more than once. So if you weight the previous average so that the 3 is twice as important as the other two is (1 + 7 + 3 + 3) / 4. In the same way, when you take the weighted average of geometric points, if you increase a point's importance, you have to increase the divisor equal to the extra level of importance you give it. To illustrate: If you're simply averaging two points A and B, then the formula is (A + B) / 2. The divisor is equal to the number of points you're averaging. If you want A to count twice as much as B, then it's (A + A + B) / 3. Again, the divisor is equal to the number of points, even though one of them is really the same point twice. Here, the "little a" is 2 and the "little b" is 1. So it could be rewritten as (2*A + 1*B) / 2 + 1. If you don't increase the divisor like this, you won't get an average, and your new point will wind up in the wrong place.

I didn't understand what he said at the end. (The language he was speaking). Also, what did he say?

It was Spanish. I would suggest looking at the above comment.

what is weighted average?

Imagine that you want the average of two points, X and Y, but you want Y to be twice as important. So rather than saying (X + Y)/2, you average X, Y, and Y, saying (X + 2Y)/3. Now, the technical definition: The weighted average of vectors (fancy word for points) X_1, X_2, X_3, . . ., X_n with weights A_1, A_2, A_3, . . ., A_n is (A_1 X_1 + A_2 X_2 + A_3 X_3 + . . . + A_n X_n)/(A_1 + A_2 + A_3 + . . . + A_n).

Hi I am kinda new to this, so what does little a, little b and little c represent?

a b and c are the weights of the triangle

I forgot!! In one video, they had a recommended free software or something, and I wanted to try that out! It may be past this point..if anyone finds it, can you tell me which video? Thanks!

In this video (https://www.khanacademy.org/partner-content/pixar/environment-modeling-2/animating-parabolas-ver2/v/tony-derose) Tony mentions Blender, which is a free 3D modeling and animation software.

Main content

Pixar in a Box

Course: Pixar in a Box > Unit 3

Lesson 2: Mathematics of subdivision

Weighted average of three points

Google Classroom

First we'll review weighted averages of two points and extend the idea to three points.
Practice weighted averages of two points in Environment Modeling if you haven't seen it before.

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Madeliv
Posted 8 years ago. Direct link to Madeliv's post “I don't understand why: `...”
I don't understand why:
M is the weighted average of three points, A, B and C. What is the position of M of using weights of 0, 0 and 0, respectively?
results in "M is undefined" rather than M defaulting to the center of △ABC triangle (which is the case for all other situations where the values are equal, like: 1, 1, 1, or 5,5,5 etc.). If there is no weight, it should be in the center of the triangle, right?
Button navigates to signup pageComment on Madeliv's post “I don't understand why: `...”
(7 votes)
Answer
- Zuarrie
  Posted 8 years ago. Direct link to Zuarrie's post “Because, according to the...”
  Because, according to the mathematical formula, the average is (aA + bB + cC)/(a + b + c) = (0A + 0B + 0C)/(0 + 0 + 0) = 0/0 = Indeterminate.
  Comment on Zuarrie's post “Because, according to the...”
  (4 votes)
coppernugget
Posted 8 years ago. Direct link to coppernugget's post “At 1:29 he says that to m...”
At
1:29
he says that to make the expression a proper average you have to divide by a+b. Why? By doing that you change the value don't you?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Bryan Ray
  Posted 8 years ago. Direct link to Bryan Ray's post “When you're taking an ave...”
  When you're taking an average of several numbers, you add all the numbers up, then divide by the number of numbers. So the average of the set [1,7,3] is (1 + 7 + 3) / 3.
  
  When you're taking a weighted average, you count one of your numbers more than once. So if you weight the previous average so that the 3 is twice as important as the other two is (1 + 7 + 3 + 3) / 4.
  
  In the same way, when you take the weighted average of geometric points, if you increase a point's importance, you have to increase the divisor equal to the extra level of importance you give it.
  
  To illustrate: If you're simply averaging two points A and B, then the formula is (A + B) / 2. The divisor is equal to the number of points you're averaging. If you want A to count twice as much as B, then it's (A + A + B) / 3. Again, the divisor is equal to the number of points, even though one of them is really the same point twice. Here, the "little a" is 2 and the "little b" is 1. So it could be rewritten as (2*A + 1*B) / 2 + 1.
  
  If you don't increase the divisor like this, you won't get an average, and your new point will wind up in the wrong place.
  Comment on Bryan Ray's post “When you're taking an ave...”
  (5 votes)
Mike SIms
Posted 6 years ago. Direct link to Mike SIms's post “0:03 - Now that you have ...”
0:03

- Now that you have a feel for how t works,
0:05

we're ready to calculate our intersection point I
0:08

between our ray CP and our line segment AB.
0:11

Recall from the previous video that
0:13

the slope intercept form of the line AB
0:15

is y equals negative three x plus 11
0:18

and the parametric representation of the ray CP
0:21

is the function R of t equals one minus t
0:25

times C plus t times P.
0:29

Different values of the parameter t
0:31

locate different points on the ray.
0:36

The intersection point that we're after
0:38

is one such point on the ray so there
0:40

must be some value of t, call it t star,
0:44

such that I equals R of t star.
0:49

This is really two equations, one for the x-coordinate
0:52

of I and one for the y-coordinate.
0:55

These two equations are I sub x equals R sub x
0:59

of t star, which equals one minus t star
1:03

times C sub x plus t star times P sub x.
1:09

In the same way I sub y equals R sub y of t star,
1:14

which equals one minus t star times C sub y
1:18

plus t star times P sub y.
1:22

In this particular case C, our camera position,
1:26

has coordinates zero, zero
1:28

and P has coordinates two, 1/2.
1:33

So we have I sub x equals t star times two
1:38

and I sub y equals t star times 1/2.
•
1:42

I is also on the line segment AB meaning that
1:46

I satisfies the slope intercept form for AB,
1:50

that is I sub y equals negative three
1:53

times I sub x plus 11.
1:57

So we have three equations and three unknowns,
2:01

I sub x, I sub y and t star.
2:06

We can solve the system of equations
2:08

by substituting the first two equations
2:10

into the third to get an equation just in t star.
2:14

1/2 t star equals negative three
2:18

times two times t star plus 11.
2:23

Solve this for t star, then plug that value
2:26

of t star into the first two equations
2:28

to get I sub x and I sub y.
2:32

And that's how it's done.
2:33

Before we continue get some experience using this
2:36

kind of parametric function in the next exercise.
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(6 votes)
Answer
NADIA
Posted 8 years ago. Direct link to NADIA's post “what is weighted average?”
what is weighted average?
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(3 votes)
Answer
- Zuarrie
  Posted 8 years ago. Direct link to Zuarrie's post “Imagine that you want the...”
  Imagine that you want the average of two points, X and Y, but you want Y to be twice as important. So rather than saying (X + Y)/2, you average X, Y, and Y, saying (X + 2Y)/3.
  
  Now, the technical definition: The weighted average of vectors (fancy word for points) X_1, X_2, X_3, . . ., X_n with weights A_1, A_2, A_3, . . ., A_n is (A_1 X_1 + A_2 X_2 + A_3 X_3 + . . . + A_n X_n)/(A_1 + A_2 + A_3 + . . . + A_n).
  Comment on Zuarrie's post “Imagine that you want the...”
  (5 votes)
Katie Wrona
Posted 8 years ago. Direct link to Katie Wrona's post “I forgot!! In one video, ...”
I forgot!! In one video, they had a recommended free software or something, and I wanted to try that out! It may be past this point..if anyone finds it, can you tell me which video? Thanks!
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Peter Collingridge
  Posted 8 years ago. Direct link to Peter Collingridge's post “In this video (https://ww...”
  In this video (https://www.khanacademy.org/partner-content/pixar/environment-modeling-2/animating-parabolas-ver2/v/tony-derose) Tony mentions Blender, which is a free 3D modeling and animation software.
  Comment on Peter Collingridge's post “In this video (https://ww...”
  (6 votes)
Daniel H
Posted 8 years ago. Direct link to Daniel H's post “I didn't understand what ...”
I didn't understand what he said at the end. (The language he was speaking).
Also, what did he say?
Button navigates to signup pageComment on Daniel H's post “I didn't understand what ...”
(4 votes)
Answer
- Clarissa Simmerer
  Posted 8 years ago. Direct link to Clarissa Simmerer's post “It was Spanish. I would s...”
  It was Spanish. I would suggest looking at the above comment.
  Comment on Clarissa Simmerer's post “It was Spanish. I would s...”
  (2 votes)
Catherine A.
Posted 7 years ago. Direct link to Catherine A.'s post “Hi I am kinda new to this...”
Hi I am kinda new to this, so what does little a, little b and little c represent?
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(3 votes)
Answer
- Destroyer
  Posted 7 years ago. Direct link to Destroyer's post “a b and c are the weights...”
  a b and c are the weights of the triangle
  Button navigates to signup page
  (2 votes)
Kiersten Kirkindall
Posted 8 years ago. Direct link to Kiersten Kirkindall's post “How is M the weighted ave...”
How is M the weighted average of A, B and C
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(3 votes)
Answer
hoopoothepinkielover
Posted 5 years ago. Direct link to hoopoothepinkielover's post “Ok the way he explains th...”
Ok the way he explains the math makes WAY more sense then what the other guy in Environment Modeling did, and now I understand it better, not entirely but better
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(3 votes)
Answer
Seahawk
Posted 3 months ago. Direct link to Seahawk's post “ugh i hate math”
ugh i hate math
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(3 votes)
Answer

Video transcript

Here we are again with the interactive replacement curves