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### Course: Pixar in a Box > Unit 11

Lesson 1: Modeling grass with parabolas- Start here!
- Introduction to parabolic arcs
- 1. String art
- String art construction
- 2. Midpoint formula
- Midpoint formula
- 3. Parabolic arcs
- Parabolic curve matching
- 4. Modeling grass
- Design challenge: Modeling grass
- 5. Animating grass
- Design challenge: Animating grass
- Getting to know Tony DeRose
- Hands-on activity

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# 2. Midpoint formula

Explore how midpoints are used to describe the "string art construction.".

## Want to join the conversation?

- So, the more strings, the curvier the parabolic arc?(16 votes)
- If by "more curvy" you mean "smoother" (the points where it changes direction are less visible to the naked eye), then I believe you are correct. Increasing the frequency of shifts will distribute the changes over smaller and smaller intervals, making each change less apparent.(19 votes)

- At2:25, he starts talking about creating the second order midpoints. The way he talks about it, is computing the midpoints between each subsection. Would we not get the same result, if rather than using each subsection, we actually construct each 1/4 point, between the end points?

For example, you get the midpoint of on the segment AB by taking (A+B)/2. What he is suggesting is to then go to the next order you calculate the midpoint on each subsection so subsection A((A+B)/2) and B((A+B)/2).

But are these also not the "quarter points (1 and 3)" on segment AB.

So rather than going midpoint [2,1] = (A+(A+B)/2)/2 we could go midpoint [2,1] = (A+B)/4...

Expanding the first formula, we end up with (A+(A+B)/2)/2 = (A+A/2+B/2)/2 = (3A/2+B/2)/2 = 3A/4+B/4...

It may appear that I have either made a mistake, or am not sure of what I was understanding intuitively... (A+B)/4 = A/4+B/4...(8 votes)- You are absolutely correct. If you calculate a midpoint, then the midpoints between the end points and that first midpoint the you get a "quarter point" and a "three quarter point", which can be calculated directly with the equation you suggest. Later, you'll see we can use the equation
`(1 - t)A + tB`

and plugging in values of t from 0 to 1, to get a point any proportion along the line AB.

The reason he explained it this way was to show that just by using the midpoint formula repeatedly, you can get multiple, evenly spaced points along a line.(7 votes)

- so great! but i'm not understand very much.

but I think it so amazing. in your country have a lot of amazing subject. i'm from viet nam but in viet nam, we don't study a lot about computer. :( but student in Vietnam love it.

in grade 8 to 12, last year we have computer lesson. but at that subject, we study only about word, exel or paint. it so boring.

and now, in my country, Vietnam, we don't have computer lesson for grade 8 -> 12. Minister of Education and Training think we have to study more for math and English, so...of cause, they cut computer lesson of timetable in school. it so sad! How to we have a basic knowledge about computer for our future? it so important!!(5 votes)- use khan alot

play around with whatever devices you have access to

try taking online courses(3 votes)

- So anyone can do KA even if they are in 12 grade do you think that I should try it out?(5 votes)
- Yes definitely! Even adults can learn from and get a lot from this site.(2 votes)

- at1:20what does sub mean when he says b sub x and b sub y?(4 votes)
- It's short for subscript, which means writing the letter smaller and lower down. "b sub x" means you write
`b`

followed by a smaller, lower`x`

. In this case, this refers to the x-coordinate or point b.(2 votes)

- at1:10, What does he mean when he says "A sub X and Y" and "B sub X and Y"?(2 votes)
- One way of understanding "Ax" would be to read and think of it as "x coordinate of A".

So, Ax and Ay are the x and y coordinates of A.

Equivalently, A has the coordinates (Ax, Ay).(3 votes)

- I don’t understand it?(3 votes)
- At2:45he says that you can just create midpoints of midpoints and so on. A midpoint is build by computing the avarage x and y coordinate of A and B. So if we say the our three points to build the arc inside of it are A B and C, we can just create intervals of the X and Y axis of the edgepoint B to both A and C and add them to the point B for creating matching points for building the arc with. I think this would work if you use a parameter called t in this case and you use it for the point thats between B and A in the following way : PointInbetwenn(Px,Py)=(Bx+t*(Cx-Bx),By+t*(Cy-By) and for the point between B and C you use the same procedure just with using 1-t instead of just using t.

This would be the procedure I just thought of that is able to create infinite points on whatever positin you would like to have them.

I'm sorry if I wasnt able to express myself the right way, I'm no native speaker, but I think I explained my idea understandable, even thought it's not very structured(2 votes)- I think I understand what you're saying and you are correct. In fact, you are describing what is done later in the tutorial, using a parameter t, to define a weighted average anywhere along a line.

https://www.khanacademy.org/partner-content/pixar/environment-modeling-2/mathematics-of-parabolas2-ver2/v/weighted-average-two-points(2 votes)

- I don't understand what the letters are! It's too complicated!(1 vote)
- The letters are called "variables". They're used to represent a number, when that number is unknown.(4 votes)

- It said this lesson was for 5th grade+ but I'm in 8th and I'm very confused.(2 votes)

## Video transcript

- In the previous video, we saw how to use three points to define a parabolic arc. Later we'll talk about how to define the width, the color, the motion, and the variety that would be needed to create a convincing field of grass. But first, let's talk about what math we would need to create a computer program that our artists could use. A program like this one, for instance, where I can pull points around, and the parabola updates. So what kind of math is required underneath this program? First we need to talk about computing the locations of the points that we're going to connect together with the string art lines. Now, some of these points are a little bit easier to construct than others, and the easiest ones are the ones here, at the midpoints. So this point here is on the midpoint of this leg, and this point here is on the midpoint of this leg, and so on. So let's look at the mathematics of midpoints. So here I have a line segment, A B. The midpoint of that line segment is going to be somewhere in the middle. Like this yellow point here. Let's call that point Q. Now the question is, if I know the coordinates of A, suppose the coordinates of A are A sub x, and A sub y. Similarly suppose the coordinates of B are B sub x, and B sub y, then the question is, what are the coordinates of the midpoint Q? Well, if you think about it for a minute, you'll realize that Q is halfway between A and B in the x direction, and halfway between A and B in the y direction. We can compute a point like that by simple averaging. It is the x coordinate of Q is going to be the average of the x coordinates of A and B. So that's going to be Ax plus Bx over two. The y coordinate is going to be Ay plus By over 2. That's a very simple calculation. ^I can abbreviate it and make it look a little bit simpler, ^by writing Q as A plus B over 2. ^What this means is, ^take the average of x and the average of y. ^So these two expressions just mean the same thing. Now our string art construction requires more than just a pair of midpoints. But we can create as many points as we want, by computing midpoints of midpoints. For instance, here is just a pair of midpoints, and now I'm going to compute a midpoint on this subsegment, and another midpoint on this subsegment. Similarly for these two subsegments. So that gives me three points on each leg, and I can create as many as I want just by continuing this midpoint of midpoint process. Next, you'll have a chance to experiment with these ideas.