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### Course: Pixar in a Box > Unit 6

Lesson 2: Mathematics of depth of field# Algebra of focus

In this video we develop a formula which tells us how big the circles of confusion are when an object is out of focus.

## Want to join the conversation?

- i dont get any of this(10 votes)
- Does this equation work for concave lenses or just convex lenses(7 votes)
- Are we doing math or pixar in a box(6 votes)
- since when did computer science turn into math?(6 votes)
- i would like to take this time thank are sponsor of the day jgftthrhjkooiunhbvgdhfthtjynjj,kijghggfghjljivdfghjklkjhgfd(5 votes)
- In the next practice "Circle of confusion calculation" at the last question there is a error. Because in the problem says "the aperture diameter is 20 mm", so DE is equal to 10 mm in this case. But this problem takes a radius value of 20 mm...(4 votes)
- Can you be more clear?(4 votes)
- Haha...What :D I didn't really get any of this. Pixar in a Box should refilm this bit and make things clearer.(3 votes)
- I did not understand anything(3 votes)
- what is the letter math for?(3 votes)

## Video transcript

- Okay, now that you're
comfortable with this diagram, I think you are, we can develop our final formula. Remember, we need a formula that tells us what BC is in terms of things we know, like the focal length of the lens, the size of the aperture, and
the distance to our object, P. The key to this formula is to notice that there are similar triangles
lurking in the diagram. To figure this out, let's
simplify our diagram to show only the two
triangles we care about. Notice the triangle ABC is
similar to the triangle AED. That means that the length BC divided by the length DE is equal to
the length AB divided by AE. Let's call this Equation One. We know what DE is, that's
the radius of the aperture. But we don't know AB or AE. I'm also gonna add two
new points, F and G. This gives us two new right triangles, ABF and AEG, which are also similar. This means that AB divided by
AE equals FA divided by AG. But, FA is just the difference between i and i prime,
and AG is the distance i. So, we can rewrite this
as AB divided by AE equals i prime minus i, divided by i. And now, let's substitute
this back into Equation One, which gives us BC divided by DE equals i prime minus i, divided by i. Finally, we just solve for
BC, which gives us our answer. BC equals DE times i prime
minus i divided by i. Ah, ha, we have an equation
for the Circle of Confusion. (chiming) If you wanted to, you
could also substitute i and i prime from the
simple lens equation to write out the radius
of the Circle of Confusion in terms of F, O, and O prime. I'll leave that for you to work
out in this final exercise.