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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.

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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.