# Relationship between number and focal length

Using a zoom lens also reduces the number of times you need to change the lens The main advantages of prime or fixed focal length lenses are their size and . to focus very close to the subject and reproduce them at a life-size ratio on. Knowing what the focal length means, especially in relation to your camera, is very important when The higher the number, the more zoomed your lens will be. Understanding the Relationship and Effects of Focal Length, Depth of Field and Somewhat counter intuitively, the smaller the number, the wider the opening.

And so this gives us a sense of what the image will look like. In this case, it is larger than the actual object. What I want to do is come up with a relationship with these values. So let's see if we can label them here. And then, just do a little bit of geometry and a little bit of algebra to figure out if there is an algebraic relationship right here. So the first number, the distance of the object-- that's this distance from here to here, or we could just label it here. Since this is already drawn for us, this is the distance of the object.

This is the way we drew it. This was the parallel light ray. But before it got refracted, it traveled the distance from the object to the actual lens.

Now, the distance from the image to the lens, that's this right over here. This is how far this parallel light ray had to travel. So this is the distance from the image to the lens. And then we have the focal distance, the focal length. And that's just this distance right here. This right here is our focal length. Or, we could view it on this side as well. This right here is also our focal length. So I want to come up with some relationship.

And to do that, I'm going to draw some triangles here. So what we can do is-- and the whole strategy-- I'm going to keep looking for similar triangles, and then try to see if I can find relationship, or ratios, that relate these three things to each other.

### f-number - Wikipedia

So let me find some similar triangles. So the best thing I could think of to do is let me redraw this triangle over here.

• Relationship of the f/n to the focal length and diameter

Let me just flip it over. Let me just draw the same triangle on the right-hand side of this diagram. So if I were to draw the same triangle, it would look like this.

And let me just be clear, this is this triangle right over here. I just flipped it over.

## Object image and focal distance relationship (proof of formula)

And so if we want to make sure we're keeping track of the same sides, if this length right here is d sub 0, or d naught sometimes we could call it, or d0, whatever you want to call it, then this length up here is also going to be d0. And the reason why I want to do that is because now we can do something interesting. We can relate this triangle up here to this triangle down here. And actually, we can see that they're going to be similar.

And then we can get some ratios of sides. And then what we're going to do is try to show that this triangle over here is similar to this triangle over here, get a couple of more ratios. And then we might be able to relate all of these things.

So the first thing we have to prove to ourselves is that those triangles really are similar. So the first thing to realize, this angle right here is definitely the same thing as that angle right over there. They're sometimes called opposite angles or vertical angles.

They're on the opposite side of lines that are intersecting. So they're going to be equal. Now, the next thing-- and this comes out of the fact that both of these lines-- this line is parallel to that line right over there.

And I guess you could call it alternate interior angles, if you look at the angles game, or the parallel lines or the transversal of parallel lines from geometry. We know that this angle, since they're alternate interior angles, this angle is going to be the same value as this angle.

You could view this line right here as a transversal of two parallel lines. These are alternate interior angles, so they will be the same. Now, we can make that exact same argument for this angle and this angle. And so what we see is this triangle up here has the same three angles as this triangle down here.

So these two triangles are similar. These are both-- Is really more of a review of geometry than optics. These are similar triangles. Similar-- I don't have to write triangles.

And because they're similar, the ratios of corresponding sides are going to be the same. So d0 corresponds to this. They're both opposite this pink angle. They're both opposite that pink angle. So the ratio of d0 to d let me write this over here.

So the ratio of d0. Let me write this a little bit neater. The ratio of d0 to d1.

So this is the ratio of corresponding sides-- is going to be the same thing. And let me make some labels here. That's going to be the same thing as the ratio of this side right over here. This side right over here, I'll call that A. It's opposite this magenta angle right over here.

That's going to be the same thing as the ratio of that side to this side over here, to side B. And once again, we can keep track of it because side B is opposite the magenta angle on this bottom triangle. So that's how we know that this side, it's corresponding side in the other similar triangle is that one.

They're both opposite the magenta angles. We've been able to relate these two things to these kind of two arbitrarily lengths. But we need to somehow connect those to the focal length. And to connect them to a focal length, what we might want to do is relate A and B. A sits on the same triangle as the focal length right over here.

So let's look at this triangle right over here. Let me put in a better color. So let's look at this triangle right over here that I'm highlighting in green. This triangle in green.

And let's look at that in comparison to this triangle that I'm also highlighting. This triangle that I'm also highlighting in green. Now, the first thing I want to show you is that these are also similar triangles. This angle right over here and this angle are going to be the same. They are opposite angles of intersecting lines.

And then, we can make a similar argument-- alternate interior angles. Well, there's a couple arguments we could make. One, you can see that this is a right angle right over here.

This is a right angle. Depth of field can be described as depending on just angle of view, subject distance, and entrance pupil diameter as in von Rohr's method.

As a result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same angle of view since a smaller format requires a shorter focal length wider angle lens to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced—depth-of-field effects will require smaller f-numbers when using small-format cameras than when using larger-format cameras. The larger number of elements in modern lenses allow the designer to compensate for aberrations, allowing the lens to give better pictures at lower f-numbers.

Even if aberration is minimized by using the best lenses, diffraction creates some spreading of the rays causing defocus. Light falloff is also sensitive to f-stop. Many wide-angle lenses will show a significant light falloff vignetting at the edges for large apertures. Treating the eye as an ordinary air-filled camera and lens results in a different focal length, thus yielding an incorrect f-number.

Toxic substances and poisons like atropine can significantly reduce the range of aperture. Pharmaceutical products such as eye drops may also cause similar side-effects.

Tropicamide and phenylephrine are used in medicine as mydriatics to dilate pupils for retinal and lens examination. These medications take effect in about 30—45 minutes after instillation and last for about 8 hours. Atropine is also used in such a way but its effects can last up to 2 weeks, along with the mydriatic effect; it produces cycloplegia a condition in which the crystalline lens of the eye cannot accommodate to focus near objects.