Camera Maths, or Why The Crop Ratio Works

I still think there's something odd about APS-C cameras. 'Odd' means 'It doesn't quite do what my old full-frame film camera did with the same settings'.

Time for some maths.

We will work all these examples with a full-frame (35.8 x 23.8 mm) sensor with a 50 mm lens at f8, and an APS-C (23.6 mm x 15.6 mm) with a 35mm lens at f8. Shutter speed is on Auto.

Let's look at how much light is getting in.

The f-number is the ratio of focal length to diameter of the shutter pupil. So the diameter of the full-frame shutter pupil diameter is 50/8 = 6.25mm giving an area of 3.142*3.125^2 / 2 = 15.34 sq mm. Which is a proxy for how much light is getting in. The APS-C will have a shutter pupil diameter of 35/8 = 4.375 mm, giving an area of 3.142*(2.19)^2 / 2 = 7.53 sq mm. The APS-C has 7.53 sq mm of light falling on 365.8 sq mm of sensor, or 2.06%. For the full-frame, it's 15.34 / 853 = 1.8%.

Slightly more light per sq mm falls on the APS-C sensor. If we want the same, we will need 7.53*1.8/2.06 = 6.6's worth of light, which means a pupil radius of sqrt(6.6/3.142) = 1.45 mm, a diameter of 2.9 mm and an f-number of 35/2.9 = 12. Or I could increase the shutter speed by 14%. Shutter speeds don't do that. ISO's don't either.

Let's look at depth of focus.

According to Wikipedia, depth of focus is roughly

2u^2 Nc / f^2

for a given circle of confusion (c), focal length (f), f-number (N), and distance to subject (u). The circle of confusion is conventionally 0.05 mm. For the full-frame lens and a subject 5 metres away, the depth of field is 2*(5000)^2 * 8 * 0.05 / 50^2 = 8000 mm, which is from 4m in front of the subject to 4m behind them.

The only thing that changes in this calculation when we switch to the APS-C sensor is the focal length, from 50 to 35. It changes to 16.3 m (!), 8 metres in from to 8 metres behind. That's a whole lot of extra depth of focus.

Why does the APS-C have a shorter focal length and why that one? Everybody does this, but we should understand why.

The 35 mm sensor has an area of 35.8 * 23.8 = 852 sq mm. The Fuji APS-C has an area of 23.6 * 15.6 = 365.8 sq mm. So the APS-C sensor is 43% of the full-frame area. Or the 35 mm is 2.33 times larger than the APS-C.

The APS-C sensor is showing you a smaller part of the full-frame image for a given focal length. To get the same image with an APS-C lens, we have to have a shorter focal length (shorter focal length = wider and higher picture). How much shorter?

The answer involves some school geometry. (Graphics are not my strong point.)

The focal point of the lens is behind the sensor. (I know, I learned it at school, and I'm still nodding along. Optics is magic, not physics.) The distance behind the focal point and the sensor is the focal length. (Which has nothing to do with the length of the lens. It's the length of the path the light takes between the front lens and the sensor: lenses with huge focal lengths are obtained by using mirrors. Lots of mirrors.) But I digress.

The key bit of camera geometry is the angle in red, called the field of vision. (Wrong notation in the picture.) School geometry says the it is the angle $\theta$ such that

$\tan(\theta) = \frac{\text{sensor width}}{2* \text{focal length}}$

The magic maths is this: for a given distance $D$ away, the width of the picture that the sensor will capture is $2 D \tan(\theta)$. This is why you move forwards to get nearby things you don't want in the frame, or backwards to include more of them.

Since we want the same field of vision with the APS-C as the full frame, the angle stays the same, and we can set

$\frac{\text{sensor width}}{2*\text{focal length}}$ (full-frame) = $\frac{\text{sensor width}}{2*\text{focal length}}$ (APS-C)

or 35.3/100 = 23.6/2*(APS-C focal length), 

giving APS-C length = (23.6*100)/(2*35.3) = 33 mm. 

Which is 2mm shorter than the industry-standard equivalent of 35mm.

(Now you know why nobody explains why you use the crop ratio.)

So using the same f-stop and the industry-rule of thumb equivalent lens sizes, the APS-C gives us - for the same shutter speed - a slightly brighter, ever so slightly narrower picture with way more depth of focus, than a full-frame. The difference in brightness is not enough at sensible ISOs to affect the shutter speed, even if you have shutter speed or ISO on auto, so it will be slightly brighter.

The ISO / shutter speed is way too coarse to adjust for the small change in brightness. But that depth of field can be adjusted, by halving the f-stop you would use on a full-frame, and letting the camera make the shutter speed adjustment.