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A not-so-quick look at the lens focal length

August 2, 2012

There is a lot of talk about guides to selecting lens sets. I decided to add my two cents into the mixture. How to select lens focal length when deciding on new system? I hope I don’t have to explain what is the focal length, so I will start with lens classes. First, there is classification by design. Most popular lenses are rectilinear lenses and fish eye lenses. Later have very wide-angle, but project spherical image into rectangular frame they apply sometimes extreme transformations. Former is a lot easier to work with and is more useful in most cases, resulting in more natural look. It is simple projection of image to frame. Any lens can be fixed focal length or zoom by design. Today I will concentrate on rectilinear fixed focal length lenses, or how to select them.

How do we understand the focal length? In short, when we consider a lens to be focused on infinity and we know the frame size, focal length determines our field of view. First of all, for humans horizontal field of view is a lot more important than vertical field of view. We have very wide horizontal FOV because of how our eyes are placed on head, and this in turned is caused by directions from where our natural enemies were coming. Anyway, to calculate horizontal FOV we need to know what is width of our film frame or sensor. We will start with so-called full frame or 35mm, which is 36mm wide. Then FOV can be calculated by

FOV = 2 \arctan \frac{\text{width}}{2\times\text{focal length}}.

For example, 50mm lens on full frame will give FOV= 2\arctan 0.36 \approx 39.6^\circ. Most people call 50mm on full frame normal, but what really is normal lens? It’s very hard question. Some people say that lens of focal length equal to frame diagonal is normal. Unfortunately, this isn’t too close to commonly used lenses and uses diagonal FOV instead of horizontal FOV to determine what lens is normal. In my opinion, because horizontal FOV is more important and lens project image equally high and wide, normal lens should be diagonal of smallest square into which full frame fits. In short, we take larger dimension from width and height of frame, and multiply it by \sqrt{2}. Full frame is 36x24mm, 36mm is wider, so normal lens would be 36\sqrt{2} \approx 51mm. This is very close to commonly used 50mm, a lot closed to frame diagonal equal approximately 43mm.

It is again common knowledge, that if focal length is multiplied by two, it shows totally different image qualities, but small differences don’t show too much. That’s why I propose the following equation to calculate lens class

C(f) = \log_2\left(\frac{f}{\max\{\text{frame width},\text{frame height}\}\sqrt{2}}\right) =\log_2\frac{f}{\max\{\text{frame width},\text{frame height}\}}-\frac12 .

Now, taking into account the class, we can introduce names for major lens classes:

  • C(f) \in (-2.5,-1.5) are ultra wide-angle (9-18mm on full frame), with C(f)=-2 (13mm on full frame) being perfect ultra wide-angle lens
  • C(f) \in (-1.5,-0.5) are wide-angle (18-36mm on full frame), with C(f)=-1 (25mm on full frame) being perfect wide-angle lens
  • C(f) \in (-0.5,0.5) are normal (36-72mm on full frame), with C(f)=0 (51mm on full frame) being perfect normal lens
  • C(f) \in (0.5,1.5) are short telephoto (72-144mm on full frame), with C(f)=1 (102mm on full frame) being perfect short telephoto lens
  • C(f) \in (1.5,2.5) are long telephoto (144-288mm on full frame), with C(f)=2 (204mm on full frame) being perfect long telephoto lens
  • C(f) \in (2.5,3.5) are ultra long telephoto (288-576mm on full frame), with C(f)=3 (407mm on full frame) being perfect ultra long telephoto lens

Values of C(f) outside of (-2.5,3.5) are very hard to get because of physics. Usually wider lenses are fish eye lenses and lenses with longer focal length must be made using reflex technology. Notice, how most popular choices of lenses (24mm, 50mm, 105mm, 200mm) are close to “perfect” lenses, so cases where class function is integer.

You can also introduce more fine-grained classes, so called minor lens classes:

  • C(f) \in (-1,-1/3) is useful for photographing architecture or street photography, with C(f)=-2/3 being perfect architecture lens (32mm equivalent)
  • C(f) \in (-1/3,1/3) is generic lens
  • C(f) \in (1/3,1) is best for portraiture, with C(f)=2/3 being perfect portrait lens (80mm equivalent)

and so on, you can easily define more classes.What class would be 23mm lens on Fuji X100? Fuji X100 is APS-C frame, so it is 23.6mm wide. Placing into equation, we see that this 23mm lens is class C(23)\approx -0.537. It means, it is almost widest possible normal lens, very useful for architecture or street photography. On full frame this is equivalent to 35mm lens, i.e. C(35)\approx -0.541 on full frame.

If you want to get only three lenses that will do for you all, you should decide what you will shoot most. If it will be landscapes, set C0=-1/3. If it will be portraits, set C0=1/3. If you cannot decide yet, leave C0=0. Then, get three lenses – ones with C(f) = -1,0,1. But how do you find what focal lens you need? Just invert equation for lens class

f(C) = 2^{C+\frac12} \max\{\text{frame width},\text{frame height}\}.

So, in our full frame case, you have 25mm, 51mm and 102mm. If you want to use medium format (say 6x6cm) to shoot portraits, use set defined by C(f) = 1/3-1,1/3,1/3+1, so it is f(1/3-1)=f(-2/3)=2^{1/6} 60 = 53mm for widest lens and 106mm and 212mm for longer lenses. Similarly if you want to use your large format (4×5 inch) for landscape, use 71mm, 143mm and 285mm. Don’t know how to use this formula? Use Google – here you will see example to calculate f(-1) for full frame.

Now of course you won’t be able to get exactly those numbers, but you should be able to get very close. You can get 24mm, 50mm and 105mm Nikon lenses for the first case, 50mm, 100mm and 180mm from Hasselblad in the medium format case or 75mm, 150mm and 300mm Fujinon large format lenses. The differences are pretty small.

You can use this as a guide. But don’t use it as the guide. Selecting focal length is only first step – your perfect producer needs to make such lens and it should be bright enough and with good quality. Don’t be afraid to pick totally different lens set.

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