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Which focal-length lens is usually used for portrait photography, and why?
Assuming a similar aperture, what difference would you see using a 50mm, 85mm, or telephoto lens for a portrait?
Would the backgrounds appear differently? For example, It's not clear to my why someone would get a 50 and 85mm prime lens instead of just say a 50mm and move closer?
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It's all about foreshortening, the effect by which the depth of the scene appears compressed. Different focal lengths just permit you to be different distances from your subject and still give the appropriate framing.
Subject distance is the key value here. If you are a kilometre away from your subject, then the tip of their nose is a kilometre away, as are their ears. If you are 10cm away from your subject then the tip of their nose might be 5cm away and their ears 15cm. These distances are suddenly important as the nose is three times closer it will appear three times larger.
Thus if you use your 50mm lens and simply get closer as you suggest, then you will enlarge relatively those features that are closest to the camera as well as accentuating any affects of the subject not being parallel to the camera (e.g. if they are leaning forward their forehead will be enlarged relative to their chin).
Beyond a certain distance the difference in depth between features become small enough that you cease to notice it. For this reason photographers tend to settle on a focal length that is long enough to prevent odd foreshortening effects, but not so long that you need a walkie-talkie to communicate with your subject. For APS-C cameras this is about 85mm.
The difference, considering you can keep all else equal (DOF, framing, metering) is the "compression" of the Z axis (depth). Your background gets enlarged, and with the same DOF that makes it appear more blurry, because the larger we view a blurry part of an image, the more blurry we perceive it.
The worst part is that the face you are portraiting gets affects, too. Because each depth gets enlarged differently and the face spans 20-30 cm of depth, you get a certain distortion. Anything below 50mm will be perceived as distortion, and the upper limit is around 135mm.
I did a test with my wife with 28,35,50,85, and 135mm and she liked 50mm so-so and loved 85-135mm. This was on a aps-c.
Here I show 50mm vs 135mm on a dog:
Notice how the nose of the dog seems short in the 135mm lens. The face gets flatter with higher.
Here's a person with 20-200mm:
http://www.petapixel.com/2012/08/12/how-focal-length-affects-your-subjects-weight-in-portraits/
So in conclusion, if you can get away with it, go for 85mm for the best balance, 135mm if you need to flatten the face a bit, and 50mm if you don't have room, or you need to enhance some features.
For a portrait on a aps-c sensor you need to be 1.6m away on 50mm, 2.72m on 85mm, and 4.32m on 135mm. With F/1.8 you get 7cm DOF, which is good to keep facial features and the nose in focus, while the back of the head gets blurry. Mind you, a 50mm 1.8 costs $100, 85mm F1.8 $300, and 135mm F1.8 costs $1700!
