Is there an accurate way of explaining face-warping when using wide angle lenses?
I realised I can't actually explain it without diagrams, so a succinct definition (no more than a few lines) would be greatly appreciated.
To fill the frame with your subject with a wide angle lens, you have to be really close. Perspective — the way lines map from 3D reality to a 2D surface like a photograph — is naturally strongest with objects that are close and flattens as things are far away. That's all there is to it.
This is really nicely explained with pictures at What does it really mean that telephoto lenses "flatten" scenes?, and How to avoid unflattering facial distortion when using a slightly wide angle lens? is likely also interesting.
It's all about relative distances.
Wide angle lenses don't distort, take a photo from the same distance as you'd use a 50mm lens and crop, and you'll see none of the trade mark wide angle look.
When you get close to fill the frame, features that stick out such as noses are relatively much closer to the camera than the rest of the face so they appear much larger. It's easy to visualise by taking the opposite case, using a super telephoto lens from a long way away. If a person's nose is 100m away, then their eyes 100.02m away, which is pretty much the same distance, hence they appear the same size for a very flattering effect.
Now imagine the case where the subject is 10cm away, the distances become 10cm and 12cm, which generates an appreciable difference in size, enlarging the nose.
It is because everything is sampled through a single small set of points. In order to get a truly flat image at any distance, the pixels of the image have to be sampled at the same distance. You would need a lens the same size as the scene you are capturing to capture everything one to one directly flat perspective.
Instead, we have a small, relatively speaking, lens that has light focused to it from a wide area down to the area of the sensor. The further off axis that light comes from, the more curvature it appears to have since the angle your looking at it from is effectively rounded. The image that is sampled directly in front of the sensor is closer to a direct alignment through the lens, but the pixels way off to the side have to be captured at a large downward angle since there isn't any lens element to capture.
For long telephoto shots, the area of your view that the scene fills is much closer to the size of the lens, so the light is sampled more straight through. In the case of a wide angle lens, the lens doesn't do anything to try and bend more light in, but the angle the light comes from is sharper.
A fish eye lens on the other hand does intentionally distort the light entering the lens to capture an even wider field of view, but at the cost of creating even more curvature.