From my understanding (and research) focal length affects how a face will end up looking on a photograph.
Here's an example of focal length distortion on a full-frame sensor:
My question is: will a photo taken with a 28mm focal length on an APS-C sensor look identical to a photo taken with a 40mm (28mm equivalent on a full frame sensor) on a full frame sensor? or will the levels of distortion remain the same as with a 28mm - full-frame sensor combo?
The facial deformations you are worried about are due to perspective distortion. Perspective is determined by one thing and one thing only: subject distance.
Here's an example of focal length distortion on a full-frame sensor:
That really is an example of how changing the subject distance to frame the same subject with different focal length lenses affects the perspective. To get the subject (roughly) the same size in each image required very different shooting distances for the 20mm lens and the 200mm lens. The 20mm shot was probably taken less that a foot away from the tip of the subject's nose. The 200mm shot was probably taken from around 10 feet (if the lens in question is an "honest" 200mm at short focus distances).
In the first case (20mm) the subject's ears were nearly twice as far from the camera as the subject's nose. In the last case (200mm), the distances between the camera and the nose on the one hand and the camera and the ears on the other was probably less than 10 percent.
If you take a picture with a 50mm lens on a FF camera from a distance of 10 feet and also take a picture with a 30-35mm lens on an APS-C camera from a distance of 10 feet both pictures will have the same perspective and framing (allowing for the rounding of focal lengths - a 50mm lens may be anywhere from about 46-53mm in actual focal length and most other lenses are also usually rounded to the nearest common focal length).
The focal length of a 35mm lens is always 35mm. A 35mm FX lens and a 35mm DX lens will both provide the same field of view on a camera with DX sensor (Again, allowing for that pesky rounding - all 35mm lenses aren't exactly 35mm. Some are 37mm, others might be 33mm).
The only reason we use crop factors is to compare the angle of view (AoV) of lenses used on cameras with smaller sensors to the focal length needed to obtain the same angle of view on a full frame body. When comparing the AoV of a 35mm DX lens or 35mm FX lens used on a DX body to the focal length needed for the same FoV on an FX camera both lenses need to be multiplied by the DX crop factor.
YES, you are correct. A photo taken with a 28mm lens on an APS-C will have the same level of distortion as a 40mm lens on a Full Frame camera provided you shoot the subject from the same distance.
It is the distance from the subject that determines the facial “perspective” distortion, not the focal length.
What actually affects the way a face will look is perspective - how close you are to the subject. The difference in the look between the first image example and the last is entirely driven by how far the camera is from the subject.
The focal length and sensor size will determine the size of the head in the final image but have no effect on the "look" of the facial features.
So your subject's facial features taken with a 28mm lens on an APS-C will look identical to a picture taken with a 28mm lens on a full frame camera and identical to a picture taken with a 28mm lens taken with a medium format camera - taken from the same spot. The difference will be that the head in the shot taken with the APS-C sensor will take up more of the captured pixels than the full frame sensor and much more than the medium format.
The "look" of the face will be the same because all three shots were taken from the same distance.
Now we normally think of a head and shoulders portrait of a person having the head take up the majority of the height of the final image and we usually like to have a certain look to our portraits. The portrait photographer makes the choice of where to stand to get the look they want. But that look is driven by the distance you are standing from your subject. It is the perspective given by that subject distance that you want. Then to achieve the head and shoulders filling the frame you need to choose a different focal length lens on our three example formats.
Say for example you want to choose a look like the one labelled "50mm" in the example above. To get that look you have to choose the right subject distance and then to get the subject's face to fill the frame you need to choose a focal length to get that. If you are using an APS-C camera you will want to choose a 33mm lens. If you are using a full frame 35mm camera you will want a 50mm lens. If you are using a 645 medium format camera you will want an 80mm lens.
But if all you have is a 28mm lens you can still get the look you want by standing in the appropriate place and cropping the APS-C image a little, cropping the full frame a more and cropping the 645 image a larger amount. They will all give you the same final look given that you took the image from the same location.
Facial distortion is evident when features like nose size and ear size are perceived as incorrect. Our view of what we look like is derived from our dressing mirror. If the photographer can replicate this mind’s eye view, facial distortion becomes moot. In most cases, if the photograph just stepped back, the problem is vanished.
OK a more scientific approach. Every image has a “correct” viewing distance associated with it. If the image is viewed from such a distance, most all problems related to perspective distortion are moot. Fortunately, most pictures are immune from perspective distortion so any viewing distance is OK. The human face is a major exception.
What comprises the correct viewing distance? This distance intertwines focal length and the magnification applied to make the final display. Suppose we make 8 by 12 inch print or display same on a monitor / TV with the same dimensions. A full frame 24 x 36mm must the enlarged 8 ½ X to obtain this size. A compact digital frame is 16 X 24mm and we must enlarge 12 ¾ X to get to the same size.
From a scientific point of view, we should use a lens with a focal length of about 2.5X normal when doing portraiture. OK to round most of the values. At this point let me add that in art, there are no rules so you are free to follow your heart. For the FX that’s about 105mm focal length. For the DX that’s 75mm. These values are approximately the corner to corner measurement of the frame size time 2.5.
The correct viewing distance minimizing distortion is focal length multiplied by magnification. We observe an image 8x12 inch shot on FX with 105mm enlarged 8.5x. The viewing distance is 105 X 8.5 = 890mm = 35 inches.
Using this same formula for a DX with a 70mm making a 8x12 display, the focal length is 75mm X 12.75 = 950mm = 37 inches.
My conclusion -- the FX with a 105 will display about the same degree of facial distortion as the DX with a 75mm mounted.
“My question is: will a photo taken with a 28mm focal length on an APS-C sensor look identical to a photo taken with a 40mm (28mm equivalent on a full frame sensor) on a full frame sensor?”
Answer DX 28mm X 12.75 = 350mm = 13.7 inch viewing distance. FX 40mm X 8.5 = 340mm = 14 inch viewing distance. Conclusion – They will look about the same.
Perspective is by capture ratio, not focal length per se, so you are exactly right. If a 40mm lens has the same capture size as a 28mm lens on a larger format of capture, then the perspective will be identical. Assuming the cameras have the same x/y ratio (e.g. 35mm is 24x36mm, i.e. 1:1.5 or 2:3), a good way of determining the proportional focal length is to take the size of the diagonal. You can figure this out using the Pythagorean theorem, since it's a triangle:
x^2 + y^2 = z^2 (z is the diagonal)
24^2 + 36^2 = 1872
z^2 = 1872
z = sqr(1872) = 43.3 mm
If your crop sensor camera has a diagonal of, say, 30mm, then 30/43.3 = 0.69 and if you multiply that by a focal length on 35mm, you'll get the equivalent focal length. For example, a 100mm lens on full frame or 35mm film capture is the equivalent of a 69mm lens on this fictitious crop-sensor camera and will have the same perspective.
Now, you didn't ask about this, but depth of field is another animal. Depth of field is fixed by focal length. A 28mm lens has the same depth of field on all formats. This means, with smaller sensors or film, depth of field is greater at a given capture perspective than you get on larger formats. This means very deep depth of field is easy to get on very tiny sensors, but isolating the background is very difficult. On very large formats, the opposite occurs.
