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Why does harsh light give darker shadows? I thought a shadow area is where the light is unable to reach, so shouldn't the shadow areas have the same darkness regardless of the strength of the light?

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Shadow does not always mean darkness, it can be partial darkness. –  dpollitt May 3 '13 at 17:21
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There are two factors here. One is the relative brightness the other is how directional the light is. Normally, light comes from multiple sources or is highly reflected. In this case, shadows are soft or non-existent because light hits where the shadow is from other directions than the one that is blocked.

Harsh light is generally very bright and very directional. The intensity of the light means that the dynamic range (the difference between the brightest and darkest part of the image) of the camera is going to require more light to be present in the shadow area to get a good exposure (since it can't overexpose the bright part). Compounding that, since the harsh light is very directional, it casts a harder shadow with less light being reflected or diffused to fill the shadow from other angles.

This is why you often see professional photographers using diffusers and reflective umbrellas with their flashes and using multiple flashes that are off camera in order to spread the light out and make sure light reaches the areas that would otherwise be dark shadows from a point light source (an intense light emitted from one point in space).

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A shadow is where less light is reaching a given area, in comparison to another area. Few spots have zero light, as there is usually a reflection off of the surroundings that is contributing light. It is likely less light, but there is light.

Harsh light gives darker shadows because of the contrast (the in comparison thing). There is a much larger contrast between light and dark areas.

It can be difficult to properly expose such situations, as you wish to avoid blowing out the brightly lit areas, but still provide detail in the poorly lit areas.

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When we say "darker shadows," we're really commenting on the contrast between the bright parts and the shadows. Under harsh light, you are correct, the shadows may be just as well lit (or even brighter) than they would be under soft light. But the bright parts are so much brighter that the shadows are darker relative to the bright parts. Instead of being 3 EV darker than the bright parts, they may be 7 EV darker.

Generally in photography, we think about exposure in relative terms... "highlights, midtones, and shadows." Photometrically, the "shadows" of one image might be brighter than the "highlights" of another image. But because we adjust the exposure depending on the light, the absolute brightness isn't usually relevant.

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