The warm and cold rule is a bit too simplified. To really understand how it works, you’ll have to think like this:
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There are usually a main light source and an ambient light source. The ambient light source can be a seperate independent light source, or it could just be the main light source reflected off of various environment surfaces.
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The main light source and the ambient light source could either be the same color, or different colors, depending on what the ambient light source is. Examples:
-If you have a strong pale amber main light source that’s from the sun outside the window, shinning directly onto a person standing right by the window, then that’s a warm main light source. The sunlight will also bounce around the room (floor, walls, ceiling…etc) and that becomes ambient light, and will light the shadow side of the person. If the entire room is painted a neutral color like white, then the ambient light will remain warm, because it’s simply the main light reflected off of a white environment.
-If the room is painted a different color–for example blue, then when the main sunlight comes into the room and bounces around, what reflects back to the person’s shadow side will be blue (and since the main light itself has some amber in it, that amber will mix in with the blue a little). To make it a bit more complex, if there is a lamp inside the room and it’s a red colored lamp, then you’ll get red mixed into that blue (and the little bit of amber) on wherever the red lamp’s light rays will reach.
The best way to think about light/shadow color relationships is this: The shadow side of the object isn’t receiving the direct light, but it is receiving EVERY OTHER LIGHT SOURCE–be it bounced ambient light or another weaker light source, and whatever colors those are, the combined result is your shadow color.