It’s really no different than scanning in a colored piece of paper on a scanner and then printing it. The color value is read by the scanner, the computer translates that into commands for the printer to mix colors to match it.
In this case they use something called a spectrophotometer (color light scanner) to determine the color value of your paint chip. That tells the paint mixing machine how much of which pigments to add to a can of white paint to achieve that color. But unlike CMYK printing, these mixing machines can have a dozen or more colors they can mix to match what was scanned.
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