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If I take a photo with a camera facing straight down onto a flat surface (no tilt), will 1 cm of the surface be represented by the same number of pixels at both the edge of the camera and the center of the camera?

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I presumed 1 cm at the center of an image would occupy more pixels, and tried to verify it by photographing a measuring tape and measuring the pixel coordinates at each centimeter mark, but I couldn't detect any difference. I'm not sure if it's because my phone camera automatically adjusts for it, or if the difference is too small to be seen.

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Yes, if you ignore barrel/pincushion distortion.

Distortion aside, a camera projection obeys the pin hole camera model. Under this model straight lines in the scene are always rendered as straight lines in the image. Now imagine photographing some squared paper head on. There's no way the squares in the centre of the image can be larger or smaller than squares at the edges whilst also obeying this straight line rule, thus all squares must be the same size in the image.

Real lenses always have a degree of barrel/pincushion distortion so in practice elements near the edges can appear larger or smaller. However you can usually correct this distortion in camera (enable lens profiles) or during RAW conversion.

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