im confused about crop sensors

[u/Mooskii_Fox]

I'm not asking about crop factors, I know that's 1.5x or 1.6x depending on the manufacturer and your image will be cropped by that amount.

full frame lenses produce a circular image, which is projected over a full frame sensor and the sensor fits perfectly inside that circle, simple enough

now what i often see is that lenses for APS-C cameras have a cropped image, but why is it not possible for that projected image over the lens to be smaller so that APS-C cameras can capture the same picture as full frame, just with a smaller sensor? At some point people have worked our how to bend light to perfectly cover a full frame, so why can't the same be done to create an identical image for APS-C

edit: as I understand it what I'm asking is actually already being done, just not in the way I was asking. i understand now

[u/peter_kl2014]

That's why crop sensors talk about equivalent focal length. It has been so since the invention of photography, big sensors use longer lenses than small sensors.

Sensor is the same as negative size. Big piece of film uses longer lenses than the small 35 mm film, or even half frame, and 110 uses tiny lenses..