I think that, in order to have this be a projection of a square, the space between the interior right angles of the space from which it was projected would have to be not just curved, but also twisted, like a Möbius strip, such that a person "walking" the square and starting from the rightmost angle leftward would start walking as if they were on your screen (their head coming out away from the screen), but then they would need to have their perspective twist so that they are now walking on the "underside" of the figure (their head now pointing into your phone). This would allow them to perceive the two "external" turns as "internal" turns, as well. Then it just needs to untwist on the way back. We just can't see the twisting, because the lines have no width.

"That's ....like.....just your perspective, man"

You could just use polar coordinates