Q: Do these images represent the same thing? The first one is a disk model of Euclidean geometry, and the second one is the warping of a regular grid through a glass ball. If they are the same, why? It seems like the second image is in the realm of spherical geometry. Why are the geodesics circular arcs if it's supposed to represent Euclidean geometry?
This is a disk model of Euclidean geometry (where geodesics are circular arcs).
It looks exactly like the grid on the glass ball in this image.
On
I'm not sure what ℝ²→ℝ² map your "disc model of Euclidian geometry" is using.
The "warped by glass ball" map is called refraction, and is explained by Ray Tracing math.
To be honest, I am not sure whether an ordinary refractive glass sphere would show the whole plane, but this can be determined using ray tracing.
How do you know the first one is all circular arcs?
They are not the same. The lines all meet precisely on the perimeter of the disk model of Euclidean geometry, but not so in the glass orb, to take just one of several examples.