Why does a fish see the outside world compressed into 96 degrees?

[rishch]

There's a para in my textbook which doesn't make much sense to me:

An observer or a fish under water looks up to see a compressed view of the outside world. The 180 degree view from horizon to horizon is seen through an angle of 96 degrees (twice the critical angle). A lens, called fish eye lens used in special photographs, similarly compresses a wide wiew.

What on Earth does that mean? Where did 96 degrees and twice the critical angle come from?
Can someone explain how they got that?

 

[K^2]

Refraction. Ray hitting water from outside at nearly 90° is going to continue at critical angle under water. That means that cone of ± critical angle gives you a 180° view of world above water.

 

[Ibix]

It can be easier thinking of this one backwards - treat the fish as the source. Draw a point below a line. The line is the water surface and the point is the fish. Draw rays coming out from the fish to the surface, refract them, and see where they go. You can draw as many as you like, but the obviously interesting angles are all that you really need.

Rays coming from above the surface to the fish look exactly the same. Remember that the diagram is symmetric about the vertical.

 

[rishch]

Yes I get that but how is it 2c?

 

[K^2]

Here, $\small \alpha$ is the critical angle. So the full FOV is $\small 2\alpha$.

 

[rishch]

Thanks a lot. I tried a bit more and got it by saying how the inside angles of the triangle are 90-alpha and then getting it.