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Refraction of a Parallel Beam of Light through a Hemisphere.

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A parallel beam of light is incident on the surface of a transparent hemisphere of index 2.0 at an angel of 45 degrees to the axis. Show whether or not the central ray in the beam is totally reflected at the plane surface and find the position of the image formed by refraction at the first surface and the plane surface. Indicate the paths of the subsequent paths of the three incident rays.

In regards to this question, the text only discusses paraxial rays and the formulas for the calculations. I'm unsure how to proceed in the case where the incident rays are not paraxial.

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In the solution the refraction of a parallel beam of light incident on the plane surface of the hemisphere is shown with diagram and the position o9f image is calculated.

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A parallel beam of light is incident on the surface of a transparent hemisphere of index 2.0 at an angel of 45 degrees to the axis. Show whether or not the central ray in the beam is totally reflected at the plane surface and find the position of the image formed by refraction at the first surface and the plane surface. Indicate the paths of the subsequent paths of the three incident rays.

The Snell's laws of refraction is valid for the curved as well, as very small part ...

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