Determining Angle wrt to normal of a curved surface

[chlorine addi]

Okay, sorry another question. I know I'm being a total idiot but I'm in a brain-stick (I just made that up) and I can't seem to get out of it. Anyways, I'm doing a little thought experiment about waveguides, which act like fiber optics. So I have a tube and a light ray travels through it. It starts from the center of the entrance of the tube, which lies in the x-y plane, and the center is the origin. If the fiber optic was straight, the light ray would travel 9.52 degrees with respect to the z axis away from the origin. If the fiber optic was straight, the light ray would hit the edge of the tube at blah-blah distance in the z direction and blah-blah distance in the y direction and have hit 80.48 degrees with respect to the surface it hit.

But what if instead the fiber optic was curved? Curved in a circle of known radius? Then the light ray would hit the edge of the tube earlier, correct? And at a different angle with respect to the normal surface of the tube? How would I go about figuring out that angle?? Ugh, I was never good at geometry.

I've included a picture with the post to help make sense of my nonsensical ranting. I want the angle "a" from the red line to the blue line in the right-hand picture. NOT the angle from the red to the pink, that is just 80.48.

I hope that makes sense and thanks SOOOOO much to anyone who can help me. Geometry is my weakest math subject, haha.

P.S. sorry the picture is bad quality, I'm impatient :)

 

[chlorine addi]

i have a hunch that with the right materials involved (water inside the tube and quartz of thickness 0.15mm outside) that light can escape from a curved waveguide but not from a straight waveguide. that's why I'm asking haha. i just can't do the geometry!