The Vorpal Swordsman
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The Parabola

The parabola is the fundamental curve on which the telescope mirror is based.  Mirrors based on this shape reflect parallel rays of light so that they all converge at a point called the "focus". This project began out of a curious desire to know the equation of the parabola of a given f/ ratio. It turns out that just the f/ ratio is insufficient to derive the equation of the parabola, but that's another story.

So far, we have gleaned the following:
For a paraboloa:
y = ax² + r

The equation of the tangent at (q, f(q)) is:
y = 2aqx - aq² + r
Since the equation of a line is generally:
y = mx + b, then for the tangent
m (the slope) is 2aq which is f'(x) at q or simply f'(q)
b (the y-intercept) is therefore -aq + r

The equation of the normal to the tangent at (q, f(q)) is:
y = -x/2aq + (aq² + 1/2a + r)
parameters m and b for the normal are:
m: -x/2aq (negative reciprocal of the slope of the tangent)
b: aq² + 1/2a + r

Note that r always appears alone in terms. To simplify, take r to be 0. The only consequence is that the bottom of the parabola sits on the origin. (assuming a > 0)

Next, we need to find the equations of the reflected rays.
Wish me luck!

This Excel spreadsheet contains the results of my efforts so far. I've included a few false starts, too (but you'll have to look for them).
Note: I would recommend using your browser's "save target as" feature (or equivalent) to download the spreadsheet to your local hard drive. It should contain no macros, since it had none when I posted it. Downloading it thus gives your anti-virus software a chance to scan it.