Where is the focus of a parabola? Alternatively, what formula should I use to create a parabola with a given focus?
Take the equation y = ax2. Obviously the focus will be on the y-axis, but how far up? We know all the incoming rays will meet there, so let's choose a convenient one. A great choice is a ray that is turned by 90°. That is, it comes in vertically, hits the parabola, and heads towards the focus horizontally.
Since the angle of incidence equals the angle of reflection, the slope of the parabola at the point the ray hits is going to be 45°. (I spent like 20 minutes using gnuplot and the gimp trying to illustrate this before giving up. Just visualize it.) Where on a parabola is the slope 45°? Slope is also rise/run, so the the slope in the y = mx + b sense is 1. Where is the slope 1?
The equation was y = ax2. The instantaneous slope is the derivative, y' = 2ax. We want that to be 1.
2ax = 1
x = 1/2a
Substitute in to find out where on the y-axis this is.
y = ax2
y = a(1/2a)2
y = 1/4a
Now let's say I want to make a parabola with a focus that is 6 inches from the bottom of the curve.
1/4a = 6 inches
a = 1/24 inches
Therefore my equation in inches should be: y = .0416x2. (I think. Contradicting my claim about how easy this is is the fact that I actually got this wrong on paper, TWICE, before posting this.)