## Friday, January 11, 2008

### Parabolae Foci

I keep having to solve this problem. It's not hard, but to "save time" I usually google it and find too much information and not enough explanation. Then I end up solving it myself. And it's so easy that each time I figure it out I'm like "there's no need to write this down--it's obvious BY INSPECTION". And then I forget it. So here it is:

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.)

1. Inches. WTF.

Of course I mean CENTIMETERS. What self-respecting Citizen-Scientist uses inches?

I want the focus to be ~15 cm. Therefore a = 1/60 and the equation is y = .016x^2.

2. I do everything in SI, so it's meters rather than centimeters.

I spent like 20 minutes using gnuplot and the gimp trying to illustrate this before giving up.

Uh, pencil and paper?

3. IKTIATB I mean illustrate it on La Intarweba.

Speaking of units, how can I make a paper template of a parabola? I'd like to generate the line on the computer and print it out, but can I specify units like that? Or am I going to have to do some kind of crazy meters = N * pixels conversion?

Alternatively I can plot the points by hand, but....ugh.