Parabola for a solar powered death ray mirror?

February 2, 2006 1:02 PM

Bendable six foot mirror into parabolic sunlight reflector.

I have obtained a mirror! Yay for me.
It is made of a fairly bendable plastic and has excellant reflective quality. It measures 76-3/4" by 4-5/8". I would like to create a frame that it will hold it in a parabola. Think along the lines of a hot dog solar cooker(although mine is all one mirror instead of many smaller ones lined up side-by-side). Or this sort of device.
How do I come up with the exact parabola, knowing the dimensions?
In other words, given a particular linear dimension, is there some mathematical way of deriving the parabola that length would create?

posted by UnclePlayground to science & nature (14 comments total)

The easiest way is to lay it out mechanically using a T-square and a string.

As far as the length, I'm a little confused. Parabolas go on forever.
posted by smackfu at 1:11 PM on February 2, 2006

This site looks potentially helpful; if not, Google "construct parabolic reflector" for a bunch of other sites.
posted by TedW at 1:11 PM on February 2, 2006

As far as the length, I'm a little confused.

I have one meter of material. How big a parabola can I make with this? How tall and how wide should it be?
posted by ROU_Xenophobe at 1:28 PM on February 2, 2006

Yeah, what Rou said.
I like the t-square/string method. Pretty straight-forward and I can wrap my head around DIY aspect. But it requires that you know the distance from the bottom of your parabola to the top beforehand. I don't have that information. Only the total length of parabolic line from end to end. I'd like to make the largest parabola with the focal point furthest from the bottom as is possible with the materials that I've got.
posted by UnclePlayground at 1:40 PM on February 2, 2006

The arc length along a parabola is the integral of sqrt[1+f'(x)²] from a to b. Let a =-b/2 and b = b/2(to get the nice, symmetric parabola you want, rather than just half of one), let the paraboloid be simple in that it is the same parabola defines it in both dimensions x and y, and (I assume, probably in an unwarranted fashion) that the very simple parabola f(x)=x² will do. That's

length = ∫_b/2^-b/2 √(1+4x²), which is 1/2 * [b sqrt(1+b²) + arcsinh(b)], solve that for b, plugging in either your short or long dimension (I'm not sure which, really, without giving it some more thought).

There's your formula.

But Wait. This makes no sense. You can make any parabola have any arbitrary arc length by just solving this integral and getting a domain from -b/2 to to b/2 over which the parabola is defined, but I don't think it helps you at all. You need to knowwhich parabola to use, as there are infinitely many (maybe radio theory will help you out here).

For instance, letting length equal your 76.75, a parabola of the form x² from -6.11965 to 6.11965 (inches) has an arc length of 76.7495. But so what? How did that help you?

Well, that does help you. It tells you that a parabola of the form f(x)=x² is probably to damn steep.

So you need to experiment with different parabolas, of the form f(x) = k x² for differnt values of k, and using the fact that arc length is the integral from -b to b of sqrt[1+f'(x)²].

(typing math equations on MeFi sucks).
posted by teece at 1:43 PM on February 2, 2006

Get an old satellite dish.

Glue bendy mirror to dish.

Focal point is where the rf receiver module would sit.

Profit! (or severe burns if you're not careful :))
posted by 5MeoCMP at 2:06 PM on February 2, 2006

It doesn't really matter. Take a look here for deep-focus vs. shallow focus. As long as your curve is smooth and even, changing the curve will just change where the focal point is (and therefore how the rest of your apparatus will be set up), but a variety of curves will "work", that is, a variety of curves will have focal points within a few inches to a few feet of the mirror. Does that make sense?

You may want the shallow curve type, like a satellite dish, because my guess is that the plastic reflects better at nearly perpendicular angles of incidence than it does at steep angles. That is, I think a shallow curve will reflect more light toward the focal point with your particular material.
posted by jellicle at 2:07 PM on February 2, 2006

Wow, teece. You really seem to know what you're talking about. Too bad I don't. ;)

I guess I just wanna know what shape (parabola) to bend this wicked-cool mirror into and know where (focus) to place stuff (marshmallow, G.I. Joe, etc.) that'll cook (catch fire) when the sun is out.

(Don't you adults worry none, I'm a responsible 37 year old kid looking to relive my younger years. Maybe I'll share the experience and a few pictures later on if it works. Vicarious childishness through MeFi, ya know?)

/not into slashes, more into (parabolas)
posted by UnclePlayground at 2:08 PM on February 2, 2006

I'm thinking the T-square method suggested by smackfu, and some trial and error, would be best.

I'm guessing the mirror is somewhat flexible, but not really floppy. This will be a limiting factor, since you'll probably need a fairly gentle curve. Use a standard piece of 4x8 plywood, and start with a focal point in the dead center of the board. Draw the curve, and see if you can bend your mirror into that shape. If not, move the focal point farther away from the bottom of the curve, and try again.
posted by MrMoonPie at 2:24 PM on February 2, 2006

UnclePlayground, I don't know that the math will help you unless you're really comfortable with it.

But say you make the plastic into a parabola of the form f(x) = 0.1 x². That parabola will be 35.87 inches across, and the top of the parabola will be 32.17 inches above the vertex, and the focus is 2.5 inches from the vertex.

A parabola of the form f(x)=0.5 x² will be 17.1356 inches across, have 36.7036 inches from vertex to top, and have a focus 0.5 inches from the vertex.

A parabola of the form f(x)= x² will be 12.2393 inches across, have 37.4503 inches from vertex to top, and have a focus 0.25 inches from the vertex.

All three of these fit the constraint that you have 76.75 inches of mirror on the long end.

I can compute that for any parabola you want, but can you compute that? If not, the math is no good for you, and you should just get that plastic and experiment. (Although feel free to email me off list if want).

It looks like you want a shallow parabola (k < 1), and the focus will be pretty near to the vertex. You can see why satellite dishes are very shallow parabolas -- if not, the focus is too close to the dish to fit the necessary electronics in, I bet.
posted by teece at 3:20 PM on February 2, 2006

I just wanted to approach this from the other side: why not decide on a generically shallow parabola (for the reasons suggested by teece) and then figure out the focal point that works best for your purposes afterward?

I figure any curved mirror is going to be "wasting" a lot of reflected light and scattering it all over the place, but with a certain amount of the light focused toward one point. So you don't have to be too precise in your construction, because you know that whatever parabola you create will have a focal point, and you just need to find it.

You might want to check out a similar project at cockeyed.com. Rob at that site is very clever and helpful, and replies to e-mails from strangers, in my experience.
posted by chudmonkey at 6:32 PM on February 2, 2006

I've been playing around a little with this freeware parabola calculator (Windows only). You might find it helpful. You can't directly enter the focal length and length of arc, which are what you're really interested in, but you can arrive at the desired values by trial and error. You enter diameter (width of desired segment) and depth (which is the distance from the end of the segment to the vertex. Focal length is then computed and displayed and there's a "View Linear Distance of Parabola Segments" menu item which gives length of arc. You can't save the plot directly but you could screen-grab it and scale it up in an image editor.
posted by TimeFactor at 8:55 PM on February 2, 2006

You could just let it droop into a curve, this make a catenary, not a parabola, but it should still make a good cooker. In fact, for a cooker you want the focus to be the width of whatever you're cooking, not a point. I remember unloading a 4'x8' sheet of 16 ga. stainless steel from the back of my truck at work; it drooped into a nice curve and just about blinded me.
posted by 445supermag at 9:15 PM on February 2, 2006

Here is a picture of one I saw in Tibet. They are all over in the countryside.
posted by geekyguy at 4:24 AM on February 3, 2006