parabolic vs XX reflector math/curve

I noticed that my new SL PP 4AA Lux has a "deep" reflector,
as does our Dorcy 1W that we bought last year.
I'm used to the simple parabolic reflector found on most common flashlights and dish antennas that - as I recall from basic math - focuses all incoming parallel lines onto a single focal point (or the reverse for a flashlight).

So - what's the name, math, or curve involved for these "deep" reflectors that seem to create (as my SL PP does) a very narrow spot of intense light.

Same thing, a parabola - just more of it.

Weep said:

Same thing, a parabola - just more of it.

Click to expand...

that would just make it wider - like a radio telescope reflector.
Here is a link I found regarding stage lighting - http://www.thestagecrew.com/Pages/Chapters/stagecraft_lights/lighting_parts/reflectors.html

so it seems like these LED flashlights deep reflectors might be more related to the Ellipsoidal Reflectors just from the cross section of the geometry involved.

An ellipse would be a poor choice for a flashlight reflector, because it causes rays to focus to a single point and then diverge quickly beyond that point.

The deep reflectors found in Luxeon flashlights like the Dorcy 3D or the Streamlight 4AA are parabolic, but with a short focal length and therefore a strong curvature. Mathematically, parabolas have only one "type"; like circles, the differences between them are strictly a matter of uniform scaling.

For reflector design, an additional parameter is the cutoff radius that determines how wide the reflector will be. For reflectors with a short focal length, you will get a deeper reflector than you would get with a longer focal length and the same cutoff radius.

In overly simplified terms, you are dealing with relative errors in comparing a small reflector to a large one. Both are described similarly in mathematical terms, and both have a focal point. Emphasis on 'point.' Light sources, be they bulbs or LEDs are not points. The portion of the light source that is away from the focal point is projected away from center. This is the case with large light sources and small reflectors, e.g., SureFire L2/L4 with their 5W Luxeons and small reflectors. Compare that to a smaller light source in a big reflector, say Aleph 3 with its smaller Lux III in a big reflector. There, the relative error is much smaller, meaning that the size of the light source is small compared to the size of the reflector.

Hope that makes sense -- it's difficult to describe.

Scott

asdalton said:

An ellipse would be a poor choice for a flashlight reflector, because it causes rays to focus to a single point and then diverge quickly beyond that point.

Click to expand...

I also thought the parabolic would be the only choice, but the deep tunnel-like reflector sorta looked like the ellipse cross-section - even though it seemed wrong... I'll have to find some parabolic formulas and use my son's graphing calc to see how it works.

Here's a math link I finally ran across - http://mathworld.wolfram.com/Parabola.html
It appears that the parabola is more flat as the focal point moves farther from the curve vertex (backplane) and the parabola gets more narrow as the focal point is moved closer to the vertex as might be seen in the SL PP 4AA Lux. So, a flat parabola is better for collecting more rays as with a large dish antenna, but a narrow tunnel like curve is better at enhancing a generating point source into a tight radiated beam pattern.
All this math makes my head hurt - I'll be quiet now...