Parabolic Mirrors - Numberphile

NUMBERPHILE

Videos by Brady Haran
Animation by Pete McPartlan
Thanks CJ Smith for help with error spotting.

[TomRocksMaths]

I'm going to claim we definitely planned to release this at the same time as Steve Mould's mirror video...

[phizc]

Is it the International Day of Reflection or something? Mirror videos from Numberphile, AlphaPhoenix, and Steve Mould, all within 12 minutes. 😃

[AttilaAsztalos]

One thing to note that the parabola maintains this focusing property for rays that are NOT coming in horizontally but at some angle - as long as the rays are parallel, the focus point continues to exist, but it moves off the zero axis in the opposite direction. Which is why those "satellite dishes" that have their receiver very obviously not on their centerline still work just fine - they're just not looking where they are apparently pointing... :)

[sameadslighting]

Most modern spotlights (especially in theatre) use ellipsoidal reflectors. Parabolic reflectors are still used in fixtures called PAR Cans, but ellipsoidal reflectors allow fixtures to become more light projectors, not just a source of light. Awesome video!

[banderi]

Minor correction at 8:40:
The vector from point A to point B is, counter-intuitively, point B minus point A, not the reverse!
You can think of it in terms of "what displacement does A need to have to reach B" and you can keep in mind even in 1 dimensions, e.g.:
To go from 3 to 2 ---> 2 - 3 = -1 ---> decrease by one!

[arcuscotangens]

I remember reading about this in my maths text book. I spent that day proving to myself that it is actually true.
If memory serves, I used the derivative of the parabola to calculate the reflection of an incoming beam of light at any point, and found the intersection of all the reflected rays.
Not a terribly elegant solution, but that's what I came up with.

[unspeakablevorn]

Correction at 14:20: the focus of the parabola is actually much further in, about 1/4 of the way from the vertex to the shown point. this is why the angles are coming out unequal. Really, the focus is on the line that connects the two points where the mirror makes a 1/8 circle angle with the axis of symmetry.

[wheresmyoldaccount]

The "Whispering Wall" dam in South Australia is an amazing example of the parabolic effect, in this case using sound, that I have experienced first hand. You can have a conversation over 100m as if the other person is standing right next to you - as in really next to you like maybe 1 or 2 metres away.

[agmessier]

When Brady asked about a distant star, he and Tom remarked that the light rays are *almost* parallel. I always see it described this way, but I think it's a missed opportunity. The ideal shape for focusing a point (rather than a truly collimated) light source is an ellipse. Instead of saying the light rays are nearly parallel, you could say that the paraboloid is nearly elliptical, as the eccentricity approaches 1.

[jonathanross]

An interesting additional thing to prove, is that the path length of all the light beams to the focus is the same for all paths, so the signal retains coherence.

[sgctactics]

That is straight up the most hardcore speedrun of the equivalent of a 2 hour college lecture, done with only a few layers of paper and a sharpie with some words in between. And yet I think I grasped everything that was meant to be explained completely.
A+ sir

[jondoolio]

What I love about math is that I was able to prove the same thing but through a different method. I feel like my method was a bit more complex compared to this, required differential equations, but it proved the same thing

[tamasdhgebrq]

Conic sections and reflection (waves of light, sound, etc.) are an interesting topic.
1. The light of a point source in the center of the circle is reflected back into itself.
2. Light rays coming from one focus of the ellipse converge at the other focus.
3. Parallel rays of light going inside the parabola meet at the focus of the parabola.
4. Rays of light reaching the outer surface of the hyperbola and going towards one focus meet at the other focus.

[miroslavzikic]

For someone who ground and polished his own parabolic mirror to make a complete Dobsonian telescope, this episode was quite familiar and satisfying to watch.

[adityavardhanjain]

I read the title, immediately paused all my work and grabbed my earphones. Some things just sound that interesting!

[michaelgibbons]

Tom is such a great math communicator. We love Tom in this house

[Fogmeister]

1:13

“All possible parabolas”

There is only one true parabola!

StandUpMaths would take issue with that statement.

😂

[tuppb]

Nice video!

However, that LED flashlight (torch) is using *refractive* optics -- not reflective optics. It is not utilizing a parabolic reflector to focus its beam.

Note that the LED emitter is located in the back of the "reflector" and not forward at a focal point. Also, notice that the reflector isn't even specular.

Due to the structure and size of LED modules, there are few LED lighting fixtures that utilize parabolic reflectors.

[alphamath]

Great video!! Now I can show my students why we learn about parabolas & where it is applied in real life

[rkalle]

It's not only the bouncing that angles into the focal point ... but the distance the wavefront of the beam is travelling is equal, too.