Why shadows (almost) always trace out hyperbolas (but it depends on where you live).
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- Опубліковано 23 тра 2023
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►Full answer (spoilers)
The path traced out by the tip of the shadow can be...
1) A circle (if you are on one of the poles during their summer)
2) An ellipse (if you are in the Arctic or Antarctic circle, but not on either pole, during a day where the sun doesn't set)
3) A line (this happens if you are anywhere on Earth besides the poles, during either the Spring or Autumn equinox)
4) A parabola (this happens if you are in/on the Arctic or Antarctic circle during a day where the sun BARELY sets, meaning it sets and rises in the exact same location. This happens twice a year for any point in those circles. If you are ON one of the circles, then it will happen during the Summer or Winter solstice for the Arctic and Antarctic circles respectively).
5) A hyperbola (If you aren't in one of the places listed above on those specific days, then you'll get this shape. This is what happens at most places on Earth throughout most of the year).
►Resources
Interactive tool: www.geogebra.org/m/AHEgBVUd
Video showing linear path on Equinox: • Equinox 20th March 202...
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Wow the guy from the comedy sketches knows quite a lot about maths
I was thinking the guy from the math videos sure is good at comedy! :D
jack of (clever and complicated equation to prove 2 is 2 and how it is with the explanation from that one massive book)
Which math?
@@resipsaloquitur13 All 'em. All the maths.
@@neithvoidI think it's called mathmatica. I could be very wrong
0:02 Zach: "If you go outside-"
Me: **clicks off**
Zach deserves a prize for how smoothly he was able to throw a burn into the first 20 seconds.
Zach dropped😤
😊
It’s becoming harder and harder to take these videos seriously when I’m used to hearing the same voice in NSFW comedy skits
@@andrerenault lol right?
But is he actually correct please prove me wrong ua-cam.com/video/7s5bpeHNwIw/v-deo.htmlsi=tk1mQQETslxVocF4
The more creative and lesser-known ways to apply maths and sciences are why I enjoy the two so much. There's never really an end to how we can utilize concepts from either echelon, and that only goes higher the more we learn more about our surroundings and beyond. Cheers.
That was really cool how the double cone shape appeared naturally in this problem! I'll definitely remember this.
My go-to had been the trajectory of particles passing by the sun (or really any situation with an inverse square law): if your speed is slow enough then you'll fall into an elliptical orbit, but otherwise you'll trace out a hyperbola. But it's not easy to actually connect that to conic sections, whereas here that connection is so elegant!
That flashlight demonstration absolutely FLOORED me. 😮🔦
Where were you when I was in high school?! 😭
I really like the flashlight demo. That helped solidify things!
Imagine how mind blown people were when the discoverers of these mathematics shared them with people of their time. I am equally mind blown. This is by far the best explanation of the sun's position in the sky over the year I have ever seen. My highest praises to you Zach
"Here we got the earth, chilling" made me immediately want to fire ze missiles
6:49 There's a place in Hawaii where this happens and at noon, the city (or island, idk) looks like a video game on pictures because all the shadows are exactly below the objects that have a "below" and the rest don't have one at all.
This happens because that place is located exactly at the sub-solar point. This does not happen every noon. Vsauce has a great video that covers this topic.
YAY! A full-length video! I refuse to watch #shorts.
that rare setting when the sun traces a parabola sounds like the kind of date and place considered sacred by the ancients
Really missed this type of videos. Brilliant!
Wow!!! Thanks for this illustration with the flash light.
I had to write a 5 pages paper for a project in my analytical mechanics class. The topic was the two-body problem under the Kepler potential (that is a potential that's prop. to the invers of the distance). One part of the project is to investigate what types of orbits that can be obtained given the initial conditions and the system (the two bodies) and it turned out to be conic sections. It's just so wonderful how we are surrounded by patterns.
Very cool. We studied Kepler, Copernicus and Ptolemy but I never noticed the conic sections in their works. I’m assuming you used a computer program with Kepler’s data? Also, what is your major?
@@jakeelsner2963 Because the problem was restricted to two bodies, no computers were required, the differential equations governing the problem were solvable by hand. I would however need a computer if three bodies were involved because then the system is chaotic. As for Keplers data, it was mentioned how he used it to conjecture the laws but the approach i used to prove them didn't rely on any data. It was just setting up and solving some differential equations.
As for my major, I used to be a physics major but i switched last year to pure math.
Omg, I started watching your videos in 2018 or 2019 and because of your second channel, I am physically unable to tell, if you want to explain something, or you're just telling a joke, thank you for your amazing content❤❤❤
Dude! Your videos hit on a different level! 🍻
I love when you upload to this channel
Wow, that was fantastic. Thanks!
Wow, hyperbolas just got a whole lot more fascinating! Thanks for sharing this hidden gem of knowledge!
Great video!
I love these math videos, more pls zach
well done!👍
The sass at the beginning is off the charts and I love it
wow, this was pretty cool to know. the visuals were really helpful
zach u make such cool vids 🔥
Thanks! I never thought about how the flashlight is a cone before!
Very cool! Bonus points for using Elastigirl as a prop.
I have been binge watching all your vedios since last few days. Really enjoyed your content ❤. Thanks for reawakening my interest in maths.
Woah! I just now realized that "conic sections" is referring to sectioning (i.e., slicing through) cones. When I was in high school and my teacher said we were going to begin studying conic sections, I thought she was referring to the sections of the textbook dealing with conics. That was 25 years ago. I've since earned a EE degree and have been immersed in mathematics for most of my life at this point. Yet, I'm just now putting it together after seeing your video thumbnail. I feel both dumb and enlightened. Well done!
Hi Zach, can you make a video about bode plots ? Thanks
Oh that's a good idea
I really wanted to see how conic sections change with eccentricity, but couldn't find anything that was satisfying and intuitive. Like at which point does a parabola becomes an ellipse? All of a sudden, found this video. Thank you so much for this! This is exactly what i wanted to see!
Always nice to see he’s still hilarious on the main, educational channel
Finally new video🎉
isn't the axis of rotation of rotation is tilted with respect to sun
Yes it is. But still, no matter what that tilt is, the sun is going to appear to do circles perpendicular to that axis of rotation. The tilt is why there can be days in the arctic circle where the sun never sets, or why the sun is never directly above us unless you're in between the tropic of cancer and tropic of capricorn. You can see why in this image. en.wikipedia.org/wiki/Tropic_of_Cancer#/media/File:Axial_tilt_vs_tropical_and_polar_circles.svg
@@zachstar that means the axis 9f rotation shouldn't necessary at north pile ,right
I cant imagine how narrow-minded I'd be without educational videos like that one. If not for UA-cam I still wouldn't know where Math can be applied. Thanks for educating us!:)
This video was great! I’m sending it to my mom so she can use it in her class next year.
Finally a video!
Can someone make a concept art of what kinda weapon a hyper-bola would be. Like, bolas are basically clackers on longer strings, so what would a hyper-bola be?
4D bola.
That flashlight thing was really cool...
7:00 Not just the equator - the equinoctial line is straight irrespective of latitude by virtue of the axis of rotation being tangent to the minor axis of the elipse that is our orbit, though the equatorial line is the only one that passes through the gnomon of a horizontal dial. I even made a timelapse of the line in March 2021, and I definitely live nowhere near the equator!
12:35 Aha, I should have waited until I had time to finish the video😁 Nicely presented.
Edit: 14:28 I _really_ should have waited! Your next foray into dials should be the analemmics, they will give endless scope for your evident graphical presentation skills and you can nicely illustrate the equation of time from our tilted elliptical orbit. All the best.
maybe rephrase that in less science terms
Owlman is about to get a spike in his viewership.
Well deserved.
The same owlman from DC?
Superb❤
Inside the arctic circle you can have an elliptical sundial... At least one actually exists from what i know
You forgot to include the case where you are in the arctic/antarctic circle during winter, in which case the set of shadowed points over the course of a day is empty.
It is so hard to take you seriously after all those sketches 😂
Zach is the type of guy who makes a video about conic sections.
7:29 which explained away my confusion.
At this point this is his second chanel
Well, it's a pretty flash light, but it's really an electric torch 🔦
3:23 **gasp**
So amazing dude really found it useful you got a new regular viewer.....from BHARAT that is INDIA ❤❤
Very interesting. Thanks.
Check out a paper I wrote called “Swept Conics”. I think it was at the 2010 International Optical Design Conference. But maybe the 2006.
Hammer a nail at one of the focal points of a conic and sweep it about that axis. Lots of fun stuff. I got a patent on some of this stuff for applications to optical systems, like converting a point source to a diffraction limited line or arc of light. I was surprised to see that Bang and Olufsen has a speaker (Beo 5) that uses this exact concept for an acoustic lens. They place a speaker at the focal point of a tilted ellipse and it spreads the sound out into a 180 degree arc.
Conics are fun.
Thanks for the tutorial here.
Babe wake up, Zach just dropped a new vid.
If a tree falls in a forest and nobody's is around to hear it, its trunk shape is a hyperbola.
"The surface of the earth is a flat plane" lol
Sweet vid 😎
God in the first 20 seconds you already roast us
Me in southern alabama wondering why the sun is above me at noon sometimes 💀💀
1:31 I felt so smart and so stupid at the same time
1:30 Very cool
Funny when he said if you've never seen a cone cut in half in real life "then you're wrong"
i like how he sounds semi-passive-agressive for the whole video
The reflector in the flashlight has the bulb in the Loci of a PARABLOIC Dish to have the light leave straight out. It is So a candle is a better demonstration tool.
Wow this was a very good video. Sun dials are definitely cool.
This video ended up teaching me more about earths rotation than the conic sections
for some reason i feel like i know zach personally just from the way he talks
I wonder if I could help improve the translation of this video into Norwegian. I see the auto-translate translate "plane" into "aircraft"(should be "en flate", not " et fly"), "wall" into "wool" (should be "veggen, not "ullen"), "which shape will it make" into "which "shape will it do" (should be "hvilken form vil den lage" not "hvilken form vil den gjøre", "cone" (kjegle) and "point" (punkt) goes untranslated. and are interpreted as names.
Just learned it few weeks ago at my Analytic Geometry classes, thats cool as fuck, math is really cool sometimes, but its also hard
Woooo!!! We got whiteboard Zach!
Do you have other examples
Now on a _flat_ earth, of course, the shadow of the stick's tip always describes a circle. (Sorry flerfs, you did this yourself.)
What about the lesser known conic section of two intersecting lines? From the equation x^2=y^2, with two intersecting line solutions of y=x and y=-x.
5:13 "This is the idea for most people" My brother half the world lives in East/South Asia
It's just the Western world (which to be fair has the most UA-cam viewers in general) that revolves on that axis
To all highschoolers he is only explaining conics in algebra 2 and mentioning that you might touch the other stuff if you pursue mathematics as a career which is very unlikely
also this stuff is not practical nor convenient just something to have fun exploring if you can understand it
@Paul O'Reilly no no you got the right idea
its just connecting practicalness with this mathematics and the issue is this type of mathematics doesn't do much for an individual. Sure might make the brain smarter but whats the point if there is no funding for further research? Without funding technology starts to stagnate
If conics are sections of a cone, are quadrics a hypersurfice of some four-dimentional solid??
Hmm there ain’t no South Star for the Southern Hemisphere but you can work out south from the Southern Cross
wait this isnt a shitpost skit, its actually a really nice and helpful video
Eccentricity!
The relatibility between sun shadow and conic sections is unexpected. You made me a little less dumb.
He got me in the first second
hey! your globe is spinning the wrong way! D=<
Great video btw :)
You left out point. You can slice a conic and get a point, with as many cuts as give you a circle......
I can read!!
2:10
5:05 bro excluded everyone in china and India, some how they don't count as "most people"
Hey I've got that shirt from Flammy!
Application of hyperbolas sounds off in this case, coz when was the last time you looked for the shadow path shape? it's barely applicable to the sundial mechanics and at most useful for some artists nowadays. Energy collectors definitely doesn't look for shadows but the sunny path and it's sphere sections.
can u tell me the animation software,Pretty please :)
If you are talking about for the 3D software I was using it's called Runiter.
Difference with Zach Star Himself channel gets increasingly vague in this vid
Wait, isn't the axis of rotation of earth in 66° angle with the plane of it's revolution around the sun?
But what happens to australia? Does the earth rotate around the sun there?
Nice
Zach, are you doing grad school for something with applied mathematics?
No but I want to at some point. Right now I just self study for fun.
Technically, the relationship between the Earth and the Sun changes slightly through the day. So the shape will probably be a slightly warped hyperbola, starting out as one and finishing as another that's a tiny bit different.
You need a Nobel 🎉🎉
Flat earthers gonna love this video
I'm in Miami :)
☀😎AWESOME VIDEO!!! Loved it, so clearly explained!!! NEXT should be: "Analemmas" Here's the question: "If you were to walk out every day exactly at noon, assuming no clouds, and noted the sun's position... what shape would sun appear to trace?" I won't spoil the answer, and if you don't know, before you google the answer make a guess, Circle? Ellipse? Well, maybe... but not likely in your location on Earth... It's not what most people would expect.
Okay one more WOW! And what blows people's minds, I know it did mine, okay... so after you know the shape, then you're informed: you've already seen the analemma if you've ever looked at an old maps or globe... Ok, mind blown? 🤯 and suddenly you say... "OH BLEEP! THAT'S WHAT'S IT'S FOR" yep... and you never thought twice about it, didn't realize what it was or how they were used. We forgot all about the old tech, back in the day this was useful information that people used before we had our modern technology, and THAT is why they put them on the maps... Analemmas are COOL too 😎 Cheers