Why shadows (almost) always trace out hyperbolas (but it depends on where you live).
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- เผยแพร่เมื่อ 23 พ.ค. 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).
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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 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 th-cam.com/video/7s5bpeHNwIw/w-d-xo.htmlsi=tk1mQQETslxVocF4
Zach deserves a prize for how smoothly he was able to throw a burn into the first 20 seconds.
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!
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
That flashlight demonstration absolutely FLOORED me. 😮🔦
Where were you when I was in high school?! 😭
Really missed this type of videos. Brilliant!
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❤❤❤
"Here we got the earth, chilling" made me immediately want to fire ze missiles
Wow!!! Thanks for this illustration with the flash light.
YAY! A full-length video! I refuse to watch #shorts.
I love when you upload to this channel
Dude! Your videos hit on a different level! 🍻
Great video!
Wow, that was fantastic. Thanks!
that rare setting when the sun traces a parabola sounds like the kind of date and place considered sacred by the ancients
Wow, hyperbolas just got a whole lot more fascinating! Thanks for sharing this hidden gem of knowledge!
well done!👍
wow, this was pretty cool to know. the visuals were really helpful
I love these math videos, more pls zach
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.
Finally new video🎉
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.
zach u make such cool vids 🔥
Always nice to see he’s still hilarious on the main, educational channel
Finally a video!
I have been binge watching all your vedios since last few days. Really enjoyed your content ❤. Thanks for reawakening my interest in maths.
Very cool! Bonus points for using Elastigirl as a prop.
The sass at the beginning is off the charts and I love it
This video was great! I’m sending it to my mom so she can use it in her class next year.
Thanks! I never thought about how the flashlight is a cone before!
Superb❤
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!
I cant imagine how narrow-minded I'd be without educational videos like that one. If not for TH-cam I still wouldn't know where Math can be applied. Thanks for educating us!:)
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
Sweet vid 😎
That flashlight thing was really cool...
Owlman is about to get a spike in his viewership.
Well deserved.
The same owlman from DC?
Wow this was a very good video. Sun dials are definitely cool.
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.
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.
7:29 which explained away my confusion.
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
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
1:30 Very cool
God in the first 20 seconds you already roast us
Zach is the type of guy who makes a video about conic sections.
"The surface of the earth is a flat plane" lol
This video ended up teaching me more about earths rotation than the conic sections
Just learned it few weeks ago at my Analytic Geometry classes, thats cool as fuck, math is really cool sometimes, but its also hard
It is so hard to take you seriously after all those sketches 😂
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.
Inside the arctic circle you can have an elliptical sundial... At least one actually exists from what i know
Woooo!!! We got whiteboard Zach!
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.
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.
Well, it's a pretty flash light, but it's really an electric torch 🔦
i like how he sounds semi-passive-agressive for the whole video
At this point this is his second chanel
Funny when he said if you've never seen a cone cut in half in real life "then you're wrong"
3:23 **gasp**
If conics are sections of a cone, are quadrics a hypersurfice of some four-dimentional solid??
wait this isnt a shitpost skit, its actually a really nice and helpful 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.
Do you have other examples
Me in southern alabama wondering why the sun is above me at noon sometimes 💀💀
Eccentricity!
1:31 I felt so smart and so stupid at the same time
for some reason i feel like i know zach personally just from the way he talks
Now on a _flat_ earth, of course, the shadow of the stick's tip always describes a circle. (Sorry flerfs, you did this yourself.)
He got me in the first second
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 TH-cam viewers in general) that revolves on that axis
The relatibility between sun shadow and conic sections is unexpected. You made me a little less dumb.
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.
hey! your globe is spinning the wrong way! D=<
Great video btw :)
Hey I've got that shirt from Flammy!
Interesting showcase of the azimuthal grid of vision. Here's some additional gravy on the movement of the sun and perspective: th-cam.com/video/ZRVF-WZWJ0E/w-d-xo.html
Hmm there ain’t no South Star for the Southern Hemisphere but you can work out south from the Southern Cross
If a tree falls in a forest and nobody's is around to hear it, its trunk shape is a hyperbola.
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.
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.
☀😎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
HEY ZACH Plzz answer my question I will be grateful to you.ANY BOOK RECOMMENDATIONS.
It helped me a lot. I am a 10th grad student . Below is question.
Please read every line before answering my question and try to answer as best as you can.
I don't want to study math without true understanding.Conceptual knowledge is not understanding. All the Books that I have read till date describe things in an abstract manner without any context. And even worse, they always make assumptions that turns the material inaccessible.For a long time, the way I have done math in general is to sometimes just accept things as they are. For example, Pythagoras Theorem states the way to find the hypotenuse of a right angle triangle. How (the proof or 'why' it works)? Doesn't matter, just do it and you find the answer. Why forumula works.This has scaled up to my current position .
While I enjoy math, and am very good at it from conventional standards (getting A/A* predicted on tests, and breezing through the books), I constantly feel that I dont really know where it comes from. Yes, the basics are there (a circles chord when bisected passes through the centre, proof by standard pattern spots in integration, etc) but I always find myself asking WHY thats true. I get the answer in the end but I dont really know what I am doing. This problem really shows itself when I deal with abstract questions (or 'fun' questions) you find in olympiads and the like. The solutions seem so simple and I cant help but appreciate them but for whatever reason I can rarely do them. The way I tried to overcome this is by covering topics I had just accepted to always work and understand their core concepts.
And this relates to my question. I cant stop myself from asking WHY that work.
Any book recommendations
I can read!!
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
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
2:10
I'm in Miami :)
But what happens to australia? Does the earth rotate around the sun there?
Difference with Zach Star Himself channel gets increasingly vague in this vid
You left out point. You can slice a conic and get a point, with as many cuts as give you a circle......
You make a difference 😉