Dear high school/college students, Here's an application of hyperbolas that you never learned

This video is sponsored by brilliant. If you go outside and plant a stick in the ground, we'll assume a flat ground locally and then trace the tip of the Shadow throughout the day. What shape will it make? Now, if you said hyperbola, cool, you can read titles, no one cares. Show more

But that's not even the full answer. Hence why this video is a video and not a short. The detailed answer is down in the description, if that's all you care about. But the main thing we need to answer this question is just the basic concept behind conic sections, just the fact that, depending on how you slice a cone with a plane, you'll either get a parabola, a hyperbola, an ellipse or a circle. Pretty much all you need to answer this question. And if you think you've never seen in real life a cone getting sliced by a plane, then you're wrong, and here's why. Okay, sorry for the change in video quality. Show more

Had to use my phone for this part, but here we have a flashlight. Now the actual light in this is like an inch down and because of the circular top, when I turn this on, it is going to create a cone of light, just the light getting cut off by that circular top. We've all seen this before. Here we have a plane. We're going to use the wall to slice that cone. Now if I hold the flashlight vertically like this, it's going to create a vertical cone and the wall is going to slice it vertically, as we just saw. When a plane slices a cone vertically you get a hyperbola, as we can see here: linear asymptotes and the curve in the middle. It's hard to hold it still, but this is a hyperbola. If you have a lamp near a wall- you've probably seen this a million times because the lamp is kind of the same idea- you have a light beneath the circular Hood. It's going to create a cone of light. You're going to get this shape. Now, unfortunately I can't angle the wool, but I can angle the flashlight, The Cone. So there will still be a relative angle between our plane and Cone. So it's all the same thing. Now, if I angle the flashlight slightly, we're still going to have a hyperbola. Okay, if the plane slices our cone at a slight angle from vertical, you still get a hyperbola. The parabola only comes up at one specific angle, so that's kind of hard to find, but there is one angle. We're going to get a parabola, no longer the linear asymptotes. And then, if I angle it anymore, what's going to happen is, essentially, the ends of the parabola are going to snap together to create an ellipse. So there we have our next conic section, and then, if I angle the flashlight straight on the Whiteboard, of course we get a circle. So there you go. You have seen a cone sliced by a plane before, and this was very important for what we're about to see now. The other thing that's crucial here is we need to understand what the Sun appears to do relative to us. Yes, we are spinning as we revolve around the Sun, but that's not how we're going to think about this. We just need to think about what the sun's apparent path is, and that path is a circle, the. The Sun appears to make circles around the Earth, but more specifically, it's appearing to make circles around the Earth's axis of rotation. Okay, so here we got the Earth chilling. Now this is the North Pole and we're spinning about that. So the axis of rotation is pointing where the North Pole is pointing. If you were on the North Pole, there's you, you're on the North Pole, the axis of rotation is pointing straight up. So if you looked straight up, then the sun will appear to do circles around where you're looking, that axis of rotation. That's it. If you were to lay on your back on the North Pole and just stared straight up, you would see the sun doing circles around that axis of rotation. It would never be directly above you, it would just be doing circles around you, assuming it was the summer. As the year went on, the circles would become wider until it went all the way around the Horizon and then it would dip below the Horizon and it would be night time for the entire winter. So we have one very specific answer: if you put a stick on the North Pole with the sun appearing to do horizontal circles around it, above and around it, then the tip of the Shadow is going to trace out a circle on the ground. Show more

That's our first answer. But we're not on the North Pole. So the axis of rotation is not going to be straight up. For most of us it's going to be, you know, some angle, something between straight up and the Horizon, and whatever it is, the sun is still going to appear to do circles around that. But because of the angle, those circles are going to be tilted, which you can see in this image here. Look at the top left part. That says 50 degrees north latitude. This is what most of us see. 50 degrees north latitude is basically the border between Canada and the US and it goes through a lot of Europe- France, Germany, BelgiEtc. Show more

But this is the idea for most people. The sun always appears to make pretty much perfect circles around us, no matter where you are, but those circles are tilted for most parts of the world, and different times of the year don't change the angle of the Tilt for a given location on Earth, as you can see from the three different circles for the varying seasons. All that changes as the year goes on is the circles appear to be higher or lower in the sky. Show more

By the way, these circles are always perpendicular to the axis of rotation, so that means that axis- the North Pole- is pointing this way. That's where you'd look to see the North Star if you lived on this latitude line. And also notice the sun only Rises exactly in the East and sets exactly in the west during the equinoxes. On the bottom 90 degrees north or the North Pole, we see circles that are horizontal because the axis of rotation is straight up. That's what we just saw on the equator. On the top right is where you'll find what most people probably imagine, those actual nice arcs. Show more

However, notice that only on the Autumn and spring equinox will the sun actually be directly over your head at noon and do a perfect Arc from east to west. So on those days, if you planted a stick outside on the equator, the shadow would Trace out a line from west to east. So a line is another answer to our question, but that's only in specific cases, one being you're on the equator during an equinox. The rest of the year the sun will never be directly above your head, even at High Noon. And if you live on the 50th parallel, notice that the sun is never directly above you. If you're in Seattle or Berlin and the sun is out, walk outside, Point straight up and you will never be pointing directly at the sun, even if it's noon on the summer solstice. Show more

You'll only ever experience the sun directly above your head if you live in this part of the world between the Tropic of Cancer and the Tropic of Capricorn. Now the vertical arcs happen on the equator, because if this is the equator right here and that's you just hanging out looking straight forward, well here's the North Pole facing that same direction and again, the sun always appears to do circles around that perpendicular to it. Show more

So that's why you're going to get those nice vertical circles if you're on the equator, and it's not just the Sun that does this. If you're on the equator looking North, just at the Horizon, you're looking in the direction of the axis of rotation, which means you're also looking at the North Star. It's going to be right there on the horizon where the sky meets the Earth. And if you were to record the Stars throughout the night, you'll find they appear to do circles around that axis of rotation, around the North Star. That's on the horizon. If you're anywhere else in the world, you'll see the same thing, but that North Star will be higher up in the sky, but you'll still see those circles around it. And that's also exactly exactly what the sun is doing, just circles around the axis rotation or around the North Star. They're just much wider circles. And, by the way, this also all applies to the southern hemisphere. But you just be looking at the south star. But now let's put this all together. Imagine this Red Stick here is planted exactly on the North Pole, as we saw. That means the sun will appear to move along a circular path above it, around that axis, and to find where the tip of the Shadow will be at any given point in time, we just draw a line from the Sun at that moment to the top of the stick and then follow through to the ground. Show more

Wherever it hits. The ground is where the tip of the Shadow will be at that time. So, as the sun does an entire rotation, the Rays traced out at each moment from the Sun to the top of the stick will Trace out- hey, look at that- a cone. And if we continue those lines through the top of the stick all the way to the ground, then we get, oh, another cone. Thus the circular shadow on the ground is simply the horizontal ground intersecting a vertical cone. That's it. But typically the sun circular path is going to appear to have some tilt to it and it will set like we see here. Show more

This is almost like the nice arcs at the equator, but there is some tilt and since the sun is very far away from the earth, we can create that same cone from tracing the Rays from the Sun to the top of the stick for the entire circle. Then, following through, we find that shadow is going to be whatever shape you get when a plane or the ground intersects a cone at a slight angle, which we know is a hyperbola like we see here now. It might seem weird to some of you that the apex of the cone is on the tip of the stick, and you're right, it wouldn't be a perfect cone Center there, but the sun is very far away, so this still works. It is an approximation, but a pretty accurate one, and if we're somewhere in the world with even more tilt still, you're likely to get a hyperbola. Then we can also get an elliptical Shadow. This happens when you're near one of the poles, but not on them, where the sun's apparent path is slightly tilted, almost horizontal, but not exactly, and it doesn't set. This can only happen in the Arctic or Antarctic circles, as within those there are times of the year where the Sun never sets and during those days you will get an ellipse, just the ground intersecting a cone, but at a nearly horizontal angle. Someone made this nice resource here where you can play around with different latitudes and Sun locations to see how the Shadows path will change. Show more

Here's 45 degrees north latitude, Northern United States, and as the sun goes along its circular path we can see the shadow tip traces out a hyperbola as the season changes. It stays a hyperbola until the Equinox, when it becomes a line, but just for that day. Show more

And then, as we move into the summer, it goes right back to being a hyperbola again, but facing the other direction. If we go to zero degrees latitude, the equator, we again see that nice vertical Arc which still casts a hyperbolic Shadow until again you reach the equinox and get a line. Then after the Equinox it goes right back to a hyperbola. In fact, no matter where you are on earth other than the poles, you'll find that on the Autumn and spring equinox the tip of the Shadow will Trace out a line from west to east. So that's the real answer in regards to the line. It will happen no matter where you are on earth except the poles, but it only happens two days a year, on the Autumn and spring equinox. And the last conic section we haven't seen is a parabola which can happen if the sun just barely sets, essentially as in, it just touches down on the horizon and then immediately starts to rise in the same location. That's where the shadow ends. Both kind of meet at Infinity before snapping to an ellipse. And this can only happen again if you are in or on the Arctic or Antarctic Circle. Show more

If you're there but not on the pole, the sun will appear to have a slightly tilted circular path and as summer goes to Winter, the apparent path will start to get lower and lower until one end just barely dips below the Horizon, giving us that parabolic shape. Then we quickly hit the Equinox, where still we get that straight line Shadow and then afterwards hyperbolic, until the sun's apparent path completely dips below the Horizon, leaving it dark for the winter until it comes back up. Show more

And finally, of course, the actual real world application of this would be sundials, which do exist, based on what we've seen here, where the path of the Shadow along one of those conic sections- usually hyperbolas- tells us not only the time of day but also the day of the year, where the more curved the hyperbola, the closer you are to the summer or winter solstice. Show more

And when you have a straight line, no matter where you are besides the poles, you know it's the Autumn or spring. Equinox boom. Sundials are cool. And then, before I end this video, If you enjoyed what you saw, I highly recommend checking out brilliant, the sponsor of this video. Brilliant is an educational platform home to thousands of lessons in math, science and engineering, with new lessons being added monthly, and a big focus with brilliant is real world applications, as they show you exactly how to apply the formulas and Concepts within their lessons to the real world, and this gives you a much deeper understanding of even the more technical topics. These applications often include interactive exercises and easy to understand visuals, which, in my opinion, are the most important thing for Effective learning when it comes to math and science Concepts, and you can try everything brilliant has to offer free for a full 30 days. Just go to brilliantorg zakstar or click the link in the description below, plus the first 200 of you to sign up will get 20 off brilliant's annual premisubscription, with that gonna end that video there. Thanks, as always, to my supporters on patreon social media links to follow me or down below, and I'll see you all in the next video. Show more