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Curves we (mostly) don't learn in high school (and applications)

Apr 09, 2020
This video was sponsored by Curiosity Stream and contains over 2,500 documentaries and non-fiction titles for curious minds. Here we have the graph of sine of sine X plus cosine y equals cosine of sine X plus cosine no I'm just kidding or maybe not but as far as I know I'm just kidding now the reason I show this is really because it's funny and the only reason I could find this curve was because it was in a book called Curse for the math curious, which is a more technical book, but for math and thesis, you can enjoy it anyway.
curves we mostly don t learn in high school and applications
Various

curves

and mathematics have

applications

. I'll get to the more advanced ones later, but we

learn

ed several of them out loud.

school

like parabolas or sinusoids or logarithms and more stuff like that, but often any randomly chosen curve doesn't really have any

applications

or at least non-obvious ones, it just looks pretty or cool or really weird or has some unique property, for example, take the Weierstrass function I hope I'm pronouncing that correctly. This is a curve that is continuous everywhere but not differentiable anywhere. If you've done calculus, you know that a curve like this with a sharp corner or one with a cusp is continuous but not differentiable at that sharp point.
curves we mostly don t learn in high school and applications

More Interesting Facts About,

curves we mostly don t learn in high school and applications...

Well, this function is continuous, there are no holes or breaks anywhere, but again it is not differentiable anywhere, so we can say that it is infinitely irregular and if you zoom in you will see that it behaves like a fractal. This is not very applicable, but it was the first function discovered. having this property continuous everywhere differentiable nowhere is even less useful, although the Einstein curve is a very simple set of parametric equations. I'm sure everyone can understand this and what you get when you plot it is Einstein. I just found this on Wolfram Alpha, but the fact that even a computer can determine this is pretty impressive and they have more of the health curve or yoga curve in warrior stance etc. so it's not very applicable but it's definitely artistic.
curves we mostly don t learn in high school and applications
However, there is one type of curve that has applications when it comes to art like Photoshop and video editing software more like that and those would be Bézier

curves

. Now, this is a complete family of curves that is found as a function of a certain number of control points, for example, with two control points we can construct a linear Bezier curve which is. It's not too exciting, but with three control points we can do a quadratic Bézier as follows: first imagine two additional points moving from one to two and then from two to three respectively in the same amount of time.
curves we mostly don t learn in high school and applications
Those spinning numbers are basically percentages of the path completed and it always matches now. I'm going to do the same thing again, but I'm going to add a third point that moves along that purple connecting line from one end to the other. Furthermore, in the same amount of time, the path that the point traces is a quadratic Bézier curve. for a cubic Bezier you need four control points with a bunch of other connection point cut lines following this and then it continues with

high

er order curves which I won't show and you just need to look at them a few times to see how they work.
To Jason Davies for coding this. By the way, something very useful to keep in mind is that the location of the control points determines what the curves will look like in their entirety, and also, in most cases, the curves in red do not touch each of them. control points like this for that quadratic Bézier now, if we have these four points, the associated cubic Bézier curve would exceed this here, it may not touch points 2 and 3, however, if we draw tangent lines right at the endpoints here , then they will always come across some. other points and this makes it a little bit easier to predict what these curves will look like because if I move one point, we still get the same behavior bringing this into computer graphics in Premiere Pro when I use the pencil tool and I just click on two points.
I get a linear Bezier curve, but instead if I click once and then click and drag a second time, I get a quadratic Bezier curve that is defined by these three points, no, the fourth one doesn't do anything yet and then I can adjust the shape. I want it, but that fourth point appears when I click and hold again because now we get a cubic Bézier curve defined by these four points, so what we have here is a quadratic attached to a completely separate cubic Bézier curve, something as well as a piecewise function. These two points are therefore for two totally different curves, but by forcing them to be on this tangent line and move together, we don't have to worry about a corner here, it will always be smooth and, believe it or not, the cubic and Quadratics are the main curves you'll see in computer graphics, connecting them over and over again allows for much more complex shapes, and since a Bézier curve is defined by the location of those control points, placing them strategically allows us to create basically what whatever we want and yes, this is used to create fonts, computer graphics and more.
It might have been a new type of curve for some of you, but the next one you're probably more familiar with is the lemniscate, although you've probably never seen it like that. The lemniscate is basically a figure eight, but here we have a specific type called a Bernoulli lemniscate, it has two focus points and mathematically it is defined as all the points where, if you take the distance of each focus and multiply those values, you get a constant without matter where you are. they are on the curve, but you can also make the shape like this: first we place a rod from the left focused zero point one then from there to zero point negative one and from there to the other focus to get this link here now imagine the focus points They are fixed, they can't move, but the other joints can, since the rods are rigid, this one can only move along this circle coming off the screen and this other one can only move along this circle now if you place a point in the center of that yellow circle. rod and allow everything to spin, that point will sweep the Bernoulli lemniscate.
This is common in robotics, for example, to look at all the connections and see which path is swept by some point and a famous example of this is the Watts link. Here is the animation. You will find on Wikipedia that the setup consists of three rods and with this setup the center point can only move in an almost straight line up and down. This was actually used in Watts' patent for the steam engine in the late 18th century and still holds up. The link applies to suspensions for different types of vehicles to ensure no horizontal movement.
Now another curve that you can create with something physical is the cycloid, since this is the path that a point on a wheel traces. I'm sure many of you know where it is. goes, but this relates to Burke Easter's chrome curve. This curve answers the question of two points given at different heights. What shapes slide without friction? It minimizes the time it takes for an object to slide from point A to point B. Naturally, many people say shorter path or a straight line, but that's not the answer here, even putting a straight drop at the beginning, which increases the length, it actually reduces the time, but the absolute minimum time comes when you take a piece of that cycloid, turn it around and make it bigger here. and use it as a slide, that shape will win every time.
Now this is derived through something called calculus of variations. In calculus, you

learn

to find points that have maximum or minimum Y values, where if you simply move AB in any direction from that point the Y value in this case decreases well Variation calculus is used when you need to find entire curves that have an optimum something that something could be the distance where this straight line is minimum or it could also be the time where the Kista chrome bar curve is minimum and if you move that curve in some way the associated time will increase which means that the original curve is done the minimum, but it is not only the time and distance that you can optimize, for example, take the catenary curve that I have talked about.
This before, but still, real quick, if you were to ask a typical

high

school

student what shape this is, we have a cable hanging from two fixed points. I think most of them would say a parabola and that's very close, but it's actually a catenary curve whose equation is this here now how is it derived through the calculus of variations and that's because this minimizes something and that something is potential energy. I forgot to state this explicitly, but since potential energy is a function of height, minimizing this essentially means that a wire wants to hang. the lowest one is fine now again, so if you take these two fixed points and draw each curve of this length, the length of the wire or at least the part between these two points and each curve with that length goes from here to here and you add up all their potential energies at each small point they would all have some value but this catenary curve is the one that has the minimum value and that is the shape that the cable takes in addition to just hanging cables the catenary curve is seen in architecture for example Due to the structural properties it offers, now the regular parabola appears when you throw an object through the air at least close to the surface of the earth, but the good thing is that even this minimizes something that can be found with the calculus of variations is not obvious at all , but it turns out that if you find the kinetic energy or 1/2 MV squared at some point and the potential energy MGH, then you subtract them and do that for each point along the curve and add or actually integrate the results that The value will be minimum for this parabolic curve.
You can do the same for each curve that may exist between points A and B and assuming that the total time is the same for all of them, if you add all the kinetic energies less potential at each point, the minimum sum would be for the parabola, which is the path that takes the object. It's a totally different way of thinking about motion rather than F being equal to MA. Now the shortest distance between two points on some surface has a more specific name and that is it. a geodesic that isn't usually very easy to find if I asked you to find the shortest path from point A to point B.
That's not too exciting because it's obviously a straight line, but that simplicity goes away when you have curved surfaces, so now I can. I don't really cover this very well, but here is a curved surface if you want to find the shortest path or geodesic from point A to point B, which requires you to move along the paper, how do we find that well? It is not very easy to do. However, in this case, it is because that geodesic path will be exactly the same and the reason for this is that geodesics are preserved under certain smooth transformations, so as I knew that this was the shortest path online Straight, of course, then I know, no matter how.
I warp this, this will still be the shortest path from point A to point B. I also talked in a previous video of my sweet oatmeal that you can wrap a piece of paper around a cylinder without having any distortion or wrinkles or anything like that , so if I'm willing to find, hey, what's the shortest path from point A to point B moving along the cylinder. It's going to be the spiral moving along the side here and that's because again I knew this was the shortest path. This is a smooth transformation that preserves the geodesic, therefore.
This is still the shortest path and, again, not easy to find in general, but this was a big part of Einstein's general theory of relativity because photons, for example, travel along geodesic curves in space. -time, in addition to what we have seen. So far, there are other famous curves, some applied and some just interesting, but I may have to save them for another time for those who want to continue learning about how mathematics is applied to the real world, although I recommend checking out the sponsor's curiosity channel today. video, a documentary that they have that extends beyond what we saw here is the secret life of chaos that explores mathematics, of course, chaos theory, this details some of the patterns that appear in nature, as well as the mathematics itself, and you would be amazed at so many applications, there are two streams of curiosity about chaos theory that are home to thousands of other documentaries and non-fiction titles, so if you want to learn about black holes in astrophysics or future methods of transport up ancient history and more, this deck will have exactly what you're looking for, it's also available on a variety of platforms worldwide including Roku, Android, Xbox one, Amazon fire, Apple TV and more, and it's only $2, 99 per month, moreover, if you go to the stream out of curiosity, forward slash, takstar or click on the link. below and use the promo code Zak Star, you will get your first month of membership completely free, so risk-free and just try it,and this will give you unlimited access to the best documentaries and non-fiction titles that I know many of you will find very interesting.
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