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

Curves we (mostly) don't learn in high school (and applications)
this video was sponsored by curiosity stream hold over 2500 documentaries and nonfiction titles for curious minds here we have the graph of sine of sine X plus cosine y equals cosine of sine X y plus cosine X and although it looks crazy it actually has some interesting

applications

no I'm just kidding or maybe I'm not but as far as I know I'm just kidding now the reason I'm showing this is really cuz it's funny and the only reason I was able to find this curve was because it
curves we mostly don t learn in high school and applications
was in a book called curse for the mathematically curious which is a more technical book but for the math and thesis you may enjoy it anyway several

curves

and mathematics do have

applications

I'll get to more advanced ones later but we

learn

ed several these in

high

school

right like parabolas or sinusoids or logarithms and more like that but often just any randomly picked curve doesn't really have

applications

or at least not obvious ones it just looks pretty or cool or really weird or
it has some unique property for example take the Weierstrass function i hope i'm pronouncing that correctly this is a curve that's continuous everywhere but differentiable nowhere if you've taken calculus you know a curve like this with a sharp corner or one with a cusp is continuous but not differentiable at that sharp point well this function is continuous there are no holes or breaks anywhere but again differentiable nowhere so we can kind of say it's infinitely jagged and if
you zoom in you see it behaves like a fractal this isn't super applicable but it was the first function discovered to have this continuous everywhere differentiable nowhere property even less useful though is the Einstein curve it's 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 could determine this is quite impressive and they have more
like the santé curve or the warrior pose yoga curve and so on so not very applicable but definitely artistic however there is a kind of curve that has

applications

when it comes to art like Photoshop and video editing software more like that and those would be Bezier

curves

now this is an entire family of

curves

that's found based on a certain number of control points for example with two control points we can construct a linear Bezier curve which isn't too exciting but with three
control points we can make a quadratic Bezier as follows first imagine two extra points moving from one to two and then two to three respectively in the same amount of time those spinning numbers are basically percentages of the completed path and they always match now I'm gonna do the same thing again but add in a third dot that moves along that purple connecting line from one end to the other also in the same amount of time the path that dot traces out is a quadratic Bezier curve then for
a cubic Bezier you need four control points with a bunch of other connecting point slash lines that follow this then continues to

high

er-order

curves

which I won't show and it does just take watching these a few times to see how they work credit to Jason Davies for coding this by the way a very useful one thing to notice is that the location of the control points determines what the

curves

will look like entirely and also in most cases the

curves

in red don't touch every single one of
those control points like this one for that quadratic Bezier now if we have these four points the associated cubic Bezier curve would beat this here it might not touch points 2 & 3 however if we draw tangent lines just at the ends here then those will always intersect some other points and this makes it a little easier to predict the look of these

curves

because if I move one point around then we still get that same behavior taking this to computer graphics in Premiere Pro when I use the pen
curves we mostly don t learn in high school and applications
tool and just click two points then 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 which is defined by these three points no that fourth one doesn't do anything yet and then I can adjust the shape however I want but that fourth point does come in when I click and hold again because now we get a cubic Bezier curve defined by these four points so what we have here is a quadratic attached to a completely separate cubic
Bezier curve kind of like a piecewise function these two points are thus for two totally different

curves

but by forcing them to both be on this tangent line and move together that we don't have to worry about a corner here it will always be smooth and believe it or not cubics and quadratics are the main

curves

you'll see in computer graphics connecting them together over and over allows for much more complex shapes and since a Bezier curve is defined by the location of those control
points placing them strategically allows us to create basically whatever we want and yes this is used to create fonts computer graphics and more that might have been a new type of curve for some of you but this next one will probably be more familiar and that's the lemniscate although you've probably never seen it like this the lemniscate is basically a figure eight but here we have a specific kind called the lemniscate of Bernoulli it has two focus points and mathematically this is
defined as all the points where if you take the distance from each focus and multiply those values together you get some constant regardless of where you are on the curve but you can also make the shape like this first we attach a rod from the left focused zero comma one then from there to zero comma negative one and from there to the other focus to get this linkage here now imagine the focus points are pinned they cannot move but the other joints can since the rods are rigid then this one can
only move along this circle that goes off screen and this other one can only move along this circle now if you put a dot in the center of that yellow rod and allow everything to rotate that dot will sweep out the lemniscate of bernoulli this is common in robotics for example to look at all the connections and see what path is swept out by some point and one famous example of this is Watts linkage here's the animation you'd find on Wikipedia the configuration consists of three rods and
with this setup the center point can only move in a nearly straight line up and down this was actually used in watts patent for the steam engine in the late 1700s and still the stay the linkage applies to suspensions for different types of vehicles to ensure there's no horizontal motion now another curve you can create with something physical is the cycloid as this is the path that one point on a wheel traces out I'm sure many of you know where this is going but this relates to the Burke
Easter chrome curve this curve answers the question of given two points at different heights what shapes slide with no friction minimizes the time it takes for an object to slide from point A to point B naturally many people say the shortest path or a straight line but that's not the answer here even putting a straight drop in the beginning which does increase length actually reduces time but the absolute minimum time comes when you take a piece of that cycloid flip it upside down and make
curves we mostly don t learn in high school and applications
it bigger here and use that as the slide that shape will win every single time now this is derived through something called the calculus of variations in calculus one you

learn

how to find points that have maximum or minimum Y values where if you just move AB in any direction from that spot the Y value in this case decreases well calculus of variations is used when you need to find entire

curves

that have an optimal something that something could be distance where this straight line is a minimum
or it could also be time where the bar Kista chrome curve is a minimum and if you wiggle that curve in any way The Associated time would guaranteed go up meaning the original curve is in fact the minimum but it's not just time and distance you can optimize for example take the catenary curve I've talked about this before but still real quick if you asked a typical

high

school

student what shape this makes we have a cable hung from two fixed points I think most of them would say a
parabola and that's very close but this actually is a catenary curve whose equation is this here now how this is derived through the calculus of variations and that's because this minimizes something and that something is potential energy I forgot to explicitly state this but since potential energy is a function of height then minimizing this essentially means a cable wants to quote hang the lowest okay now back in so if you took these two fixed points and drew every single curve of this
length the length of the cable or at least the part between these two points and you every curve with that length going from here to here and added up all their potential energies at every little point they would all have some value but this this catenary curve is the one that has the minimum value and that is the shape that the cable takes besides just hanging cables the catenary curve is seen in architecture for example because of the structural properties that it offers now we're regular
parabola does show up is when you throw an object through the air at least near the surface of the earth but what's cool is even this minimizes something that can be found with the calculus of variations it's not obvious at all but it turns out if you find the kinetic energy or 1/2 MV squared at some point and the potential energy MGH then subtract them and do that for every point along the curve and add or really integrate the results that value will be a minimum for this parabolic
curve you can do the same thing for every curve that could possibly exist between points a and B and assuming the total time is the same for all of them if you added up all the kinetic minus potential energies at every point the minimum sum would be for the parabola which is the path the object takes it's you a totally different way to think about motion rather than F equals MA now the shortest distance between two points on some surface has a more specific name actually and that's a
geodesic which isn't usually very simple 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't really cover this that well but here curved surface if you want to find the shortest path or the geodesic from point A to point B requiring that you have to move along the paper how did we find that well that isn't super easy to
do however in this case it is because that geodesic path is gonna be the exact same thing and the reason for that is geodesics are preserved under certain smooth transformations so because I knew this was a shortest path straight line of course then I know no matter how I deform this this is still gonna be the shortest path from point A to point B I also discussed in a previous video my candy oatmeal that you can wrap a piece of paper around a cylinder without having any distortion or crumpling
or anything like that so if I willin to find hey what's the shortest path from point A to point B moving along the cylinder it's gonna be the spiral that moves along the side here and that's because again I knew this was the shortest path this is a smooth transformation which preserves the geodesic therefore this is still the shortest path and again it's not easy to find these 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 now on top of what we've seen so far there are several other famous

curves

out there some applied and some just interesting but might have to save those for another time for those who want to continue

learn

ing about how math applies to the real world though I recommend checking out curiosity stream the sponsor of today's video one documentary they have that extends beyond what we saw here is the secret life of chaos which explores the
mathematics of of course chaos theory this goes into detail about some of the patterns that appear in nature as well as mathematics itself and you'd be surprised with so many

applications

there are two chaos theory curiosity stream host thousands of other documentaries and nonfiction titles as well so whether you want to

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interesting so again links are below and with that I'm going to end that video there thanks as always my supporters on patreon social media links to follow me or down below and I'll see you guys in the next video