The applications of hyperbolic trig | Why do we even care about these things?
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- Опубліковано 14 жов 2021
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oh sht, you're alive
Yo flammable maths i love your channel
haha...
MUHAMMAD AVDOL?!
YES I AM
dammit, now we gotta cancel the funeral arrangements
In my first job I was tasked with calculating the length of a catenary so as to figure the amount of tarmac on a road bridge. It took me two days until using multiple trig substitutions and the result had square roots everywhere. I calculated a length of 102.7 ft. I showed it with pride to the Engineer and he took a piece of string, ran it over a technical drawing, measured it and said Yep! I felt daft having gone through all that and then to make it worse a new graduate said: Why didn't you use a hyperbolic substitution? What's that I asked ... feeling even more stupid. I went home and looked them up and was stunned at how easy they made things. I quit my job and went to college :-)
Damn, you need to tell us more 😂
@@helloworldfromvn It's nice to share 🙂
when was this@@alphalunamare? Are you still in college? Did you go to college for your bachelor's or master's?
Oh my cosh this is such a good video
Fun fact: Take a parabola and "roll" it on a flat surface; the focus of the parabola traces out a catenary curve.
I’m interested but don’t quite understand what you mean by “roll”
@@trilogeee just like how you roll a ball over the ground. Imagine you have a very large physical shape that is a perfect parabola. If you start pushing it, it will keep on rolling like how a ball or an egg does. Because the concavity of a parabola doesn't change (and is positive), you can keep rolling it indefinitely.
Similarly, a square wheel rolls smoothly on a catenary. (I made a video about that, for anyone curious: ua-cam.com/video/PfLz7haFvkk/v-deo.html )
@@mostly_mental nice video! I watched it all the way through and it was very clear. If you're willing to accept constructive criticism, I'd just change the way you calculated the integral. It's not ideal to have to pause and read for yourself while watching a video.
But a great nonetheless :D
@@skylardeslypere9909 Thanks. The integral wasn't the important part, so I didn't want to interrupt the flow, but I can see how that would be distracting. I'll try to avoid that going forward.
I hope UA-cam is paying you very well. When first starting to read a textbook of a given topic (electrical engineering in my case), I always found it hard to give the first step. I mean, no motivation is usually provided, so I think “why am I learning this?” or “what can I do with this?”. Providing applications as you do in many videos is the motivation many people need. So thank you a lot for these free videos.
Another useful thing about hyperbolic trig is that cos(ix)=cosh(x) and sin(ix)=sinh(x),
so for a function sin(theta) = 2, there are no real solutions, but if we let ix=theta, sinh(x)=2, therefore x = arcsinh(2), theta=ix = arcsinh(2)i
Wrong. sin(ix) = i sinh(x).
And theta = - i arcsinh(2i)
@P. BPI: You should proofread your argument for it to be complete.
Regards
Although an i is missing in some places there, once replaced where missing, the reverse also works:
cosh(ix)=cos(x)
sinh(ix)=i sin(x)
Hyperbolic trig functions show up in statistics, too. For instance, Fisher's z transformation of the correlation coefficient is the inverse hyperbolic tangent. This happens because the transformation was derived using a differential equation that is satisfied by the hyperbolic tangent function. The ubiquitous logistic function is essentially a differently scaled hyperbolic tangent. Inverse hyperbolic sine mimics the natural logarithm for medium to large values but behaves like the square root near 0 and is defined for negative values.
After binge watching your other channel, you feel completely different on this channel. I almost forgot that you used to make Science videos.
he DID?
Fantastic timing, i was just thinking about this while leaving the library.
1. cosh is the EVEN part of e^x and sinh is the ODD part of e^x. This lets us exploit symmetry as coshx + sinhx = e^x
2. Chebyshev: differential equation, polynomials of first kind, type-I and type-II filters all use hyperbolic cosine. Also, say you want the inverse cosine of 1.2 (i.e. out of the range -1 to +1), you go for cosh inverse. This is directly applied in Chebyshev applications
1. is also a good example of how a function can be split into an even function and an odd function that add to each other. [The general case is f(x)=(f(x)+f(-x))/2+(f(x)-f(-x))/2]
That’s what I was trying to say
Simply amazing and top quality. I loved it!
Around 5:00. Minimal surfaces dont (necessarily) minimize the surface area (there are some that do minimize it). In fact, the catenoid doesnt minimize the surface area of the soap between the rings, it just locally minimizes it . If you take a larger part of the catenoid and replace it by a cylinder the surface area gets smaller.
Edit: I am not sure about the soap, it might burst when large enough, but the process I described above reduces the surface area of a "mathematical" catenoid.
Edit2: I have found a video of what actually happens when you pull the rings too far apart: ua-cam.com/video/XqKDZB9nxDI/v-deo.html
Interesting!
A get that curve between the 2 rings must have a longer path than a straight line, but if you think about a ring around the catenoid, parallel to the rings, its smaller with a catenoid. So, in full 3D, the catenoid does have less surface area.
@@kindlin Thats not right. Lets say we cut the catenoid off at the heights -a and a (it should be "centered" at height 0). Then we get an expression like
A_1=C*(cosh(a)sinh(a)+a)
for the surface area. If we do the same for a cylinder from height -a to height a we get
A_2=D*a*cosh(a)
where C,D are constants. Now cosh(a)/sinh(a)-->1 for a to infinity, so A_1 behaves asymptotically like cosh^2(a) and A_2 behaves like a*cosh(a). With that we can see that A_1/A_2 goes to infity for a to infinity.
That shows that the cylinder has a smaller surface area for big a.
@@cheiron8163
You're way over complicating this. A catenoid forms between the 2 rings, easily shown and google'd, which itself is comprised of a continuously rotated hyperbolic curve. The mid point of this curve is necessarily closer to the centroid of the straight-lined ring-cylinder, and so the surface has a smaller diameter and area.
@@kindlin 1. I am not talking about the ring stuff, but about a mathematical catenoid (that goes on forever, otherwise i would not talk about limits).
2. Yes, your argument explains intuitively why the catenoid has a smaller area for small height, but it cant be applied in all cases.
3. Your argument "your proof is complicated, so it is wrong, as i found an easier answer on google" is irrational (not to be rude, but you cant just reject an argument because its "complicated").
4. If you dont trust my proof, here is a thread in math.stackexchange about the topic: math.stackexchange.com/questions/1505835/understanding-an-example-for-minimal-surface-doesnt-imply-least-area (its more complicated, but maybe helpful)
5. If you want to verify it yourself, just take the formula for rotational surfaces and use it from height -a to a like I did. You can search up the integral of cosh^2 or maybe use the definition and factor the exponentials.
Maybe you trust your own calculations more than mine :)
6. Feel free to ask questions if you have any.
Please make a full course on mathematics from zero level(elementary) to the advance level ! We all want to learn mathematics from you.
The way you teach mathematic is awesome. This course will be very helpful full for the young generation.
And there is no such well constructed course available online with a teacher like you!
Kahn academy is pretty good ngl
@@dannyCOTW like legit or just pretty good
Just grab a high school algebra book, then study geometry and trig, then move to calculus and whatever you are interested in.
Awesome video! I got all the way through differential equations in college and we never talked about hyperbolic trig at all. This was a great intro it!
Not all high schools teach it I heard
Please keep going with making your videos. They are great. I lreally like your puzzles and these random mathematical subjects in particular. So much respect.
I literally just learned about hyperbolic trig 2 days ago and wondered why it would be useful lmao perfect timing
I just finished doing my homework on quadratic air resistant, first time working with hyperbolics. Honestly, I'm starting to lose fear to them and enjoy understanding their implications. Thanks!!!
These videos bring interest naturally! Thank you for showing the applications.
Sin and cos is imaginary exponentials
sin(ix) = isinhx
@@aahaanchawla5393 I remember feeling quite proud of myself when I "discovered" this relation myself! (Hyperbolic functions were not in the school syllabus)
@@sohangchopra6478 I discovered it when I was searching for a way to find trigonometric functions of complex numbers. I watched someone do it, then forgot about it, and hen derived it myself using some algebra and euler's identity.
Good stuff! You managed to hit a topic that was pretty much neglected so far by all other math channels I follow. Was definitely worth the wait ;)
Your channel is so interesting, I'm happy your video was recommended to me. I'm a physics student and I enjoy your videos. Keep it up!
Funny how one of your skits was in the suggested videos section to the side when you pulled up "Catenoid".
Your channels are growing fast.
Quality video. I learned a bit even though most of it went over my head.
Idk if this was also the case for other engineers world wide but they didn't really teach us about this in introductory math and I feel like it's a pity. I was curious to know exactly what shape did strings described when hanging them and learning about h trig functions and their properties has been very stimulating for me. Thanks for the awesome explanation
Omg as an engineering student, I would say this video is absolutely amazing!!
Perfect timing. We started talking about this in class today and I was SO LOST
Hey Zack, don't forget about the Meniscus, when a fluid is added to a graduated cylinder. great video.
oh cool thats cosh?
Beautiful explanation! Well done!
You should have explained that for both circular and hyperbolic functions, the argument is the area between the x-axis, the curve in question, and a vector that goes from the origin to the point of the curve. Thus, these are area functions. In the case of a unit circle, the arc is equal to the area, but it is possible to define hyperbolic and elliptical functions (dependent on the parameter e, which is the eccentricity), which have the arc as their argument and not the area.
Your videos are very informative man! Thanks!
Great video and great to see someone who benefitted from college. Those tough homework problems that take hours and hours to solve really pay off. They give you the confidence to tackle more difficult problems.
This video is amazing, thank you Zack
This is absolutely fascinating. After Watching this video I have imediately wanted to solve some DEs where hyperbolic trig functions :)))
Nice video. Like many other things that could have been covered in college, my first exposure is on YT. I ran into the cosh a few times, but I don't recall that it was ever explained well. It seemed to be presented as an ad hoc black box that made a solution easier to write. A link to the Desmos page with the tractrix at 07:20 would have been nice. The link in the description is to the calculator without an example.
The curve of a tooth of gear also falls in this category. Very cool stuff.
You’re channel is absolute
awesomeness
There is an additional correspondence between trig and hyperbolic functions you didn't mention. A formula for the sine of a complex number z is sin z = (e^(iz) - e^(-iz))/(2i) and for the cosine of a complex number z we have cos z = (e^(iz) + e^(-iz))/2 where ^ represents exponentation and i is the square root of -1. Those formulas are very similar to the formulas for sinh and cosh (in fact they are identical if we remove the i from the sin and cos formulas).
Another cool link between hyperbolic trigonometry and regular trigonometry is that a cosine/sin/tan with an ix argument is just a hyperbolic cosine/sin/tan function and vice versa.
Excellent work. Keep up.
Awesome video! Thank you!
I am in Calc one, barely understand the basic trigonometry. And you bring up this new thing called hyperbolictrig. I seriously need to study trigonometry over the winter break
❤️ Doubts solved...
Nicest explanation sir🔥
A wonderful video. I've subscribed to this channel, and to your Spanish one. A quibble, though, regarding the suspension-bridge cables depicted in the thumbnail: the shape of uniformly-loaded cables (like the ones in suspension bridges) is indeed a parabola, rather than a catenary.
Cool and informative video. I learned something new. Thanks.
This really only relates to your intro but it has long bothered me that circle and the hyperbolas are oddly not directly comparable despite both being the rotations in their respective algebras. It's the square in minkowski space that double covers the circle in euclidean space when you rearrange from A²=C²-B² to the equivalent A²+B²=C². You have to do the conversion twice to relate the rotations which doesn't seem right algebraically. It has always felt to me like neither is quite the "right" way to represent things and there should be a better way. Perhaps it's something to do with our units and the way our perception works such that we square root sound and light intensity when we sample it and square it for output or transformation. Or, Einstein and others were right and all frames are equally right or wrong and there's no universal frame. Probably that, but it chafes.
Also central forces proportional to the inverse of the distance square are awesome, quadrics are some of the most beautiful trajectories.
My favourite: surface gravity waves whose relation of dispersion involves a tanh! Thanks for the nice vid :-)
I always wondered about the hyperbolic trig functions on my calculator. Now I know more about them.
Very much informative...very good for visualization...👍👍
Very good video. Thanks!
just reached university, I was having nightmares with hyperbolic, glad you're alive
Interestingly, I just learned about hyperbolic sin and cos a few days ago in my differential equations class!
I really hate to see respected STEM channels resort to hyperbole. You're better than that Zach.
@Hisham Malik sinh and cosh
Yay, real math content. And I *_have_* been curious about this.
Please do a video on different job roles like software developer, data scientist, data analyst etc., describing them ..similar to your past videos on different engineering branches ...
Great video!
Excellent vid!
Hello, please tell me what softweres you use to do animation and vizilisation of mathematical phenomena.I want to teach my student like you .it is easy them(students) to understand.
Excellent video.
Also the hyperbolic trig functions can be expressed using complex numbers and there regular counterparts
Tanh(x)=tan(xi)/i
Thanks for the video! Hyperbolic trig, somehow we missed that in calc.
amazing vídeo!
Do more say more just like this ☺️☺️☺️ your video is awesome nobody tells this much of math and science keep going you are doing a great job on planet Earth 🌎🌎🌎
The first necessary observation of soap film as a material is that is cannot sustain shear and must resort to principle stresses. Catenoids balance the tension in orthogonal directions at each point (element) such that there is no shear. These resulting hoop stresses cinch the cylinder into its characteristic shape.
Loran C comes to my mind a system to find your location at sea. signals transmitted by two seperate radio stations simultaniously you have a chart showing showing you on one line from one station and then you look up your ine fr the other station and where they cross is your location, now done by compute I am sure.., the I believe its still in use today but I may be mistaken.
also you use the same principal listen when artiliary is fires and the difference between two separate locations you can calculate where it came from . My discription may not be very accurate but its been quite a few years since I was involved..
Very fascinating indeed
So cool!
Nice ! Can you share the desmos graphs, if there are any ?
Finally, my favourite functions. Nice to see then get some representation
Nifty ! as usual.
This video is too short! Fascinating stuff
Qn: do they do the physical test first and draw the graph and then find the eqn which is never taught on how to do, or do they first mathematically guess it and then test via physical experiment? I have often wondered if I were to draw a graph, how does one work backwards to find its equation?
Another cool application of hyperbolic trig is the path of an accelerating mass in a spacetime diagram. As it approaches light speed it traces out a hyperbolic curve between the time axis and the 45 degree line representing c, just like a tracrix. In this case I guess it would represent the universe conserving energy in spacetime?
Great video
I'm having trouble thinking about the minimization of the surface area between two rings--the problem is that I end up with opposite results depending on how I think about the integration, and they both seem intuitive. In both cases I'm imagining a vertically-oriented shape, like in the video. I go about finding the side surface area (i.e., not including the top and bottom ends) in one of two ways:
METHOD 1: Start with a horizontal cross-section (say, at the bottom of the shape) and then translate that up to the top of the shape. The surface area would be the sum of each "ring", or the perimeter of each of those infinitely thin cross-sections. To minimize this area, it would make sense to minimize the radius of each of these rings, meaning that we would want the curve to be as "pinched" as possible. At the extreme, this would be an arbitrarily skinny tube connecting the top and bottom circles, and the surface area (not including the outer ends) would approach the area of the top and bottom circles.
METHOD 2: Start with a vertical line or curve connecting points on the top and bottom circles, and then rotate that curve around the vertical axis (like in the video). The surface area would be the sum of the areas of each infinitely narrow vertical area. To minimize the area in THIS case, it would make sense to have each vertical curve be as short as possible, meaning it would be a straight line, or a straight-sided cylinder.
Again, both of these methods seem intuitive to me, but they seem to lead to opposite results. Can anyone shed line on this? Thanks!
I ll try my best, I am not sure that I can explain it well though.
The answer is somewhere in the middle, both of your methods can reduce the area, but are not guaranteed to find a surface with minimal area.
If you rotate the graph of a funtion f around an axis and want to get the surface area from height a to b, you can use the formula:
Area = Integral_a^b sqrt(f(x)^2 * [1+f'(x)^2] ) dx
Now your first method reduces the factor f(x)^2 (thats the squared radius of the ring at height x). Your second method reduces the factor [1+f'(x)^2] (for a straight line f'(x)=0, so you just multiply with [1+f'(x)^2]=1).
Now to find a shape with minimal surface area you have to find a function f, such that the whole integral is minimal (if we just focus on rotational surfaces). But for finding such a function you cant just focus on one of the factors in the integral (maybe that makes some sense).
Yeee another video!!
13:55 I hope the course doesn't say that suspension bridges are catenary curves. The load from the bridge deck means they're parabolas, *not* catenaries, since the forces from the downward loads they're carrying dwarf the impact of the weights of the cables themselves.
thank you very much. i needed video like this to start studying this sh...
In orbital mechanics, there's a lot of hyperbolic trig as well. The moment an orbit's eccentricity goes above one, it goes from being an ellipse to a hyperbola, and it's all hyperbolic trig to solve those.
Powerlines hang in catenary. I make a living getting wire correct.
what benefit is gained from making the calculations of hanging cables? Less strain on the posts?
You missed the opportunity of referencing relativity and Minkowski geometry.
Can you tell me where you do these equations animation?
You are the best
8:33 how did you get this? I've done my own integration, wolfram too and they all give different answers. Yes I remembered +c
Edit:
I differentiated your integration and also used desmos and wolfram alpha and none of them end up being the -sqrt(l^2-x^2)/x we started with
5-6 days late, but Happy Birthday Zach!🥳🍷
7:11 this is art
nice sharing, keep it up
This channel just show me how much I love maths
dude im studying this right now thanksss
thanks man
Can you make a video about Nyquist theorem for signal analysis
The tractrix got me thinking: I need to see if someone has made a video on the math behind backing a trailer into a spot using a truck. Seems to me that there would be a tractrix not just from the steering wheels to fixed wheels of the vehicle but also the vehicle’s fixed wheels to the trailer’s fixed wheels.
What was the site used at 10:04 to check the size of Greenland?
The goat is back
I wish your videos existed when I was learning math in school.
Can you make a video on
-Architecture
-Biotechnology
-Engineering Physics
-Petroleum Engineering
-Cryogenics
-Cryogenic Sleep
That a lot of videos
I dont think he wants to compete with SciShow
@@readjordan2257 maybe he accidentally placed this comment here instead of at SciShow
@@danielyuan9862 but they already have at least one video on those topics. So he cant be.
If he said "a new" or "another" maybe.
@@danielyuan9862 however, if zach star talked about these things and it didnt deviate from his usual style, id definitely watch. But im not sure how he could unless it is all a hodgepodge video
Hey, have a look at the Heine-Borel theorem. I think you'd like it. It would be similar to your Bolzano-Weierstrass theorem video which I enjoyed. Thanks!
How does the name of this program ? I mean where can I create the content of this video
Thankyou
is the back wheel of the bicycle following the front one while traveling straight a tractrix?
Imagining a "hyperbolic spring" is actually rather helpful-if it were to apply force in the same direction as it was pulled, you could see the infinite, asymptotic, non-periodic motion approximating the unit hyperbola.
Thanks