The applications of hyperbolic trig | Why do we even care about these things?
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- Опубликовано: 25 ноя 2024
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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 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
Oh my cosh this is such a good video
I hope RUclips 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.
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
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)
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: ruclips.net/video/XqKDZB9nxDI/видео.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.
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?
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.
Fantastic timing, i was just thinking about this while leaving the library.
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: ruclips.net/video/PfLz7haFvkk/видео.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.
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.
Simply amazing and top quality. I loved it!
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.
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
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.
I literally just learned about hyperbolic trig 2 days ago and wondered why it would be useful lmao perfect timing
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.
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.
Omg as an engineering student, I would say this video is absolutely amazing!!
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!!!
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
Hey Zack, don't forget about the Meniscus, when a fluid is added to a graduated cylinder. great video.
oh cool thats cosh?
These videos bring interest naturally! Thank you for showing the applications.
The curve of a tooth of gear also falls in this category. Very cool stuff.
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.
Perfect timing. We started talking about this in class today and I was SO LOST
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).
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
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’ve stumbled across your videos again and want to say these are clear, concise, and fun. Keep up the great work.
Also central forces proportional to the inverse of the distance square are awesome, quadrics are some of the most beautiful trajectories.
I really hate to see respected STEM channels resort to hyperbole. You're better than that Zach.
@Hisham Malik sinh and cosh
You missed the opportunity of referencing relativity and Minkowski geometry.
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 ;)
just reached university, I was having nightmares with hyperbolic, glad you're alive
You have opened a new perspective to shapes and math in general for me; Blessed are the meek. 🔥
This is absolutely fascinating. After Watching this video I have imediately wanted to solve some DEs where hyperbolic trig functions :)))
Also the hyperbolic trig functions can be expressed using complex numbers and there regular counterparts
Tanh(x)=tan(xi)/i
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!
My favourite: surface gravity waves whose relation of dispersion involves a tanh! Thanks for the nice vid :-)
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.
Beautiful explanation! Well done!
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.
I always wondered about the hyperbolic trig functions on my calculator. Now I know more about them.
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.
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.
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?
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.
This channel just show me how much I love maths
Fun fact: the gateway arch in St Louis was designed using a hyperbolic trig function, and its max height (in m) is exactly equal to the maximum of the hyperbola.
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
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.
7:11 this is art
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?
Thanks for the video! Hyperbolic trig, somehow we missed that in calc.
My reaction after seeing the thumbnail: Oh, there's all the torturous devices I don't ever wanna find lying on the living room floor.
Me, a theatre major, one minute into the video: 👁👄👁
You’re channel is absolute
awesomeness
Interestingly, I just learned about hyperbolic sin and cos a few days ago in my differential equations class!
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..
Your videos are very informative man! Thanks!
Additional to the important applications in Special Relativity which others already mentioned, you also forgot a very imporant application in General Relativity / cosmology: The function sinh (to the power of 2/3) describes the expansion of the universe! See e.g. the article "Lambda-CDM model" at Wikipedia.
What the article doesn't mention (but it's easy to calculate that): The function coth describes the time dependence of the Hubble parameter.
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.
Zach, your earth is spinning backwards.
Nice nuance to "opposite day".
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 ...
Excellent work. Keep up.
Nice. The Sierpinski Triangle on the wall.
Universities all over the world need maths lecturers like to teach the maths with such applications then everyone would love maths.
The inverse hyperbolic sine can be used to find arc length on a parabola, arithmetic spiral, or exponential curve.
10:49 if this is a almost to true size comparison of the world I had no idea America was an island in comparison to Europe let alone any other continents. Wild.
I'm starting to think that topologists are crazy
I mean, you're right..
I wish your videos existed when I was learning math in school.
This video is too short! Fascinating stuff
exactly - why knew the applications? Zach did!
10:10 Angry flat-earthers noise
I literally got jumpscared when I saw your face and realised you were the guy who makes comedy skits
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).
Finally, my favourite functions. Nice to see then get some representation
Very good video. Thanks!
This video is amazing, thank you Zack
The goat is back
5:55 "Minimal surfaces are not that easy to find"
Kufufu I found a minimal surface when I blew my nose on some tissue and slowly opened it to find one. I am the greatest
just kidding sorta. I finally learnt a use for hyperbolics thanks Zach
Awesome video! Thank you!
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 circumference of a Buckeyball is 3.14159 Nanometers./The vertical
distance around of a Carbon Nanotube is 12 Nanometers.
i can't belive you metioned complex form for cosh and talked about the relation between cos and cosh but didnt mention cos conplex formula or cos(x)=cosh(ix)
With latlong coordinates of everything, computers should nowadays be able to adopt a 'best' projection adaptively and on demand, rotating the earth's coordinates if need be, and choosing a projection that fits the needs (conformal, equal area, etc )
Very much informative...very good for visualization...👍👍
Excellent vid!
Hey Zack make a video with flammable maths
thank you very much. i needed video like this to start studying this sh...
❤️ Doubts solved...
Nicest explanation sir🔥
Excellent video.
Can you make a video about Nyquist theorem for signal analysis
Yay, real math content. And I *_have_* been curious about this.
Yessss - my brain is so _full_
9:57 "greenland looks like it's the size of all of africa, when it's only really the size of greenland"
The pi^2 = g shirt, I want that
You should mentioned how there is a trigonometry for each of the conic sections