What are Hyperbolas? | Ch 1, Hyperbolic Trigonometry
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- Опубліковано 3 сер 2022
- This is the first chapter in a series about hyperbolas from first principles, reimagining trigonometry using hyperbolas instead of circles. This first chapter defines hyperbolas and hyperbolic relationships and sets some foreshadowings for later chapters
This is my completed submission to the #some2 contest, put on by @3blue1brown.
It would be great to have follow up videos 😢
Yeah, I agree! There’s simply not enough videos on hyperbolic trig when it describes so much, the perfect arch, a string held between two hands, and that’s just one function, the catenoid or hyperbolic cosine variant
I've never seen the Hyperbola explained in this way! WOW! Eye opening!
At last! Can't wait for the next chapters!!!
I hope you get much more attention from the math/3blue1brown community cause it's really good.
Thanks :)
@@arithknot9740theres not much material about hyperbolic trig of this kind on yt thx for this and i hope u'll make another part
What a fantastic video! The exposition is so well paced that it serves as a first rate example of mathematical foreshadowing.
May I recommend the channel "Another Roof"?
You are off to a great start. Look forward to next video. I have never looked at hyperbolas this way. thanks
Amazing explanation! Looking forward to part 2!
Oh no! Where is chapter 2! You did such a great job on this, thank you.
It's not forgotten. I'm just too terribly busy. I'm trying to pare down some commitments so that maybe I can get back to this soon. I've been so looking forward to it
Subscribed and awaiting the sequel!!
hes too busy grading my algebra 2 homework
@@jelly.timeeeThere's some truth to that 😅
@@arithknot9740 please come back 🙏🙏
Very nice video. Make more videos like this 👍❤️
Not all heroes wear capes, thank you very much for this quality video!
What a wonderful video, looking at hyperbolas a priori is something that I think would have really enlightened a lot of high school students, and even as an adult in STEM watching this video made something click like I was in 10th grade learning it for the first time. Absolutely brilliant delivery (accidental asmr?) and I can't wait for future content!!
Something fun that I find about hyperbolic geometry is how you can consider it to have a different notion of "distance" from circular geometry. A circle is formed from x^2 + y^2 = r^2, where r is the distance from (x, y) to the origin. For a hyperbola, you could instead say x^2 - y^2 = r^2, which has a few properties not normally expected from distances. For example, the diagonal lines forming the asymptotes would all have a magnitude of 0 despite not being the origin, forming the "null cone," or in special relativity "light cone." You also now get points with a negative squared magnitude, suggesting that points can have... imaginary distances‽ I think that if tachyons (hypothetical faster than light particles) existed, they would have imaginary mass, with mass normally being the magnitude of the momentum vector.
This seems like the flipside of the circular case where imaginary lengths (i*sin theta) can lead to negative areas. Coming up in a later video, there's a nice way to show the angle addition formulas for both circular and hyperbolic that shows the common structure, if you allow for having negative areas arising from complex lengths (something the discoverers of algebra weren't keen to do when completing squares and cubes)
Great job, can't tell you how much I enjoyed this video. Looking forward to the whole series being uploaded
Hey man, congrats on this video, I know I subscribed! I’m rooting for your rise in the ranks of math explainers! I’ve got some feedback:
1. Great topic, something most people have heard of (the kind who watch math videos anyway) but is not always explained. I know for me hyperbolic trig was something that just appeared at random sometimes in the middle of working on something, then I’d convert it to exponentials as quickly as possible and pray there was not some deeper detail I should know about hyperbolic trig. And I think a lot of people are in the same boat, so nice.
2. Your voice is GREAT for this, nice and baritone. I won’t name names but MOST math explainers have voice pitch that make it sounds like they’re about to cry.
3. The music was good BUT a little too loud over your voice. At times it felt like the music was fighting your voice for the attention of the listener, which made it hard to understand some parts.
4. This MIGHT just be an effect of the loud music but it did sound like your mic was muddying your voice a bit. I don’t know what mic you’re using, how it’s set up, etc. maybe you need some cheap
sound proofing? Change settings on mic? Or as a last resort get a new one (only if you’ve ruled out all cheaper options, since a good mic with bad settings or acoustic environment won’t fix things).
Great job, rooting for you!
Thank you SO much for the feedback!
You have a keen ear, so I'll explain some of what you're hearing:
1) I only learned about LUFS normalization about halfway through my video making process
2) I don't have a consistent room to record in, and some rooms sound different. I've learned through this video that this is a very significant effect, and I need to record in ONE room
3) I had covid during some of the video (the cone). I was very congested and trying not to cough.
But yes I've had other people comment on the background music. I'll have to figure out what LUFS ratio of voice to background is a good balance
Amazing explanation, congratulations! Looking forward to the next videos in the series.
bro i haven't seen better explanation on hyperbolic trig functions. Keep up the good work, King!
That is incredible, i cant believe it is your firs video, i want to marathon this play-series already hahaha.
Thank you so much :)
Beautiful video, earned my subscription. Got dissapointed that this is the only video, as I wanted to learn more about hyperbolas. Great introduction!
Good job. Most videos I have watched on hyperbolic trig skip over the geometric intuition of wtf does this have to do with hyperbolas and jump into the formulas, handwaving away the connection. That would be like if we started teaching kids circular trig by starting with Taylor Series instead of SOHCAHTOA and the unit circle. Good job taking the time to state the connection explicitly. I enjoyed the video, just subscribed, and look forward to part 2.
EXACTLY
Shame you didn't continue the series. I thought this was quite good. I've been spending a lot of time in old maths books where this kind of approach was common.
@tinkeringtim7999 I would appreciate references, if you've seen similar treatments.
@arithknot9740 Sure, it'll take some digging though. What for? Are you interested in researching math history?
@tinkeringtim7999 I am interested in the history of this topic. I haven't seen any treatments of hyperbolic trig on the basis of ratios of right triangles. I've hunted at the university library some and from what I can tell, hyperbolic trig developed alongside calculus and perhaps the e^x and ln definitions were the first definitions, and perhaps no one has cared to take it back to the geometry. But if there is indeed prior treatment of hyperbolic trig on the basis of ratios of right triangles, I'd like to know, because I've worked up a fair bit to share independently, but I'd like to give credit where credit is due.
Exemplary elaboration!!! Simple, elegant, intuitive , presenting the beauty of mathematics. God bless you.
Waiting Curiously for next video .
Thank you.
I love your video; it is clean and easy to follow, it looks professional, and your voice is pleasant. Thank you for sharing your knowledge with the world. God bless you.
Great Video! I really loved the diagrams along with the explanations!
You are amazing! You are the one who helped me understand hyperbolic trig
Loved the video! I can tell this channel will get big soon. Would love to see some stuff on how hyperbolas relate to relativity and how space-time works with a constant speed of light
I was considering that, maybe once I eatablish the tanh lines and hyperbolic rotations, then showing spacetime intervals as a practical gain, but there's definitely some groundwork to lay before then.
My primary interest is actually more mathematical than physical: I'd like to approach some different attempts at the (ir)rationality question of the Euler Mascheroni constant, and bring the viewer into the space of that unsolved mathematical question.
@@arithknot9740 well I'll be following closely and I'm looking forward to whatever you put out next!
Thank you so much for your kind words
glad to be in the first ones to stumble upon such quality content, fantastic video!
aw thanks :)
completly loved this cant wait for next part
this is so beautifulllll, fantastic video
Very clear introduction to the topic. Thank you.
perfect video hope to see next one soon
Priceless ! 👍
Clear and engaging, thanks!
bro dropped one great video and never came back
:(
Haven't forgotten. Busy life though. Definitely want to get back to this still
Fantastic video! Absolutely loved the pace and the exploration aspect of the exposition. I'm well past familiar with hyperbolas and hyperbolic space and yet you had my full atention at every moment. Just a really really great introduction to hyperbolas and a great teaser at the end. If you're thinking of making other videos, I will be here to watch them.
I wish I had this video (and it's follow ups) years ago when I was struggling with my degree. Hyperbolic trig was the breaking point for me in maths.
Eagerly awaiting part two!
paring down commitments. One byproduct is that I expect and hope to get back to this soooon
@@arithknot9740 I totally understand how 'real life' gets in the way of fun stuff like this. Good luck, and thanks for the video :)
this just made me realise why we use hyperbolic trig in special relativity this is unreal
From the other treatments I've seen, the asymptote corresponds to the speed of light. You might take a look at minutephysics' relativity series 😊
ua-cam.com/play/PLoaVOjvkzQtyjhV55wZcdicAz5KexgKvm.html&feature=shared
What a fantastic explanation
First i felt happy because i finally found someone explaining this perfectly, but then i realized that it's only a one video 😢
beautifully done !
Hi, a little idea:
You can use \cosh, \sin, \ln etc. This keeps the letters from becoming "italic"/cursive in math mode. I think it looks a bit neater that way.
Excellent
Great video!
Excited for the next chapter
Great video
Beautiful introduction!
I found this more engaging than Interstellar! Hope your heart melts and you post more such content...
Thank you for this channel. There are not enough videos about hyperbolic trigonometry taking this more fundamental approach. I hope you make a few follow up videos... especially on why it's natural that the hyperbolic trig functions contain the exponential function in the way they do. What's the connection between fixed-leg triangles and exponentials?
In short, we can parameterize the hyperbola as (x,y) = (cosh φ, sinh φ). But we can define new coordinates along the asymptotes: (a, b) = (0.5 e^φ, 0.5 e^(-φ)). Geometrically, we can see this relationship in the unit hyperbola as we draw a perpendicular from the asymptote to our point. This will be the topic of the second and third videos.
Really great video! I hope you feel inspired to make more!
Beautiful
Lovely
Finally something sensible. I am really not interested in defining hyperbolic functions in terms of exponentials- the typical way to define hyperbolic functions. Instead, I want to study the geometry of hyperbolas and then try to understand how it relates with exponentials. If you didn't know, e was defined using hyperbola before Euler changed it. Happy to see someone talking about it the way I want to learn it...
That's exactly what I have in mind. I've done a just a little bit of hunting and haven't found anyone that has developed this topic this way, so I've been doing it from scratch. It took me about a month to come up with a nice geometric proof for hyperbolic angle addition. Hopefully I can get back to this hobby soon.
@@arithknot9740 Oh, sounds exciting! Please do it, I'll be waiting for it. My fascination for this came with the idea that e^(ix) can be thought of as (e^x)^i and e^x being coshx + sinhx, e^(ix) would be cosh(ix) + sinh(ix), which is cosx + isinx - Euler's formula
So De Moivre's formula applies not only to e^(ix), but also to e^x.
Just like e^(inx) = (cosx + isinx)^n = cos(nx) + isin(nx),
We have e^(nx) = (coshx + sinhx)^n = cosh(nx) + sinh(nx)
And so
e^(ix) = (coshx + sinhx)^i = cosh(ix) + sinh(ix) = cosx + isinx
There's other fun connections to be had also. If you're that comfortable with the complex math, then you might appreciate something _far_ later I have planned in the series: the tie-in to split-complex numbers. If you've had any linear algebra, it is fairly easy to verify that the square of the matrix
0 -1
1 0
is -1 times the identity matrix. We can use this matrix as i in the field of 2x2 square matrices.
If we instead use j =
0 1
1 0
this gives hyperbolic geometry when plotted on the real and j axes instead of the circular geometry that results from i above. j^2 = 1. e^jx = cosh x + jsinh x
@@arithknot9740 Fascinating.
Let C = 0 -1
1 0
Let H = 0 1
1 0
Then, C^2 = -I and H^2 = I where I is the identity matrix.
Then (complex) exponential of matrix A can be written as:
e^(CA) = cos(A) + Csin(A)
e^(A) = cosh(A) + Hsinh(A)
Am I correct?
Another interesting observation. The matrix C transforms a point (x,y) into (-y,x). For unit circle, y = sinθ and x = cosθ. So the point (cosθ, sinθ) is transformed to (-sinθ, cosθ). Interestingly, the rotation matrix in 2 dimensions is
cosθ -sinθ
sinθ cosθ
Plug in θ = 90° and you get C. Goes on to show rotation by 90° just like i.
H transforms (x,y) to (y,x). Kind of an identity matrix because it doesn't scale the vector. But also not because it changes the point.
Some greater care has to be taken when you take e to the power of a 2x2 matrix. It does work nicely, but it's worth going through the steps to see why it works as you expect.
Let me just leave you with a few leads for now:
1) Search for split complex numbers
2) The rotation matrix by θ is
cosθ -sinθ
sinθ cosθ
which can be decomposed into its I and C parts (since cosθ and sinθ are just scalars)
I cosθ + C sinθ
The same can be done for the hyperbolic rotation matrix, rotating by an arbitrary hyperbolic angle φ:
coshφ sinhφ
sinhφ coshφ
Which decomposes into I and H components:
I coshφ + H sinhφ
This is well narrated and animated. A good topic taught from a foundational level. I like that the approach taken did not dwell on the classic trigonometry. I found the text and equations a little small. Overall, very good video.
Thanks for the feedback. I'll take that into consideration for later videos. If I may ask, on what sort of device were you viewing it?
@@arithknot9740 I started on a smartphone and switched to a TV.
Why is this only 7 minutes 😭😭 it’s such high quality
Great math video.
thanks! :)
So little views for such a great video.. very nice, I was truly enjoying myself. (My younger stupid rebel self should see me now 😅)
Hey, Look into my eyes, look into my eyes, not around the eyes, don't look around the eyes, look into my eyes. "Snaps 🤞" you're under. Now go and make another video immediately after reading this where you explain another math topic. Annnnnd you're out "🫰"
Phenomenal video, best explanation of hyperbolic trigonometry I’ve seen so far, I suppose someone with no knowledge of standard trigonometry could understand some, if not all of what was said in this video. I absolutely cannot wait for another episode in this amazing series. P.S. what software do you use to render these graphs, animations, and 3 dimensional graphs, thank you so much :)
Thank you so much. The software is mentioned at the very end. It's the manim python library originally developed by 3blue1brown's Grant Sanderson, which he released open source, called manim. I'm using the community edition.
PLEASE complete this series!
Currently working on the second one
You need to make part 2!!!
subscribed !
(1/γ)^2-(i*v/c)^2=1
i love your explaination look forward for more! i'd like to ask how you studied math to have such deep perspective, your motivation to do so. also what problem you are trying to solve, understand currently?
I study to answer my own questions. This stuff comes from my explorations into the unsolved question of the (ir)rationality of γ.
A cross between Pokemon and Star Trek. 😎
I love the video! If you want help with future chapters, I'd love to help, although I might be under qualified. Thanks again!
Kindly release the next video
This narration sounds so retro
At 3:28, I was wondering, is there a way to draw on the right triangle onto the hyperbola? The same way you can draw a right triangle inside the unit circle, and move it around?
There is, but it's not quite the same. In that method, you take a triangle at one fixed level of extension, and apply hyperbolic rotation to it. But we haven't discussed hyperbolic angle or hyperbolic rotation yet. So that's coming later.
@@arithknot9740where do the lines of triangle end up? Is the fixed length between the origin and the bottom of the hyperbola?
3:14 is that the case just for the unit hyperbola if not what it would be for hyperbola of form
(x/a)^2 - (y/b)^2 = 1 what the fixed length would be?
That's a non-square hyperbola. That would be like asking what the radius of an ellipse is.
If a=b, then it's a square hyperbola.
(x/a)^2 - (y/a)^2 = 1
x^2 - y^2 = a^2
So it has a fixed length of a.
But yeah, if a doesn't equal b, you have an x-stretch that is different than the y-stretch, and you get a non-square hyperbola. The question I left hanging was how one might get a non-square hyperbola in the triangle context, or if that even makes sense.
@@arithknot9740 Thanks for the answer!!
At 2: 47 you introduce the hyperbola but you don't explain how it relates to what has gone before. What you should have said is "Take a right angled triangle with hypotenuse h, and two sides a and b. let the side A be fixed to 1. Plot the graph showing the length of the hypotenuse h against the length of the other sideb . The equation of the line formed is derived from pythagoras . h2 = 1+b2. Now we have an explanation of the relevance of a hyperbola. But good video.
when next vid?
With four kids and a side job, it's difficult to find the time/money/energy to work on it. I've spent some time on it, but I'm hoping to make a lot of progress on it over the upcoming holiday breaks.
This is pure BS.
Trigonometric functions are secants of unit circle corresponding to a given length of arc. Using double areas of circle sectors will also do.
Hyperbolic functions are exactly same secants of unit hyperbola except arc length will not do in this case and only area of sector is used as argument ()
In American textbooks function are denoted as sinh and sinh^-1 in Europe they use more relevant notation as sh(x) and arsh(y) read "area-sinus-hyperbolic"
Yes the gudemannian function relates circular and hyperbolic trig functions, such that tan(θ)=sinh(φ) and sec(θ)=cosh(φ). Lol we're not there yet.
I'm not claiming to come up with anything new other than simply this: It would be nice to have a treatment of hyperbolic trig that starts from right triangles and ratio definitions like we do for circular trig. Some interesting geometry, algebra, and intuitions fall out as a result.
One benefit, for example, is that the forward-backward transforms that come up in calculus, such as arsinh(cosh(x)) have nice geometric methods that parallel those in circular trig, but I never see it taught that way.
lol did you died?
lol i dids not died. i just has many lots of responsabilitoeses