Why is there no equation for the perimeter of an ellipse‽
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These are my approximation equations:
perimeter ≈ π[53a/3 + 717b/35 - √(269a^2 + 667ab + 371b^2)]
perimeter ≈ π(6a/5 + 3b/4)
If you can do better, submit it to Matt Parker's Maths Puzzles.
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www.think-maths.co.uk/ellipse...
This was my pervious video featuring ellipsoids:
• Ellipsoids and The Biz...
You can buy the ellipse from this video on eBay. I've written on my two new equations and signed it. All money goes to charity (the fantastic Water Aid).
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Huge thanks to all who sent in a recording of them singing "A total ellipse of the chart." Sorry I could not include everyone. These are the people in the video:
Helen Arney
Steve Hardwick
Victoria Saigle
Andrew McLaren
Fractal
Macey
Sören Kowalick
It all started because of a request I put out on twitter.
/ 1301252952930299904
CORRECTIONS:
- So far the only times (I've noticed that) I say "eclipse" instead of "ellipse" are 5:01 and 05:26 which was just after talking about my wife who is a solar physicist. So I think we split the blame 50/50.
- It seems everyone but me recognised the Root Mean Square. I think I only associate that with current for some reason! Thanks all.
- Let me know if you spot any other mistakes!
Thanks to my Patreons who meant I could spend about a week trying to find approximations for the length of ellipses. "Are you still working on that?" Lucie would - rightfully - ask over the weekend. "I'm going the extra mile for my patreon people!" I would reply. Here is a random subset of those fine folks:
Benjamin Richter
Louie Ruck
Matthew Holland
Morgan Butt
Rathe Hollingum
Jeremy Buchanan
Sjoerd Wennekes
Barry Pitcairn
James Dexter
Adrian Cowan
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My lazy approximation would be 4a :) The more eccentric the ellipse, the more accurate it gets.
on average it's better than any, but it's practically useless
@@MaoDevHow aliens would describe me in one sentence after studying the human species.
4a is the lower limit for the circumference perimeter of an ellipse. C or 1, the circle circumference is the upper limit.
@@theglitch312 hahaha!
6:42 It is a *quadratic mean* also (very well) known as *RMS* (Root Mean Square) by Electrical & Electronics Engineers. The quadratic mean is popular closer to the highest value (Max) or greater than the centered arithmetic mean. The geometric mean, lesser than the arithmetic mean, is near the lowest value (Min), and the harmonic mean is even closer. The error graph of those means drives us to conclude that the larger axis *_b_* has more influence on the perimeter of an ellipse than the minor axis *_a_* , mainly as eccentricity increases. We also can realize that such means are the main trunk line in the search for the perimeter of an ellipse: - The first Ramanujan approximation and the first Parker approximation are some kinds of playing around with weighted arithmetic, quadratic and geometric means... yes, they can all be weighted by multiplier coefficients; - The second Ramanujan approximation, excellent by the way, is a combination of weighted arithmetic mean and the use of *_h_* has some relation to a weighted quadratic mean; - The second Parker lazy approximation is a weighted arithmetic mean, relatively good compared to the quadratic one.
"I know just enough mathematics to be dangerous" - I feel this should be a tshirt.
I'd buy one
This week I worked out that 25 grams of antimatter has the potential energy of a Megaton of TNT. So I feel like I fit into that category.
It's a way of life, that's for certain.
...with Einstein's silhouette and Matt Parker showing his square to Einstein...
I need that
I actually discovered *4(a + b) - ln(4a + 1)* at ~10AM on 08/04/2021 as my own Approximation! It only ever reaches 1.6813% (-When b = 1) error and eventually approaches -0.0297% error- 0.000% error.
I found a more general Approximation of *4(a + b) - ln(4a/b + 1)b.* It always maxes at only 1.6813% error.
@@Inspirator_AG112 That's very clean! Well done.
I think if you put 'h' inside the ln term, may be possible to find a better one.
No pi in the equation? That makes it even more awesome!
It actually approches perfection. (Correction 7 months later.)
Matt, engineers frequently use the "root mean square" to describe expressions like SQRT((a^2 + b^2)/2).
I think statisticians use it to calculate things like variance, too! Iirc cuberoot( (a^3 + b^3) / 2) helps get the skew (of a sample of size n=2). I wonder what the skew of a "radius" would be like
I see "root mean square" in a lot of audio plugins, as a way of detecting peaks in the audio (or as an alternative? I donno. It's usually a choice between "peak" and "RMS")
Very useful in machine learning - most models (mostly neural nets) are trained by taking the derivative of the "mean squared error" and following the gradient in the direction that lowers the error. Mean squared error is nice because it's differentiable - well, I guess the absolute value of the error is differentiable when the error is nonzero, but I think you'd be likely to overshoot using gradient descent on absolute value of the error.
kinda surprised he didn't know that considering he studied mechanical engineering in college.
@@josephbrandenburg4373 "RMS" in an electrical context is often a way of getting some sort of "average" because arithmetic mean in a sinusoid (AC signal) doesn't work and it ends up being useful in some areas. considering a lot of audio equipment is analog (and in odd waveforms) it would make sense to use RMS as sort of an average loudness
" if ramanujan made 1 major mistake with their mathematical career, it was having it in the past" -matt parker, everybody
Unappreciated joke
I think the mistake I made with my career as engineer on a starship is not having my career hundreds of years in the future.
This is why I love Matt.
Did Ramanujan prefer "their" as a pronoun, or did you just disrespectfully choose the pronoun that was more comfortable for you? Oh, my... I shouldn't have assumed "you" to be the correct term either.... nevermind...
"The future is now old man"
15:36 3 Blue 1 Brown's pi is sort of like the Clippy of mathematics: "It looks like you're trying to find the perimeter of an ellipse!"
now I want someone to make a 3B1B digital assistant
If Clippy were anywhere near that useful, I'd have never turned him off!
Proud to be your 666th upvote :)
@@hoebare Beast Mode! So to speak...
@@hoebare devil
I am almost 60 years old. I love mathematics and I never, never imagen if somebody could make me laugh watching a math video. Well you did. Mathematics are so amazing, fun and funny too. Thank you so much for this 20 mins. Cheers!
Same here, born 1951
you sound like my grandpa lol!
If I had a nickel for every time Matt Parker called an ellipse an "eclipse", I'd have two nickels. Which isn't a lot, but it's weird that it happened twice.
they rehearsed that song too often before recording ;)
I thought he did this more than twice, but I was not counting.
Anyone count lipses? Lips'? Lips's? Yeah, yeah. Anyone count lips's?
Definitely more than twice - he did it twice just between 5:00 and 5:30. Using Keppler's approximation and the duration of this video (21 min), I'd say, he could've confused ellipses with eclipses as many as 84 times.
I blame Bonnie Tyler.
"Who's having an ellipse that is 75 times as long as it is wide?" An Oort Cloud comet has entered the chat.
@@danieljensen2626 they are much worse. edit: If i did my math correctly, then something traveling between Uranus and Earth will have that 75 ratio. But also i feel like at this point just calling it 4a is pretty accurate
And then left and won't be back for a few centuries.
Physicists would approximate this as a line.
An ellipse has totally entered the chart.
There's a comet called Ikeya-Seki. It has an eccentricity of 0.999915. If I calculated correctly, that's 77 times more long than wide. But I think most comets are not that bad. For Hale-Bopp it's 11 something.
15:30 Matt: *slaps Pi” “This bad boy can fit an infinite series of fractions in it’
Good meme
This is the best comment.
Robert Slackware Why? Can‘t you e.g. do sth like 110100100010000...?
@Robert Slackware π is in the open interval from 0 to 3.5, so it is not infinite.
@Robert Slackware LOL! rock on, man...
13:06 Well, because an object in free fall isn't really tracing out a parabola but instead a highly eccentric elliptic orbit around the earth's gravitational centre, you might in fact need such high eccentricity
I never thaugh about that. It's only a parabola if the force feild is an infinite plane, but on a sherical one, it's an extroardinaraly eccentricical elipse. My whole life is a lie.
@@jackys_handleFor most human-scale projectile motion, the difference is so insignificant that it doesn't make a difference. Local gravitational anomalies, like a mountain or heavy mineral deposit nearby, are going to be more significant, than accounting for the difference between an ellipse and a parabola as the shape of its trajectory.
I wonder if we flatten out an ellipse, since those simple calculations usually tends to treat earths surface as flat, will we actually find a parabola?
The interesting fact I noticed about the "bouncing" approximation is that for certain values of ratio they give a 0% error
A broken clock is correct twice a day
Also, the sine function perfectly approximates the value of 0 infinitely many times, but that doesn’t make it a good approximation of 0
I would venture to guess that those certain values would be irrational?
@@BeauDiddley87 and transcendental, going on a limb here
@@fi4re Sadly that just works for analog clocks lol. Digital ones have a more nihilistic approach.
Whenever Mathematicians are scratching their heads on a problem, a wild Ramanujan appears
Wild?
@@thebiggestcauldron he is wild (commentor)
@jocaguz18 Yes.
And ... then an even wilder Ramanujan appears. This formula C = π(a+b) ((12 + h)/8 - √((2 - h)/8)) fits much better than Ramanujan's (which is C = π(a+b) (3 - √(4 - h)), when expressed in terms of h). We're onto his game!
@@dgarrard100 Gotta catch both of 'em!
"Ignore what happens a lot further that way. It's not relevant." *disapproves in Big O Notation*
Theta(n!) is so fast it even beats Theta(2n)!, if are range is 0 to 3 hehe
@@macicoinc9363 What is theta? Are you using it to mean Big O?
Oversimplified, Big O means "grows not as fast as", little o means "grows faster than" and theta means "grows roughly the same as"
@@t0mstone581 Thanks!
T0mstone wooo computational mathematics is so fun...
I remember stumbling upon this unfortunate fact when wanting to know the perimeter of a rubber gasket used for an elliptical hole at my workplace. I ultimately ended up just using a string to wrap around the edge so I could straighten it out and measure it, but still.
That's engineering vs. math in a nutshell. The mathematician will spend 18 months trying to find a better formula, the engineer will take 10 minutes to find a piece of string so they can move on with their life.
Thats what NASA does
@@Mr_Smith_369 lol damn
@@Mr_Smith_369 really big strings to measure orbits
There are tables for the elliptical integral(formula for arc length as an integral). Values for specific lengths can be interpolated using the table values for k, ( k^2 which is (a^2-b^2)). See Cal 2 texts for details
Another approach is to use the integral formula for the curve length. This integral can't be presented as a well-defined function, so you have to use a Simpson rule, for instance. With the Simpson rule, you can also estimate an error.
That was my solution, the antiderivative ends up being pretty complicated.
@@JosephEaorle but it would be exact, so the claim that there is no exact equation is false; there is no simple, exact equation; but there is an exact equation.
Yeahhhhhh maybe, but with the Simpson rule you'd get dragged down by having to write it over and over on a chalkboard.
For further Reference on the subject one should consider the Extensively studied field of Elliptic Integrals [ en.wikipedia.org/wiki/Elliptic_integral ] and for Numerical Calculation of the Integrals one could use Adaptive Gaussian Quadrature schemes like Patterson methods [ en.wikipedia.org/wiki/Gaussian_quadrature ] which provides Much Better results than Simpson Rule, or for a simply Naive but much Better than Simpson calculation one could take Romberg Integration schemes.
If you actually want the answer to "why don't we have a formula", it is simply that the perimeter of an ellipse is the line integral of its parametrisation: an ellipse is the set {(a cos(t), b sin(t)): 0
The real question here is: How do you define which functions are "usual". That's subjective.
@@qborki no it isn't. It's pretty much well defined.
@@qborki Well, I am going to make an assumption here, because I do not know this with absolute certainty, but from what I do know, its math we are talking about. I am pretty sure there is an exact definition of the "usual" function. Its probably just the one you wont understand unless you have a certain level of math knowledge.
The linear integral, which gives you the length the ellipse is unsolvable... This does not mean that there isn't a formula for the perimeter...
@@tomasstana5423 I think he ment elementary functions? Idk, as far as I'm aware of, there are no "usual functions"
Loved the little 3Blue1Brown reference.
For those who missed it, see 15:38
Yeah, what a cutie-pi :3
Lol
@@6872elpado what u mean
Saw the reference, came to the comments section looking for this comment. Now back to the rest of the video :)
Excellent, Excellent reporting! Wow! Ramanujen's brilliance was in finding something that freaking simple to do such a fantastic job. That kind of accuracy is good enough to land a probe on a comet. I enjoyed your improved lazy approximation, and I REALLY enjoyed the nice vocalist who sang Total Elipse of the Chart.
I was asked something about this at a job interview nearly 30 years ago. I was interviewing for a computer instructor and someone who worked at that college as a welding instructor asked about this and I had no idea what to say. He wanted to know because he wanted to build a horse trailer with an ellipse shaped cross-section of the top. For what he wanted, I didn't see the reason to have an ellipse -- two quarter circles with a flat piece across would be more likely to give the horse more room without bumping his head, but he was convinced that only an ellipse shape would do.
People in 100 years: if Matt Parker made one major mistake, it was having his mathematical career in the past.
And with his mathematical insight, I've got something he didn't have, I've got a quantum computer. ................................................ so even though I only know juuust enough mathematics to be hazardous I can outsource alot of it to this machine.
That's a Parker Square of a career timing
ONE major mistake?
@@motazfawzi2504 I love the idea of this, and hope things like this persist like memes online for centuries LOL
you nailed it.
That moment of realization for 2*pi*r where he says "wait a minute!" is so well timed with the realization for the viewer.
100th like :) ...send me money
BibiBosh rounded to 100? Approximately 100th? Was it 100. ? ( #BadRounding)
It was exzactly 100
Classic parker
Amazing
The quality in this video is amazing! Thank you.
I am NO mathematician, but programming, while accidentally seeing this. The information density of your beautyful feature is high AND entertaining, while i can learn in ease. I was browsing 20 unnecessary Sites to veryfy a typo in a book of Physics and found this comprehensive while deep and refreshing channel of yours. THANKS a LOT for occupying my screen, talking with purpose. I secretly like Maths in awe and i see you love it too. Being rewarded.
Take a shot every time Matt calls an ellipse an eclipse :p
makes me wanna do a parker square...
Only twice though, so you won't get many shots.
@@SumNutOnU2b Well, it's a Parker drinking game. It works somewhat okay, but not great.
An eclipse is a parker ellipse
@@wolframstahl1263 brilliant!
8:35 "His mistake was doing math in the past." Honest mistake, we'll try to do better next time.
One of the few mathematicians in the western canon that you can say that about. I feel that your joke is underappreciated.
Unfortunately, Ramanujan's mistake was deadly.
@@jansamohyl7983 being born leads to death... so we all made the mistake
@@jaredjones6570 I mean, I haven’t made that mistake yet, and I’d be kind of freaked out if you have.
Actually there were no gendered pronouns used in the video. It's hard to miss. Everything is "they".
Thanks, Matt for being so MATTematically precise in your videos.
There aren't enough comments about how wonderful that 3Blue1Brown π cameo was.
Yes! :D
I think he may be using 3b1bs open source animation software
In which second is that?
@@a.georgopoulou (: at 15:36
@@YambamYambam2 but there is no brown i don't get itt
6:45 thats called the 'root mean squared' value. Read the words in opposite order and you will know why. Very useful in kinetic theory of gases as well as calculations of alternating current.
Or 'quadratic mean.' It's interesting to note that we always have QM>=AM>=GM (quadratic, arithmetic, geometric).
@@alephnull4044 >=HM harmonic mean: 2/(1/a + 1/b) >= min(a,b) :P
I was surprised that be didn't know that
@@fares8005 Yeah. So HM would be even worse of an approximation than GM.
@@anuragjuyal7614 I guess since RMS is more common in physics and engineering. And not so much in pure maths.
I have no idea why but this has really hooked me in. I am not a mathnetician. I spent all of sunday and several hours this morning drawing elipses and circles on desmos and playing with different equations.
One of the approximations is the RMS value of a & b. The root of mean of squares one.
I just realized that my math teachers frightened me in knowing formulas of perimeter, area and volume of nearly anything, omitting to tell that one was missing.
They shielded you from a dark truth you were not yet ready to accept, that would have shattered your nascent mind
"So what are the traits of an ellipse?" "Oh well there's the major and minor axes, two focal points, an eccentricity and h." "What's h?" *leaves*
@1:50
*Insert h meme here
Incredibly incorrect and flippant answer here, but I think it's some inverse of the hypotenuse between the ends of a and b.
Considering the weight of the problem, probably Plancks constant
if put a=kb then h = (k-1)² / (k+1)² for (k>=1)
For that, first we need to delve into the nature of "π". What is π? It is the ratio of circumference to the diameter in a "Circle"(only). Now, Conics are defined by their "eccentricity"(ε) values, which too is a ratio. Conics are, the Circle (ε = 0), Ellipse (0 < ε < 1), Parabola (ε = 1) & lastly Hyperbola (1 < ε < ∞). In these only the circle & Parabola have fixed ε, each (0 or 1). It implies there is only one circle (that can be scaled up to look big) and one Parabola, while there can be an infinite number of Ellipses or (infinite number of) Hyperbolae each of a different eccentricity (ε). Just as for the definition of π (ratio of circumference to the diameter) that is valid for circle, there can be no such a thing for Ellipse. The ratio of circumference to semi-major or minor axis is a continuous variable. So there can be no π, for an Ellipse. Then why do we involve π, in the definition of circumference of an Ellipse (as some would want us to believe)? We don't need π.
Thank you for this explanation.
because you touch yourself at night
based!
Great video. I didn't know there was no exact formula. When I was at engineering school, a student in my class needed to calculate the perimeter of an ellipse for a software he was coding. I thought about it and came with a (wrong) solution, considering an ellipse is the intersection of a plane and a cylinder (of radius b. The angle between the plane and the cylinder depending on a). Then, "unwrapping" this cylinder (as it was made of paper) to put it flat and measuring the previous intersection as it was (actually, it is not) the hypotenuses of a pair of right-angle triangles, this leads to P=2*sqrt[(pi^2-4)*b^2+4*a^2]. I have just checked this formula against an online calculator that uses Ramanujan's second approximation and found a divergence around 3%.
What we learned today: Ramanujan was hot stuff
You just learned that? :D He's well up there with some of the other greats. There's even a "documentary" (more of a dramatization but regardless) of his life called "The man who knew infinity." Wouldn't say its a classic but its not terrible either.
Speak for yourself there! So brilliant and original that the Brits had to teach him to speak math like they do just so they could understand him
@@enginerdy You mean speak maths? :D
You made my day bro
I swear he must have had a mathematical IQ of like 200 or more!
“But what about orbits?” That’s when you know you married a right partner.
Sorta helps his wife is a physicist involved in satellite science. :P
@spim randsley Dammit, if only Earth had a moon as marker - save all that chalky maths stuff.
What about the perimeter of a testee?
@@pluto8404 Test these.
@spim randsley Bread + moon cheese squared. That's gotta be the solution.
15:20 scene was great 🤣🤣😅😅
I got interested in this when making bridges with geometrical shapes in a 3D program. Making a fence out of many overlapping shapes, (half-ellipses, but that's irellevant,) I wanted to know how to space them evenly on a bridge surface which was also half an ellipse. Unable to find a good lazy method, I was thankful that particular program approximated the ellipse with a relatively small number of straight segments no matter how large the ellipse was. Thus, I could easily space the fence-bits evenly on each straight section and do the turns by eye. If I do this again on a program which makes smoother ellipses, (which is most of them,) I'll certainly want to try the Parker lazy method in this video, especially because the ratio of such a bridge-ellipse can easily be 10 or more. (Y'know, I'm slightly sad because this post will spoil the number of comments. It was 5,555 before I posted this.)
I expected at least a mention of an integration approach
Yeah, I was waiting for it too...a bit disappointed that he didn't mention it
The origin of the elliptic integral.
I thought I was the only one disappointed after watching it. No mention whatsoever of the elliptic integral.
I was expecting this too, before the infinite series (like, where does that come from?)
Me too
Engineers be like "Ehh, it's close enough. Who cares....."
I can confirm this.
The correct observation; “It’s over engineered so it’ll work if we just let it ride.”
I have tried numerous ways of modeling complex curves for flat spring designs in SolidWorks CAD and failed miserably at defining them with formulae. I could use ellipses to draw segments, but trying to connect them into one poly-line with parametric segment lengths made the model geometry "blow up." In one particularly frustrating design I ended up just freehanding my desired curve and setting that as the definition for the spring shape. I was able to use the brute-force freehand curve to design bending mandrels which made just what I needed. Sometimes real-life is too complicated for computers. It bugged me that I couldn't tell my production people exactly how much flat spring material they needed to build the spring.
@@jasonspudtomsett9089 When modelling/simulating it is usually the norm to be as simple and ideal as possible. But well, all that matters is if it works lol
Wouldn't it be so much easier if Pi was 3? How accurate do we need this result? An order of magnitude? Great, Pi = 3.
Delightful, awesome video, greatly enjoyed it!
At 11:13 while you equation seems to get blasted out of the water i would like to point out that assuming where the line bounces is at 0% error, at that specific a and b values you technically have a more accurate equation.
"He knows maths. Enough to be dangerous. Matt Parker in Parker Eclipse."
Parker Duck! Let's get dangerous!
*maths 🙈
5:00
@@witerabid I'm using a mix of British and American English, whatever I feel like :D but I'll change it just for you.
@@YuureiInu 😅 I was just preempting the Brits. I usually say "math" too. 😉
The way he connects the whole thing together by stating reminding us that pi is an infinite series at the end is phenomenal
Yeah, I loved that bit. :)
Makes me wonder if we could get a nicer equation is we took away pi and put a and b into the pi series....
Pi is an infinite series if you live in world of integers. Integers are infinite series if you live in a world of Pis.
@@notabene7381 tf
Considering the quality and amount of output, with very little formal training, and dying way too young, Ramanujan must be the greatest mathematician of all time.
It would be interesting to do a similar video (area and circumference) of super-ellipse/squircle, super-shapes, lemniscate, etc. (With the infinite series for a corresponding “pi”)
For me i often define ellipses in pretty much the same way, but a=1 and b= cos(ß). Since in my application, an ellipse can often be understood as a circle with radius a, seen from an incidence angle ß. For example a rake angle. Really simple. But indeed it's weird that there is no easy approach to circumference!
Your vision is usefull for area of an ellipse but didn't help for the circumference.
I laughed so hard when Matt swept the infinite expansion under the π.
pi = 3, why bother with those stupid fractions
lmao me too
for anyone else who sees this, it happens at 15:16
Best Matt Parker moment ever!
However, PI is incomplete without its LE.
"And who's having an ellipse which is 75 times as wide as it is high?" As it turns out, there is the Hale-Bopp comet which, according to Wikipedia: Semi major axis = 186 AU eccentricity = 0.995086 Semi major / Semi minor = 203.5 Incidentally, Haley's Comet is pretty eccentric, but still below 75: Semi major axis = 17.834 AU eccentricity = 0.96714 Semi major / Semi minor = 30.4
Glad you said this. When he made that comment, I shouted "COMETS" at the screen.
TY so much for this as I was wondering about comets eccentricity's.
How did you calculate the Semi major / Semi minor ?
@@laurgao -- Using the eccentricity.
Where did you get your major/minor from? I was under the understanding that a/b=(1-e^2)^-0.5 , which gives me 10.0 and 3.93.
I actually do like the shape of your calculation!😀It looks so happy!
15:36 - that 3blue1brown reference killed me
8:33 "I know just enough mathematics to be dangerous" this surely enters my top five best statements ever to be stated
0:26 Matt - “It’s a more generalised version” and like all good mathematicians “And my goodness, is it lovely!”
3:31 Also, like all good mathematicians, he completely disregarded the actual usefulness of the focal points "light, mirrors, bla bla bla"
@@luisramos123 I'd have it no other way!
Idk if this works but when finding the perimeter of planetary orbits, you can use Kepler's equations (with true anomaly) to produce a speed-time function, and then integrate it from the bounds 0 to T, getting total distance traveled in one orbit. This is what I did for my high-school math project and it worked quite well for the planets.
Just did some math with a friend of mine lol. It’s 11pm, but we did some good work in my opinion. There are 2 equations, one simple, one more complicated. One where n = 1.5, and one where n = 1 / log(2, pi/2), or approximately 1.53493, where P = 4b((a/b)^n + 1)^(1/n). Not sure if I did the error accuracy thing right, but if I did, we should have under 0.4% error throughout with the complicated equation, and it only gets better as the ellipse becomes longer. Would love if someone wanted to recheck and let me know if I’m right lol
Interesting. I just saw this interesting video yesterday. After that, decided to try a family of solutions: 2*pi*((a^n)+b^n)/2)^(1/n). Started with n=1 and n=2. Noticed that one underestimates, the other overestimates the right answer. So, tried n=1.5. Noticed that it reduced the error to under 1% over the entire eccentricity range. Then I focused on the value that gives the exact answer as the eccentricity goes to infinite. Found exactly the same n you found. That is, n is the reciprocal of the log base 2 of (pi/2). The error is zero when b=a and when b goes to infinity. And it stays under 0.4% over the entire range.
I found these by integrating a bezier curve: a * [ sqrt(4 + (4 * b/a)² ) + 2 ] --Max 5.682% error a * [ sqrt(2pi + (4 * b/a)² ) + (3+pi)/4 ] -- Max 3.237% error a * [ sqrt(4.905 + (4 * b/a)² ) + pi/2 ] -- Max 3.200% error Edit: Found an even better one For a = 1 and 0
Looks like python code 😉
Best part of this: "I stopped searching for a function when I found that Kepler had developed an approximation."
Yup, smiled also. Einstein should've stopped searching after Newton told us what's what. But there was always a a clever-guts Albert in every schoolroom.
Nothing serious, I hope?
@@Mrbobinge Einstein's formula? What about Epstein's formula? Very successful for a long time. A lot of travelling on a plane. Also, a lot of curved surfaces.
He said "Ratio", "Major", and "Minor" in the same sentence and it wasn't about music
Music ⊆ Maths ?
@@TheYahmez yeah, i always laught inside me when someone says "i love music but hate math" :D
Everything is just applied maths
@@ali709aliali And math is applied philosophy
@@gileee No, it's the other way around.
I like your work... And way of explaining thanks man..
wtf, how can a math video be so captivating that I randomly and willingly put 20 minutes to watch it fully
There's actually some deeper math hiding beneath the surface here. The elliptic integral (which is a non-elementary integral that calculates the circumference exactly) is related to elliptic functions and elliptic curves (which were used to prove Fermat's last theorem).
I was going to comment Matt was wrong. You don't need an infinite series, just integrals.
@@revcrussell Right, an integral who's solution can only be written as an infinite series... You can also write an integral equation for Pi, but that doesn't really get you anywhere.
@@danieljensen2626 Even further: what are integrals in general, but succinctly notated limits of infinite series.
@@anteroinen4239 Good point, although some of the ones we like to use converge to algebraic or even rational numbers.
Wrong
Doesn't he say "eclipse" numerous times when referring to an "ellipse"? Maybe I'm just going crazy :-)
He does, I caught that too:))
Well everyone, atleast most of us do it.
5:00 one example I found
I wonder if it was on purpose 🧐
It’s weird I saw this comment and I found a few
Who knew there was no single equation. This is a fascinating examination of the perimeter of an ellipse. I am in awe of your wife's performance, well done. Thank you for your insights into this interesting puzzle.
I like your show, very and with good taste. Thank you for the singing and piano playing. Gracias
Parker: "And who's having an ellipse which is seventy-five times as wide as it is high?" Halley: "Hold my slide rule."
Halley's comet isn't that eccentric though....
I thought this too, but Halley's comet has an eccentricity of 0.967, which means that its orbit is only 3.93 times wider than it is high.
Also, my orbits in Kerbal Space Program...I'm usually too lazy to use the rocket equation properly, and really, *really* like solid fuel boosters for the first stage of my rockets.
C= Tau•R Wonder if some of the complexity drops if we adopt Tau instead of Pi?
@@Trevor21230 same
The term you're looking for at 6:46 is "root mean square" or rms, and is used a lot in AC electricity voltage computations.
Huh, I always called it the quadratic mean
Also molecular velocity
Yaap...it’s also used to equate kinetic energy of gas. It’s a incredible way of getting rid of negative value when finding a average.
I was looking to see if someone made this very common. Thank you.
Encountered it in molecular kinetics, average speed of particles in a gas
There is a well defined equation for the perimeter! Parameterize an ellipse and apply some vector calculus. It isn't workable by hand, but it is literally the perimeter. It is also the circumstance of a circle because of how squareroots of squares of trig functions. Take the line integral and you will get your answer.
I was expecting to find an integral that would give the path length and was surprised when none were mentioned.
Yeah but to my knowledge there’s no analytical solution
en.wikipedia.org/wiki/Ellipse#Metric_properties The ellipse circumference in general is not an elementary function.
@@badbeardbill9956 Correct. And pi is irrational number, so does it mean there's no number of length of circle?
@@leonidfro8302pi is a number a transcendental number. Means it is not countable.
I think you can make a pretty accurate one with conditionals. 1-2 range use formula A, 2-4 use formula B, 4-8 use formula C, 8-infinite use formula D.
Hell if you're clever enough and have too much time on your hands you could build one mega equation that cancels out the other formulas depending on what number range you're using, mixing in functions to give it properties rather than for any mathematical purpose just to say you have an all in one approximation lol
Title: Why is there no equation for the perimeter of an ellipse? Trick answer: There is, but it involves an infinite series. Plot Twist Just like the equation for the perimeter of a circle.
This is where I ended up in my reasoning as well, which I guess was the point of the video. My intuition was telling me that pi was to circles what some other unknown constant would be to ellipses, and then my intuition also wondered if each ellipse might have its own unique "pi"-like constant.
Best plot twist on KZhead's history
@@guillermogarciamanjarrez8934 this
@@geshtu1760 So, given a/b [which is consistent with his setting b=1, and by the way it makes more sense to use b/a -- and set a=1 -- because b can go to zero, unless you prefer that a can go to infinity] -- okay, given a/b, the perimeter equals 2*pilike(a/b)*avg(a,b)? Or perhaps 2*pilike(a/b)*a? Then the complications of figuring out the formula for pilike(a/b) are exactly the complications that he walks thru in the video. So, yes.
Ok. But is there an equation that "hides" the infinite series for an ellipse? If not, then I have a suggestion for a sequel.
"I know just enough math to be dangerous" Lol I love this. These videos are so much fun to watch (even if my friends think I'm crazy for watching maths videos in my free time)
your friends are crazy for not watching maths videos in their spare time. or, maybe they've just never tried before, cause as 3b1b discussed many times before, often people just don't know how much they love maths
I spat my meds out upon hearing that..... note to self: dont watch Parker when taking your meds
I love Matt's identity as 'StandupMaths' -- literally making Maths enjoyable to the wider public by making it into comedy. Pure genius.
@@Shrooblord doesnt it come from him doing that math/science comedy show with Steve Mould?
You have the expression. It's just an integral. You can use a Taylor expansion if you have a problem with it.
Your videos make me so interested in math.
"When are you going to get a job!" ... "In the future... I'm not gonna make the same mistake as Ramanujan..."
When I was doing my GCSEs, I was doing Graphic Design, and I was building my design, a diorama using concentric elliptical curves of clear plastic with designs drawn on them to create an interesting parallax image. I ran into an issue though, I didn't know how long I needed to cut my plastic sheets. I knew how I would work it out if they were half-circles, but not if they were half-ellipse. So I asked my teacher how to work out the circumference of an ellipse, and tbh, he was stumped - so together we looked it up, and we discovered that it was a lot harder to do than we first thought it would be
Fiiiiiiiine your videos are entertaining enough for me to put up another one while I do the dishes just now...
the name for the red 'root of the means of squares" is also called "root mean squared"--RMS. Used a lot in electronics.
I'm actually incredibly impressed by your lazy approximation, it'd seem like such a simple solution multiplying the two axes by fractional constants would have been found earlier. Great work!
I mean it's just a compromise. Sacrifice some accuracy at first for more accuracy later. But I guess in general mathematicians are more interested in symmetry.
And it is easy to remember as well, once you write 3, 4, 5, 6 in an appropriate circle thing and « fill in the gaps » with a, b, and fraction bars!
The fact that it gives the circumference of a circle as 1.95pi radians is bad starting point, but it *is* very #ParkerMaths
How would "4a - (2pi-4)b" do? I think the derivation on that one should be fairly obvious. One thing it would have been nice to see Matt Parker mention would be how the approximations do as eccentricities get large.
I agree, Parker showed himself from his best mathematical side there. I'm still not sure I'll remember this one the day I need it, but it seems the best candidate for those who want to memorize something.
That's a Parker Approximation right there. #ParkerSquare
We don't need to keep making these jokes any more, because I've generalised it: "This is a Parker N"
Parker approximations... that's two layers of haphazardness!
This is a Parker Joke
@@robinw77 that was a parker reply
I paused the video just to look up for this XD
Based on what I learned in this video, I just came up with an exact equation for all ellipses! It is: γπ(a+b)…… Where γ is (1+(h/4)+((h^2)/64)+((h^3)/256)+…)
But y isn't a constant
My Approximation is *4(a + b) - ln(4a/b + 1)b.* I found this Approximation with calculus and the help of Desmos.
For the physical interpretation of h: it’s a measure of flatness. It should lie within [0, 1] where 0 is a perfect circle (least “flat”) and 1 is a line (either horizontally or vertically, perfectly “flat”).
oww, it's that h from the standard equation of 2 degree in 2 variables?? anyways, thanks for it
I'm never not astounded at the genius of Ramanjan wow he was able to do with his just his head what a laptop was only able to do 2 times more accurate... we're talkin margin of errors in the hundredths of a percent as well jeez this guy was a beast edit: just saw his 2nd equation LMAO wtf how was that guy human
It's the difference between solving analytically and solving numerically. Not to say that Ramanujan wasn't brilliant but the two methods just have completely different outcomes, as shown by the error comparisons here.
he was a human you are not
@@sachinnandakumar1008 by numerically he means computationally making a close approximation through iterative processes, whereas analytically he means solve for a somewhat exact solution by 'traditional' mathematical methods, like algebra and calculus (not that numerical methods don't use those, of course, but that's slightly different).
I mean, he was known for pioneering achievements in sequence and series. Pretty much expected.
Creativity unbound by the labor and limitations of programming.
If you think about it an ellipse is just a variation of a circle. If a circle is x^2+y^2 then an ellipse is the exact same with with modifiers ax^2+by^2. Because we know the x and y values and we know that a single quadrant can be described using a sin equation. From there we can trace the path of the function from x1 to x0. Each quadrant is identical so the path would be 4 times the length found in the sin transformation.
6:43 did some thinking on this one, it actually makes a ton of sense!! The key thing is to split the square root so that the numerator and denominator are rooted separately. The numerator is the Pythagorean theorem applied to the major and minor axes, so the value you get is the hypotenuse for the right triangle formed by the axes. Then, that gets divided by square root of 2… where’ve we seen that before? Sin(45) and cos(45)! Dividing by root 2 basically gives us the x and y components of the hypotenuse, ultimately averaging the axes in a very unique way. I’m impressed by the cleverness of this approximation, if I could choose which one was the exact formula for perimeter it’d be this one!
It's called the root mean squared
7:00 That is usually called "root-mean-square" (not usually hyphenated but I find it easier to read and more grammatically sensible with hyphens) and comes up in a lot of places. For example, the "voltage" number for the mains electricity in homes and buildings is the root-mean-square of the instantaneous voltages of waveform across one cycle (or equivalently across n cycles or, if you pretend the waveform is infinite, across the whole waveform). It is also the conceptual origin of least-squares regression. You want to minimize the root-mean-square of the errors. Since square-root is a monotonically increasing function, this is the same as minimizing the mean-square of the errors. In general, it is a computationally friendly and integration-friendly way to indicate something similar to average magnitude.
Many engineering programs even have an RMS function, even if in most of them it is trivial to define one yourself.
Thanks, I hate it
when i first saw him being oblivious of the rms, i assumed he is joking. there is no way he doesnt know that an rms is well known average
@@Mayank-mf7xr it has nothing to do with maths, so, I'm not sure what you're expecting
@@YounesLayachi XD. there isn't a single universe where mathematicians, those too of caliber of Matt, wouldn't know of rms. that is something even a petty high schooler knows. Matt was obviously joking.
"I only know juuuuust enough mathematics to be dangerous" - Matt Parker
love ya man i just love ya thanks for the video it was great
Could you work out (or approximate) the equation for the line error percentage vs eccentricity, and then use that to modify the value you get?
I read the title by mistake as perimeter of an eclipse. And I was like “that’s a silly mistake to make” But then noticed 5:00 and I’m like okay, great, I’m not the only one.
Wdym?
@@bozzigmupp510 He says "eclipse" instead of "ellipse" at those times.
Looking at Matt's monstrosity of an equation next to Ramanujan's elegant simplicity makes me feel like there should be a sensor bar over it!
😂😂😂
Lmfao same here 😂😂
What does the Wii have to do with this?
And Ramanujan did it without the help of computers or calculators. Even without all these means he just smashes Matt's approximiation formula's. He truly was on another level entirely!
So interesting. A small point: I would have liked a quick reminder of the formula for 'h'.
At 14:20 in the video Matt shows an exact solution using an infinite series. How were those numbers generated?
Can't tell you the top, but the bottom looks like a Fibonacci sequence but multiplied instead of added
@@Kai-K Yes, the denominators are 2 raised to these exponents: 0, 2, 6, 8, 14, so it seems like kind of doubled Fibonacci numbers. And why is it suddenly 25 in the numerator?? I have searched a lot and I can not find anything about this series expansion. It is a big shame that he just throws out something like this without telling what it is. 😠We shall not guess riddles. Matt, if you read this, please explain!
@@svergurd3873 @norvegicusbass The series is found on the Wikipedia page for ellipse. It contains the ratio between successive double factorials, which explains the powers of 2 at the bottom
@@aloneitan3819 Thank you very much! Now it is clear.
@@Kai-K Huh interesting, in the MMO "Warframe"'s latest update there is an NPC called Fibonacci, and also something called the Kalymos sequence.
My approximation: "4a". Work great if a is huge compared to b. The error goes to 0 then
I wonder at wich point it becomes better than the best approximation we have
I have an approximation that works perfectly if a=0
Genius. Now try to sell it to NASA.
Oooh, you jusye made me realize how ridiculous it is to measure the approximation relatively to excentricity
(( 2rPi-4r)a/r)+4r where a is always smaller than r, wrong ?
"Who has an ellipse 75 times long than it is high?" Laughs in comet inbound from the Oort cloud.
12:58
so you know how to steal from the comments section
great job! It would be interesting if instead of looking for a general formula for the length of the ellipse, look for a generalization of the concept of pi (ratio between length and diameter a+b) and plot it as a function of h or e
Great stuff!!!
2:33 “super extreme” is an understatement. It’s literally an ellipse where the ratio of a to b is infinite
That can be achieved by setting b to zero. Essentially it's a straight line of infinite length.
@@DavidSmith-vr1nb Or 2 straight lines if b is not zero My bad, I was wrong. It's actually a parabola.
@@Owen_loves_Butters lol
@@DavidSmith-vr1nb Not of infinite lenght. If b=0, then the line is of lenght 2a. The perimeter is 4a, btw.
@@juanausensi499 the point was that the ratio is infinite, not the length Edit: my bad, misread the comment you replied to ....
6:54 Root mean square? I mean that would be the fourth most common mean after arithmetic, geometric and harmonic mean.
Yeah, the quadratic mean. I remember studying the hierarchy of which mean is greater when the values used differ from one another.
What about trimmed mean?
@@peterflom6878 That's more for messy real world data, whereas the others actually turn up in many exact formulas.
ooh whats harmonic mean that sounds fun! my first guess would be 1/(1/a + 1/b)
@@joeyhardin5903 almost. I guess you meant 2/(1/a + 1/b).
I love the editing on this…
You could also just solve the arc length. Write the equation of the ellipse in terms of y, take the derivative, put it into √(1+f'(x)^2) and then take the appropriate definite integral.
You still have to approximate it though. Because it's not possible to get the exact value even with integration. Because you can't express the Integral in elementary functions.
@@playerscience, lol, no. You can easily count exact length of a curve by mentioned formula using integrals, with only exception - some special functions. Video is kinda nice, but real (not clickbate) topic should be named "some approximations for the ellipse perimeter for those who afraid of intergrals"
@@cyrillsergeev8163 lol, no. This is one of those 'special functions'. You literally can't integrate the resulting function without using elliptic integrals which, as the previous commenter pointed out, cannot be expressed in terms of elementary functions. Unless you have access to some special understanding that eludes the rest of the worldwide mathematical community, in which case: please enlighten us, o great one.