where did the pi go? area of a superellipse
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- Опубліковано 28 вер 2024
- calculating the area of a super ellipse. Here we find the area enclosed by a pseudo ellipse, which is a generalization of an ellipse but with n. Special cases include squares, circles, and ellipses. After a change of variables, we transform this geometric figure into an integral, which involves the beta function and which has been calculated previously using the Laplace transform. The answer includes the gamma function and factorials, but where did the pi go? This is a must see for calculus and math students, enjoy!
Laplace integral: • Laplace integral gone ...
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i like how the limit as n goes to infinity approaches the area of rectangle
Finally after all these years, we can calculate the area of a rectangle. All it took was Dr Peyam to use two gamma functions.
It‘s something I find really interesting and funny. You can calculate things like areas in infinite ways and it always ends up the same.
Wow amazing!!!
This, along the fact that the π in the area of the ellipse/circle comes from the formula containing the (Γ(3/2))² are both mindblowing.
Very nice. Going f to N=infinity gives Area=4ab , which is 4 times a rectangle with base a and height b.
Wow soooo cool!!!
actually i was thinking about this some time ago, i was able to find the area quite easily like in this video but the circumference is quite more interesting. nice video tho
Finding the perimeter should be interesting as well but much more complex. I expect it would involve some elliptic integrals.
really cool... as always!
gr8 stuff. this beauty made my day :D
Nice to see shirt of celebrating women in mathematics Dr Peyam! Any notable female contributors to analysis or PDEs we can get a shoutout too? I only know of Noether for Algebra-land.
I also know ladyzhenskaya and uraltseva
Why celebrate females specifically? Why not just celebrate the best mathematicians there have been in general?
Tangent question (for future video): We have the factorial, now we want to generalize to a continuous function, ie gamma. How do we do that?
Using integrals, look up the definition of the gamma function
@@drpeyam Sure, but procedurally, how do you figure that out?
What do you mean?
If you've had this thought, surely you must've heard of the gamma function. There's loads of videos on it
@@drpeyam We want to interpolate these values, and the very aggressive rise, do I just throw functions at it until one fits, or is there a better way? One that constructs the function analytically?
Nice!
Buen vídeo 😄
I like how he just erases the 4 from 4*pi*a*b after realizing his mistake ahaha
In the Chebyshev metric, the unit circle is a square 🤪
Now calculate the perimeter 💀
💀💀💀
Beta functions and ellipses!!!
A super fun video for the superellipse :D
The take away here is to.... Be Aggressive.
How do you not have 1 million subscribers yet, I dont know ...
I know, right?
@@drpeyam you'll get there soon, you deserve it! I really appreciate your videos.
Next the perimeter?
Neat! Looking at the formula I think I can see one of the definitions of e in the limit as n -> inf ....?
Hi Dr. Peyam. Thanks for sharing. Delicious as ever!
Once a friend of mine asked about a problem.
We all agreed to measure the surface In square units to fulfill any area. For irregular surfaces, like the circle, we use the integral function to calculate it and then pi comes out.
But what would happen if we used instead, unity circles (circles of radius 1) to determine a circle area?
This way would be more straightforward since we could simply find the correct radius to cover all its area. This way a generic circle would be any actual number. We could call this area R "round meters".
The question would then be: How to measure the surface, i.e. a square. using round meters? we could fulfill all the areas using smaller circles. In this way, would be a PI counterpart for the square? How would it be?
That’s what the point of measure theory is 😁
@@drpeyam exactly!
Like ALWAYS... 😎
A M A Z I N G . . . 🥳🥳🥳
the 4x3x2 could be a 4x2x3x1 which could also be translated into a gamma function lol
Hey, thanks a lot for the cool video! Can you help me to generalize this for two different exponents in the equation of the super ellipse?
The problem is that we lose the exponent that belongs to x I think when substituting.
And after that the value of the "du" term at the end of the Integral never gets revealed 👀
دكتور پايام أعجبني إشتقاقك هذا كثيراً فشكراً جزيلاً وبارك الله فيك
Where did you get the 1 from for the integral you created with u sub? Thanks!
Can we do it in higher dimension that would be super
What are the inflection points of this function?
very nice. but how about the circumference? i bet that would be a nightmare
really cool video no wonder the cops came
Well, the area of an ellipse is no sweat. But what is the circumfence of an ellipse.
The circumfence should be the derivative of the area. Why is that not the case?
Why can you not just go backward from the area function.
The other possibility is: What is the surface area of an ellipsoid?
The circumference of an ellipse is the derivative of the area with respect to the radius, which is the parameter that you can increase such that the curve shifts uniformly normal to itself. Unfortunately, shifting an ellipse normal to itself doesn't give you an ellipse, and there's no easy formula for the area of an ellipse plus a uniform normal coating.
Essentially, it's because an ellipse doesn't grow at a uniform rate. Using infinitesimals, if we go from a circle of radius r to one of r+dr, the area is increased by a strip of length 2 pi r and uniform width dr (begins waving hands), with an area of 2 pi r dr. However, if we similarly increase the size of an ellipse, the strip will either not be of a uniform width or the new ellipse will not have the same proportions as the original one.
The perimeter of an ellipse is hard. Stand-up Maths has a video on it.
Matt Parker Stand-Up Maths has a video on the perimeter of an ellipse. That is FUUUUUUUN!
Also, you say that the circumference is the derivative of the area, but *ONLY IF THE LINE IS EVERYWHERE MOVING AT RIGHT ANGLES TO ITS LOCAL DIRECTION* . That means that your shape will become closer and closer to a circle. It will no longer be an ellipse.
How about the following expression ?
| x | = a| cos t |^p,
| y | = b| sin t |^q.
Here a, b, p, q are positive constants.
Works too
Thank you soo much man!
Really cool!!
Is there a reason you chose 2 and 3 for the denominators?
RJ
Sorry, I didn't finish watching before I asked.
RJ
If N is odd, it will not form any elliptical shapes.
It does, have to use absolute values, as I mentioned
Its a infinite series with a rigorous proof? 😁😅
I’m gonna think about it 😄
No, you don't need a quarter of the superellipse. You can take an *EIGHTH* of it because you can re-write the equation as A^n+B^n=1 and then scale X and Y scales appropriately.
I did this within the first few weeks of my Freshman year in undergrad. Good memories. :)
Here's an idea i got from the title. Define an n-ellipse as the set of all points equidistant from n fixed points. This way we would have in R2 circle is the 1-ellipse, ellipse is the 2-ellipse and what would come next?
threellipse
The 3-ellipse is unfortunately just a point.
There's in fact only 1 point that is equidistant from 3 other points, which is the center of the circumference that passes through those 3 points
@@angelaross6235 Thanks, didn't realized that