Pi for parabolas -- the universal parabolic constant. 

That would be called Pa…

for sqrt(1+u²) i would always go with hyperbolic substitution with u=sinh(t). the double angle formulas for hyperbolic functions make the calculation very straight forward and lead to the equivalent result sqrt(2)+arsinh(1) which is arguably even nicer.

Fun fact, an alternate definition of an ellipse, and even a circle, based on that of a parabola, is as such. The construction is a focus at point (0,1), and a directrix circle passing through the origin with center "due north" of the focus. The definition, similarly to a parabola, is that the ellipse is the locus of all points equidistant from the circular directeix (measured at closest point) and the focus. This is exactly the same as the regular definition of an ellipse as that equality can be extended to be the sum-of-lengths constant between the focus and the directrix center (also called the alternate focus) equalling the radius of the directrix circle. It nicely extends to a parabola as said radius goes to infinity, a sort of 1/0 infinity where the other side produces hyperbolas for "negative" radii. The eccentricity is the distance between the two foci divided by the directeix radius, which indeed equals zero for a circle and approaches one as the shape approaches a parabola.

This is a nice refresher of what analytic geometry is all about which I always found was an odd addition to the title of my calculus texts even though none of my professors ever went out of their way to point out what exactly in our calculus text qualifies as analytic geometry.

Pi constant for all circles not so surprising, but would not expect the ratio of the two quantities you initially wrote down to be a constant for all parabolas! Pretty amazing. Don’t remember seeing this in analytical geometry.

I always like to get a feel for the value of these constants. In this case P = 2.295587...

Nice, TIL that asinh(1) = ln(1 + √2)

"All parabolas are similar. And there is only one true parabola." Parabolati Confirmed?

Wonderful. I did not know there exists some constant for parabola like the case of a circle. Thank you.

Great video, professor. It rang some bells from Calc II. As always, thank you.

This video was parabolamazing, and the high quality of Michael’s videos is always constant! 👍

It's pretty obvious that all parabolas are similar when you think about eccentricity. Ellipses have 0

It feels like the "natural" parallel of pi ought to be the ratio between the arc length and the latus rectum that bounds it. That just _looks_ more like a diameter and circumference. Bringing in the distance between the focal point and the directrix is just weird to me IMO.

A touch of math and two new (to me) bands that work well together What a beautiful day!

15:23

Nice result. Love it

another great video - thanks

Fun fact: There's no closed-form expression for the circumference of an ellipse. Seeing what went into finding the parabolic constant, I no longer find this surprising!

There should be a similar constant for the unit hyperbola (eccentricity sqrt(2)). The line perpendicular to the symmetry axis at one of the foci cuts off a "cap" from one of the branches.

Thanks!

Pretty mysterious, all these transcendental constants coming out of algebraic things.

Nice sir

I would just do a hyperbolic substitution, x = sinh(u). Then you immediately get the integral of cosh^2 from -arcsinh(1) to arcsinh(1). This is a standard integral, but you can do it by parts or by the double angle formula if you want. Also you get the answer in the neater form, sqrt(2)+arcsinh(1). You can always turn arcsinh into log if you want. I think that's easier and more straight forward than splitting up the integral in a non-obvious way, doing integration by parts, and then using the standard integral of sec.

This is something they never taught me at school, or at university. I am grateful for finding out about it. I checked it out on Wikipedia. Like π it is transcendental. Thanks.

It seems to me this is less of a parabola's "pi" and more of a parabola's "tau" as the focal length is the length of the semilatus rectum. It feels more natural to me to use the ratio between the arc and the latus rectum, like pi is for a circle... I wonder why that isn't used as the parabolic constant.

About 2.2955871 for the curious.

approx. 2.296 . The curve is approx. 1.15 times the length of the straight line (i.e. 15% longer). From the picture I would have guessed that the ratio is larger, like 1.3 .

It is of interest to note that the decimal expansion of this constant is listed in the online encyclopedia of an integer sequences www.oeis as sequence # A103710 and also that P/6 is the average distance between two random points in a unit square. An article in Wikipedia on the universal parabolic constant develops the value in about five lines. It is also of interest to note that if we change the constant to sqrt(2) - ln(sqrt(2)+1) we obtain what is referred to as the universal equilateral hyperbolic constant, a ratio of areas rather than Arc lengths which is discussed in sequence number A222362. So glad that I ran across you on RU-vid Dr Penn. You are an amazing mathematician, a wonderful teacher and just about the best math presence online one could find. Thank you for all your dedication and hard work

I thought Penn to be able to explain the magic of the q-series. I am waiting eagerly.

So there is a formula for the length of a parabola in terms of this constant? Like the circumference?

Fun. Thanks.

Hi, 8:14 : you can already take into account the parity of the function at this point, 9:45 : ok. May be we can define such a constant for hyperbolas as well, provide we only consider those with perpendicular asymptotes.

Another antiderivative for the secant function is the inverse gudermannian.

I wish i paused this video before you did it and tried for myself. Great question for someone in calc 2

At 0:21 this briefly becomes a Bill Wurtz video

Very interesting. Are there also constants for ellipses and hyperbolas ?

9:55 I normally just put an x in front of the derivative if I don't know the immediate answer then fill in the rest of the antiderivative. In this case sqrt(1+x^2) ~ x int[x dx] = 1/2 x^2: 1/2 x sqrt(1 + x^2) + ... => [x sqrt(1 + x^2) + arcsinh(x)]/2 + C

nice animations at the start

I think it's pretty interesting that the silver ratio pops up here, let alone as a logarithm.

the sqrt(1+u^2) for the parabola feels related to the equation of a half circle sqrt(1-x^2)

2:18 Didn't know that was the official definition of a parabola nice to know.

9:15 it’s the area of half a circle

satisfying. Is the parabolic constant in any way related to pi? can they be mapped into each other?

Awesome, it never occurred to me that parabolas had a constant. P=2,205587149392638...

can you define this as function for any 2 order curve depending on excentricity?

Is there a similar constant, or pair of constants, for a 3rd order polynomial?

so root2 + ln(1+root 2 ) is some good number and should be seen around more

Latus rectum? Damn near latus killed 'em!

I wonder if there is a way to prove that this ratio is constant the way Ancient Greeks did for circles, i.e., by using the method of exhaustion considering 2 arbitrary parabolas p and p'

Makes me wonder when this was discovered. I know Pi is an ancient discovery, but its proof doesnt require integration. Is there another proof of this that doesnt require 17th century math?

And,.. there are also some constants to ellipse and hyperbola???

9:53. √1+u^2 can be directly integrated by parts . No need of using trigonometric function . You used it and end up with integration by parts again 😅😅

👍

Well I am expecting Pi in the final answer

I found it easier to do a hyperbolic trig sub (x=2fsinh(Φ))

Does this have a listing on OEIS?

4.59117

Have you seen Matt Parker's "One True Parabola"? To be sought out if you haven't

Is this constant used, or perhaps of use, anywhere? Similar to the way pi shows up all over physics.

A physicist would say it’s equal to 2*sqrt(2)

Does anyone happen to know if this constant is trancendental?

Which is ~2.3 :D

Nice proposal. How are we gonna call that constant? - Pa? (Pi and Pa, Pa like "Parabola) - P = Penn's constant/number ?

Kinda weird that you can express the parabolic constant using square roots and the logarithm, while there is no such simple expression for the circle constant, pi

That constant us certainly irrational. But is it also transcendental? I apologize for my ignorance.

4:18 why can we assume the directrix and F are equal distances from the origin?

Amogus in the thumbnail!

Why can we assume the directrix is at y=-f?

No F's left

If we make u=(e^x-e^(-x))/2, it will be more interesting.

Can this be generalized to an arbitrary eccentricity?

7:11 "And this part here is 2f." Proof?

>define circle using the radius >define circle constant using diameter ??????

This has a ln in its value. Why do I feel pi can be written in terms of e somehow?

should be called ψ smh

define the latus rectum

Rectum 😂

He said 'rectum'.... he,he,he

hehe. Rectum.