the parabolic trig functions

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[Kram]

some interesting weirdness about these:
- neither of them are symmetric around the y-axis
- they instead have symmetries around x=1
- cosp(x) is *antisymmetric*
- sinp(x) is *symmetric*
All of that seems weird to me.

They also have two intersection points:
2/3 (2 - sinh(3 log(1/4 (-1 ± sqrt(5) + sqrt(22 ∓ 2 sqrt(5))))

at 0.63661 and 4.3634

[Albeit_Jordan]

totally read this as 'pentatonic trig function' and thought you were taking mathcore metal to a whole new level

[boanergesaraujo]

Do we get a new trig family if we use "i" as the constant, on f"=i.f?

[sophiophile]

The hyperbolic trig functions have really useful applications in relativity. Are there any places where parabolic trig simplify physics (from a math or explanatory standpoint)?

[officiallyaninja]

Now i just need the elliptical trig functions to complete the set!

[sinecurve]

Also have elliptic trig functions as well sn(u), cn(u), and dn(u).

[BikeArea]

7:56 c1*t+c2*t should read c1*t+c2, shouldn't it?

[MatthewDickau]

Changing up the defining property results in a simpler (albeit somewhat boring) definition:

In complex numbers with the usual imaginary unit i where i^2 = -1, exp(it) = cos(t) + i sin(t)
In split-complex numbers with a unit j where j^2 = 1, exp(jt) = cosh(t) + j sinh(t)
By analogy, in a new number system with a unit k where k^2 = 0, we could define exp(kt) = cosp(t) + k sinp(t)

Which results in cosp(t) = 1 and sinp(t) = t.

[Jordan-zkwd]

Damn I really wanted to see a plot of the parabolic trig functions tbh ;-; edit: the equations provided let you plot them easily though!

[matiasmardijs]

Correct me if I'm wrong. But this video has a serious mistake. You only asked for the area condition to be true. You never asked the Acosp(t) + Bsinp(t)=c1t +c2 to be true. So the second derivative is not 0 as you pointed. Besides, for me it would have been interesting if cosp(t) and sinp(t) independently had 0 as their second derivative, and formed a base for the solutions to f''=0. But that would force cosp and sinp to be at most linear. So (cosp(t), sinp(t)) wouldn't parametrize a parabola. So all of this stuff turns out to be completely useless.

[tomholroyd]

That so amazing! Three types of conic sections. I've studied hyperbolic, and circular; parabolic is unknown to me this is so cool

[thelocalsage]

is using the relation y = 1-x^2 the best way to construct parabolic trigonometry? i’m thinking about the conic sections geometrically and i feel like the concavity of the parabola should go in the same direction as the hyperbola. some other elegant features like all the foci staying on the x-axis would make sense to me too if you think of smoothly moving the cross-section from circle to hyperbola.

[eliyahzayin]

I've done something similar but with the parabola defined by y^2=1-x, as it preserves cos as an even function and sine as an odd function. In terms of elegance, though, the best way to define a set of parabolic trig functions would be to create a general conic set paramaterized by the ecentricity and the focci position such that the regular and hyperbolic functions are special/limited cases.

[maestrobrutalizador]

Every conic can define a set of two trigonometric functions using the Minkowski space and the cone of light together with a plane at an angle that intersect it to define a conic. Then the projections on this plane of a parametrization by arc length of the conic (respect to some chosen initial point where you are measuring the length) can define trigonometric functions. Using this I get cosp t = t^2/2 and sinp t = t using as the reference point where the parabola is symmetric.

[wyboo]

we know that in the complex numbers, where i^2=-1, we get:

exp(ix)=cos(x)+i sin(x)

in the split-complex numbers, where j^2=1, we get:

exp(jx)=cosh(x)+j sinh(x)

for what nonreal k do we get:

exp(kx)=cosp(x)+k sinp(x)

going to try and work out later today

[benhbr]

So what's the connection with the DE f''(x) = 0? Shouldn't sinp ans cosp satisfy it, i. e. be linear functions?

[edwardlulofs]

Thanks. This is a super explanation of a math paper that I likely would not have understood on my own.
I found this very interesting. A good change of pace from the normal problem solving.
If I had had you for a math prof perhaps I would have become a mathematician instead of a physicist.
I liked your notation more than the papers notation but I understand why a pro would want to change the notation for publication.
😊

[TacoDude]

17:57 and 21:57, the second derivatives of these are not 0.

[ultrametric]

This would be interesting to present from the perspective of projective geometry and groups of motions. Essentially it must be a choice of metric in the Cayley-Klein sense. Also I smell line geometry :) Very fun!! Never had heard of this before.

[tommyyuan]

Why t/2? For hyperbolic and parabolic cases