In the past couple of days, I have seen a few social media mathematicians talk about Parabolic Trig Functions. Michael Penn has this video which follows this paper. I've also seen a number theorist on TikTok try to come up with something similar. While obviously good approaches, I don't really think these parameterizations should be considered "trig functions". And this is for much the same reason that we parameterize a hyperbola by area rather than arclength - the functions parameterized by arclength simply don't have nice properties.

The thing to note is that both circular and hyperbolic trig functions are basically homomorphisms from the additive reals to the conic. The circular trig functions are a homomorphism to planar rotations, and the hyperbolic trig functions are a homomorphism to real valued multiplication (this can be seen most clearly when we look at (e^u,e^-u) as the "trig functions" for the hyperbola xy=1). When viewed over complex numbers, planar rotations get absorbed in the arithmetic of multiplication which is why Euler's formula relates something clearly multiplicative (e^x) with the circular trig functions. Circles are just larval hyperbolas afterall.

Consequently, a "true" trig function for the parabola should act as a homomorphism from the additive reals to some other arithmetic natural to the parabola. This natural arithmetic on parabolas is actually addition. In fact, if (a,b) and (c,d) are points on y=x², then we can add them through the formula

• (a b) (c d) = (a b, b d 2ac)

That this is "natural" follows from the fact that there is a geometric way to reproduce both the circular and hyperbolic arithmetic and this is what you get when you apply it to the parabola (eg). There is then a very clear homomorphism from the reals to the parabola given by x -> (x,x²). This means that we get a very disappointing result in that if cosp(x) and sinp(x) are the parabolic trig functions, then we have cosp(x)=x and sinp(x)=x². These obey they additive relationship

• cosp(x y) = cosp(x) cosp(y)
• sinp(x y) = sinp(x) sinp(y) 2cosp(x)cosp(y)

If you know Lie Groups, then this geometric construction makes the parabola a Lie Group and this map is the exponential map from the Lie Algebra, just as it works with the other circular/hyperbolic trig functions. So, not a super fun result and ultimately probably disappointing, but I think it fits the parameters more naturally than these other constructions which start by defining the parameter first rather than finding the functions with certain additive properties. I haven't found a satisfying elementary geometric interpretation for this "parabolic angle" like the area under a curve. The only real things are the slope of the line from (0,0) to the point on the parabola, or just the distance from the point to the x-axis both of which seem pretty meh. (A disadvantage the other approaches do not have.)