What is the equation that fits this curve?

Question

I have a curve that looks like this (it's cyclical):

Curve

I can get a partial fit by fitting a 3rd degree polynomial, but I have a feeling there must be a better fit (something that involves sin & cos).

The fitted $a*t^3 + b*t$ curve looks like this:

Fitted Curve

Any ideas? thanks.

Answer 1

You could try to fit higher-order polynomials, which should give you the flexibility needed to ensure differentiability or some such constraint. Or if you're willing to try parametric curves, you can get a fit relatively easily, like $$x(t)=\sin(t)+t, y(t)=\sin(t)$$

I assume that you explicitly want the skewed nature of the wave in your picture. Tuning the amount of skew, height, etc. in the parametric formula is just a matter of tweaking a few parameters.

Answer 2

Assuming $\Delta \approx \dfrac{\hat\tau - \tau}{2}$, then we have:

$$ f(\Delta ) \sin \left(\frac{\pi x}{2\Delta}\right) $$

Where $f(\Delta)$ is the value of the function at $\Delta$.