I am working on an investigation where I compare different paths of a surfer paddling out to catch a wave with a perpendicular current. I found an equation for one of these ways in which the surfer keeps pointing towards the destination, which can be written in the following two ways:
$$y=x\sinh\left(\operatorname{arcsinh}\left(a\right)-b\ln x\right)$$ and
$$\frac{x}{2}(y_0(x^k+x^{-k})+\sqrt{1+y_0^2}(x^{-k}-x^k))$$
I made a model at home using some blowdryers to try and see whether I could make a similar curve.
Above is a comparison of the three, with the purple curve being the equation as the red but with a different value for b (in the first equation) and the brown- the path drawn by the object I pulled at.
My question is: Is there any way I can try and find how "well" the purple curve fits the brown one? I'm thinking of taking a few points for example on the brown one and doing linear regression to find some r value. Would that be the best way? My reason for doing this is to be able to compare how good both of the two values for b in the first equation are at fitting the brown one.
By the way, the surfer begins on the right side of the equation.