General form of the function: $$y=d\sinh^{-1}\left(\frac{ax+b}2\right)+c$$ I want the function to pass three points, $(0,0)$, $\left(\frac{t}2,\frac{g}2\right)$ and $(t,g)$, and I want the function to have a slope of $r$ at the $\left(\frac{t}2,\frac{g}2\right)$ point.
I want to rewrite the function above, but now expressed in $t$, $g$ and $r$.
Equation 1: $$y(0)=0$$ $$0=d\sinh^{-1}\left(\frac{b}2\right)+c$$ $$-c=d\sinh^{-1}\left(\frac{b}2\right)$$ Equation 2: $$\left(\frac{g}2\right)=d\sinh^{-1}\left(\frac{at+2b}4\right)+c$$ Equation 3: $$g=d\sinh^{-1}\left(\frac{at+b}2\right)+c$$ Equation 4: $$r=\frac{dy}{dx}$$ $$r=\frac{ad}{\sqrt{(ax+b)^2+4}}$$
Any comments will be greatly appreciated. Even stopping by is greatly appreciated!