I tried to plot the parametric surfaces from this paper to try to play around with the variables and potentially adapt it for other purposes. However I attempted to plot the surfaces in both Geogebra and Math3D with the parameters from the original paper, to minimal success. I have made sure to convert the appropriate values to radians, however the shapes are still completely wrong. I am not sure if I am misinterpreting the values or if there is a mistake in the equations and functions.
Geogebra: https://www.geogebra.org/calculator/t9smcgaq
Math3D: https://www.math3d.org/LqpWldNxh
The parameters I used are the ones for the nautilus.
The parametric equation in question is: $$ \begin{cases} x = \bigl( A\sin(\beta) + g(s)\cos(s+\phi)\cos(\theta+\Omega) - g(s)\sin(\mu)\sin(s+\phi)\sin(\theta) \bigr) \, e^{\theta\cot(\alpha)} \\[2pt] y = \bigl( -A\sin(\beta) - g(s)\cos(s+\phi)\sin(\theta+\Omega) - g(s)\sin(\mu)\sin(s+\phi)\cos(\theta) \bigr) \, e^{\theta\cot(\alpha)} \\[2pt] z = \bigl( -A\cos(\beta) + g(s)\sin(s+\phi)\cos(\mu) \bigr) \, e^{\theta\cot(\alpha)} \end{cases} $$ for $\theta\in (\theta_{\mathrm{start}},\theta_{\mathrm{end}})$ and $s\in (s_{\min},s_{\max})$ and $$ g(s) = \bigl( a^{-2}\cos^2(s) + b^{-2}\sin^2(s) \bigr)^{-1/2} $$
The meanings of the variables are lined out in the paper.
The issue you had was that the paper has an error in equations (9) and (10). The errata states that you need to add a $\cos(\theta)$ after the $A\cos(\beta)$ in (9) and a $\sin(\theta)$ after the $A\cos(\beta)$ in (10).
Your parametric equations then become:
$$ \begin{eqnarray} \begin{cases} x&=&\big(A\sin(\beta)\cos(\theta)+g(s)cos(s+\phi)cos(\theta+\Omega)-g(s)sin(\mu)sin(s+\phi)sin(\theta)\big)e^{\theta\cot(\alpha)}\\ y&=&\big(-A\sin(\beta)\sin(\theta)+g(s)cos(s+\phi)sin(\theta+\Omega)-g(s)sin(\mu)sin(s+\phi)cos(\theta)\big)e^{\theta\cot(\alpha)}\\ z&=&\big(-A\cos(\beta)+g(s)\sin(s+\phi)\cos(\mu)\big)e^{\theta\cot(\alpha)} \end{cases} \end{eqnarray} $$
I did that and changed your constants to exactly the constants in Table 1 for nautilus. I had to play around with $\theta_{start}$, since $-\infty$ does not work on the math3d or geogebra plotter.
This is what I finally came up with: https://www.math3d.org/maV8BHWE2. Not too bad.