Parametric Curve(s)
$$$
\forall\alpha\in\{0,2\pi\}\quad\;
0.01\leqslant\;k_{a}\;\leqslant0.5\quad\;$$$
$$$ \left \{ \begin{array}{r c l}
\theta(\alpha)& = &\alpha \\
r(\alpha)& = &k_{a}\times\left (\sin^{0.4}{(\frac{\alpha}{2})}-\frac{4}{3.5}\right )
\end{array}\right .$$$