Parametric Curve(s)
$$$
\forall\alpha\in\{0,2\pi\}\quad\;
\quad0\leqslant\;k_{a}\;\leqslant0.5\quad\;$$$
$$$ \left \{ \begin{array}{r c l}
\theta(\alpha) & = &\alpha\\
r(\alpha)& = &k_{a}\times\left(\cos^{0.64}{(\frac{\alpha}{2})}-\frac{4}{3,86}\right)
\end{array}\right .$$$