Parametric Curve(s)
$$$ \forall\alpha\in\{0,2\pi\}\quad0\leqslant\;k_{a}\;\leqslant0.227$$$
$$$
\left \{ \begin{array}{r c l}
\theta(\alpha)& = &0.2\times\tan{(0.1+k_{a}\cdot\alpha)} \\
r(\alpha)& = &0.2\times\cos{(\alpha)}+\frac{t}{3}
\end{array}\right .$$$