Parametric Curve(s)

$$$ \forall\alpha\in\{0,2\pi\}\quad\; 0.3\leqslant\;k_{a}\;\leqslant1\quad\;$$$
$$$ \left \{ \begin{array}{r c l} \theta(\alpha)& = &\cos{(\alpha)}+\pi\cdot\sin{(\alpha)}+\frac{t}{2} \\ r(\alpha)& = &k_{a}\times\frac{\alpha}{50} \end{array}\right .$$$