Parametric Curve(s)

$$$ \forall\alpha\in\{0,2\pi\}\quad\; k_{a}\in \mathbb{R}\quad\; k_{b}\in \mathbb{N}\quad1\leqslant\;k_{a}\;\leqslant15\quad\;$$$
$$$ \left \{ \begin{array}{r c l} \theta(\alpha)& = &\cos^{k_{a}}{(\alpha+k_{b}+t)} \\ r(\alpha)& = &0.23\times\sin{\alpha} \end{array}\right .$$$