So I am doing some astronomical calculations for fun and it turns out that the approximate path that the sun traces in the sky should look something like: $$ \Gamma(t) = \left(\arctan\left(\frac{c_{i}\sin\left(t\right)}{c_{l}s_{i}-s_{l}c_{i}\cos\left(t\right)}\right), \frac{\pi}{2}-\arccos\left(c_{l}c_{i}\cos\left(t\right)+s_{l}s_{i}\right)\right) =: (\alpha(t),\beta(t)) $$ where $c_i, s_i, c_l, s_l$ are constants (sine and cosine of the Earth inclination and my location's latitude) so that $s^2_i+c^2_i=s^2_l+c^2_l=1$.
The component $\alpha(t)$ should be the clockwise angle that the sun makes with the South direction at time $t$ (the azimuth minus $\pi$) and $\beta(t)$ is the angular height at time $t$. The time $t=0$ represents the local noon.
The algebraic formulation of the path is horrible. Plotting $\Gamma(t)$ for each of the 12 months (i.e. changing the parameters $c_i$ and $s_i$) the 6 resulting paths (there are 6 paths because each path counts for 2 different months, the lowest path is December-January, the highest is June-July) are relatively nice looking continuous curves:
but (due to the mean $\arccos(t)$ I guess) I had to stitch together four traslated versions of $\Gamma$ to make this picture, that is: \begin{align*} (\alpha(t),\beta(t))\quad &|\quad-\pi<t<0,\quad&\alpha(t) > 0 \\ (\alpha(t),\beta(t))\quad &|\quad 0<t<\pi,\quad&\alpha(t) < 0 \\ (\alpha(t)-\pi,\beta(t))\quad &|\quad 0<t<\pi,\quad&\alpha(t) > 0 \\ (\alpha(t)+\pi,\beta(t))\quad &|\quad-\pi<t<0,\quad&\alpha(t) < 0 \end{align*} Is there a way to somewhat "prettify" the poor $\Gamma(t)$ and plot it as a single path?