Relativistic Doppler effect

What you are seeing: the observed frequency ratio $f_\text{obs}/f_\text{src} = 1 / (\gamma (1 - \beta \cos\theta))$ as a function of viewing angle $\theta$ at fixed source speed $\beta = v/c$. The polar plot shows the same factor on radial axes: head-on approach ($\theta = 0$) gives the maximum blueshift, recession ($\theta = \pi$) gives the maximum redshift, and at $\theta = \pi/2$ there is a pure transverse redshift by $1/\gamma$ (no Newtonian analog).

Slide the speed to see the curve sharpen as $\beta \to 1$: the blueshift cone narrows toward forward beaming. The dashed red horizontal at $f_\text{obs}/f_\text{src} = 1$ marks where blueshift flips to redshift; it crosses zero at $\cos\theta = (1 - 1/\gamma) / \beta$.