English: Unit circles in various metrics arising from the alpha-max-plus-beta-min approximation to the magnitude of a vector, with an ideal L2 circle for comparison. The rest of the circles are equivalent to the portion shown here by reflections. The optimal values are ${\displaystyle \alpha ={\frac {2\cos {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}}$ and ${\displaystyle \beta ={\frac {2\sin {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}}$.