How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize.
I think we may need to divide it into 8 pieces and consider them respectively.
2026-04-04 10:59:14.1775300354
Parametrize $|x|+|y|+|z|=1$
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Then as you pointed out, there are 8 possible cases. $$ \begin{align} 0\le s \le 1, 0 \le t \le 1-s &: x=s, y=t, z=1-s-t \\ 0\le s \le 1, 0 \le t \le 1-s &: x=s, y=t, z=s+t-1\\ 0\le s \le 1, 0 \le t \le 1-s &: x=-s, y=t, z=1-s-t \\ 0\le s \le 1, 0 \le t \le 1-s &: x=-s, y=t, z=s+t-1\\ 0\le s \le 1, 0 \le t \le 1-s &: x=s, y=-t, z=1-s-t \\ 0\le s \le 1, 0 \le t \le 1-s &: x=s, y=-t, z=s+t-1\\ 0\le s \le 1, 0 \le t \le 1-s &: x=-s, y=-t, z=1-s-t \\ 0\le s \le 1, 0 \le t \le 1-s &: x=-s, y=-t, z=s+t-1\\ \end{align}$$