In a certain exercise I have been asked to find an isometry between a portion of the cylinder $S = \{ x^2+y^2 = 2: 0 < z < 1\}$ and the complete cone $S_* = \{x^2+y^2 = 2z^2: 0 < z < 1 \}$. I need to find an explicit formula. However, I have really no idea of how to start. I would really appreciate if somebody could give me a hand. I might be able to find an isometry between that cone and the cylinder and the plane $z = 0$ for example, but not between those geometric figures I said above.
2026-03-27 23:57:25.1774655845
Isometry between cone and cylinder
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What do you think to the map
$$ F\colon \text{cone}\to \text{cylinder}, \qquad (x,y,z)\mapsto (x/z, y/z, z) $$