= Two-chart smooth atlas of an ellipsoid
If $D=\operatorname{diag}(a_1,\ldots,a_{n+1})$ is invertible, $x\mapsto Dx$ sends the <ellipsoid> $\sum_i a_i^2x_i^2=1$ to the unit <sphere>. On $u=(v,s)\in S^n$, the two <stereographic projections> $v/(1-s)$ and $v/(1+s)$ omit opposite poles and have inverses $(2w/(1+|w|^2),\pm(|w|^2-1)/(1+|w|^2))$, with the sign chosen for the appropriate pole. Their overlap transition is $w\mapsto w/|w|^2$, smooth and involutive on $\mathbb R^n\setminus\{0\}$. Composing their inverses with $D^{-1}$ gives a two-chart <smooth atlas> of the ellipsoid. Its <subspace topology> is <Hausdorff> and has a <second-countable space> topology, so it is a <smooth manifold>. For $n=0$ the overlap is empty and each chart is a point.
Back to article page
