Coordinate-projection atlas of an ellipsoid
= Coordinate-projection atlas of an ellipsoid
{title2=$q_i=\varepsilon\sqrt{1-\sum_{j\ne i}q_j^2}$}
For nonzero real $a_i$, scale the ambient coordinates by $q_i=a_i r_i$. On the half where $\varepsilon q_i>0$, omitting $q_i$ is a <manifold chart> onto the <open unit ball>; its inverse restores $q_i=\varepsilon\sqrt{1-\sum_{j\ne i}q_j^2}$. These $2(n+1)$ charts form a <smooth atlas> of the <ellipsoid>, including for negative $a_i$.