Standard affine atlas of real projective space
= Standard affine atlas of real projective space
The $n+1$ <manifold charts> $U_i=\{[X]:X_i\ne0\}$ have coordinates $X_k/X_i$, with $k\ne i$. Their inverse inserts $1$ as the $i$th homogeneous coordinate. On an overlap, changing the denominator from $X_i$ to $X_j$ divides each remaining coordinate by the nonzero coordinate $X_j/X_i$. These rational <smooth transition maps> give <Real projective space> its standard <smooth manifold> structure.