Projective line

The (real) projective line is the projectivization of ${\mathbb{R}}^2$:

Equivalently, it is the space of 1-dimensional linear subspaces (lines through the origin) in ${\mathbb{R}}^2$.

Coordinates / affine chart

Choosing a decomposition into an affine chart identifies ${\mathbb{RP}}^1$ with $\mathbb{R}\cup \{\mathrm{\infty }\}$.

In homogeneous coordinates $[x:y]$, the affine chart $y\ne 0$ is given by $t=x/y$, and the point at infinity corresponds to $[1:0]$.

Automorphisms

Projective automorphisms of ${\mathbb{RP}}^1$ are induced by invertible linear maps of ${\mathbb{R}}^2$ modulo nonzero scalars, i.e. by $\mathbb{P}GL(2,\mathbb{R})$.

See:

• projective space
• projective linear group
• projectivity

See also projective line, cross-ratio, and invariants.