Projectivity

A projectivity (also called a homography / projective transformation) of a projective space $\mathbb{P}(V)$ is the map induced by an isomorphism $T\in GL(V)$, considered up to multiplication by a nonzero scalar.

Equivalently, projectivities are the elements of projective linear group $\mathbb{P}GL(V)$ acting on projective space $\mathbb{P}(V)$.

Projectivities of the projective line

For $\mathbb{P}({\mathbb{R}}^2)={\mathbb{RP}}^1$, a projectivity is determined by a matrix

up to scalar multiple. In the affine chart $\mathbb{R}\cup \{\mathrm{\infty }\}$ it acts as a fractional linear map

Such transformations can send finite points to $\mathrm{\infty }$ and vice versa.

Invariants

A key invariant of projectivities of ${\mathbb{P}}^1$ is the cross-ratio.

See also:

• projective line
• Fundamental theorem of projective geometry
• projective line, cross-ratio, and invariants