Projective line, cross-ratio, and invariants

This note is a short synthesis of basic projective-geometry ideas around ${\mathbb{P}}^1$.

1. The projective line and projectivities

• The projective line is ${\mathbb{RP}}^1=\mathbb{P}({\mathbb{R}}^2)$: lines through the origin in ${\mathbb{R}}^2$.
• An invertible linear map $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ induces a map on ${\mathbb{RP}}^1$, but $T$ and $\lambda T$ (with $\lambda \ne 0$) induce the same transformation. So a projective transformation is really an equivalence class.

These induced transformations are projectivities (aka homographies). In an affine chart they appear as fractional linear transformations (Moebius transformations).

2. Fundamental theorem

The Fundamental theorem of projective geometry says that three distinct points determine a unique projectivity: given $A,B,C$ and targets $A^{\prime },B^{\prime },C^{\prime }$, there is a unique projectivity mapping $A\mapsto A^{\prime }$, $B\mapsto B^{\prime }$, $C\mapsto C^{\prime }$.

This makes “choosing three reference points” the projective analogue of “choosing a coordinate system” on the line.

3. The cross-ratio as an invariant

Using the fundamental theorem, given distinct $A,B,C$ one can send them to $0,1,\mathrm{\infty }$ by a unique projectivity; the image of a fourth point $D$ is the cross-ratio $(A,B;C,D)$.

The cross-ratio is invariant under projectivities: applying a projectivity to $(A,B,C,D)$ does not change $(A,B;C,D)$.

4. Perspectivities

A perspectivity is defined by projecting from one line to another through a center $O$ in a plane.

• Every perspectivity is (in suitable homogeneous coordinates) a projectivity.
• Not every projectivity is a single perspectivity, but projectivities between distinct lines can be written as a composition of at most two perspectivities (and a line-to-itself projectivity may require three).

Entry points

• projective line
• projectivity
• cross-ratio
• perspectivity
• projective linear group
• invariant