Fundamental theorem of projective geometry (on ${\mathbb{P}}^1$)

Let $A,B,C$ be three distinct points on a projective line, and let $A^{\prime },B^{\prime },C^{\prime }$ be three distinct points on (possibly another) projective line.
Then there exists a unique projectivity $f$ such that

Remarks

• This is the statement that a projective coordinate on ${\mathbb{P}}^1$ is fixed once you choose three distinct reference points.
• One can choose affine charts and represent $f$ by a $2\times 2$ matrix up to scalar, which matches the “3 degrees of freedom” intuition.

Uses

• Defines the cross-ratio by normalizing $(A,B,C)$ to $(0,1,\mathrm{\infty })$.

See also:

• projective line
• projectivity
• cross-ratio