The Cross-Ratio

An invariant of perspective

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Cross-ratio: $\frac{(c-a)(d-b)}{(c-b)(d-a)}$.
• Perspective projection changes lengths but preserves the cross-ratio.
• It lets you recover true spacing, or the horizon, from a picture.

Steps

1. What projection preserves
2. What survives?
3. Place the fourth post
4. Find the horizon
5. Compute a cross-ratio
6. Unequal spacing