Enter four distinct finite signed coordinates a, b, c and d, using the same origin, direction and scale. Negative coordinates and decimals are allowed. Inputs use the registered unit 1; the result is dimensionless. Keep the point labels in the intended order rather than sorting the coordinates.
These one-dimensional coordinates already describe positions on one line, so no collinearity checker or image input is needed. Points at infinity are excluded.
Point ordering can change the cross ratio. When comparing two configurations, preserve the correspondence of a, b, c and d and use the same formula convention.
The cross ratio is preserved under a nonsingular projective transformation of the line. This makes it useful for projective-geometry exercises and checking perspective invariants when all four original and transformed coordinates are finite. Background: The cross ratio — Stanford.
This tool evaluates a four-point invariant. It does not apply coordinate transformations, calculate a midpoint or perform ordinary ratio scaling.
| a | b | c | d | Cross ratio |
|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4/3 ≈ 1.33333333 |
| 1 | 0 | 2 | 3 | 3/4 = 0.75 |
| -3 | -1 | 1 | 3 | 4/3 ≈ 1.33333333 |
| 0 | 2 | 1 | 3 | -1 |
All four coordinates must be distinct. Check this before interpreting the result: this arithmetic calculator does not enforce every pairwise inequality. Repeated coordinates can produce a numerical value even though they fall outside the stated four-distinct-points input requirement.
If a = d or b = c, the denominator is zero and the result is unavailable, not zero or infinity. Numeric-range failures also produce unavailable results and propagate through dependent calculations. Very large coordinates or very small differences can exceed numerical limits or lose precision. The displayed decimal is rounded to eight places; numerical agreement alone is not a proof of a projective relationship.