The Cross-Ratio Ruler

Distance read off a picture, by the one number perspective cannot change.

A picture squeezes distances, and squeezes the far ones most; but four points on a line keep one number that no picture can change, their cross-ratio. Calibrate a ruler on three known marks and it reads true distance straight off the picture, wherever the painter stood.

Plate I. A straight road, a sign at ten metres, and a ruler graduated by cross-ratio.

The painter's distance The eye's height

the pins follow the picture show the naive ruler

Drag the staff to 30 m. It should stand beside the third post.
Change both dials. The gaps between the posts change; their cross-ratio stays 4/3.
Show the naive ruler, and find the one place past the near mark where it reads true.
Untick the pins follow, change the painter's distance, and read the staff; then lay the ruler again.

a.Three brass pins set the ruler: on the near mark (0 m), on the sign (10 m), and on the horizon, the picture of the point at infinity. They catch when dragged near their marks.

b.The graduation is cut from the three pins alone, and crowds toward the horizon as the road does.

c.The red staff stands on the road. Drag it; the ruler reads its foot.

d.The posts stand every ten metres, and fall on 10, 20, 30 of the ruler.

¶Commentary

Perspective is projective. A painter standing at distance d from the near mark, with the eye at height h, sends the point t metres down the road to the height y = y∞ + (y₀ − y∞)·d/(t + d) in the picture. Maps of the form y = (at + b)/(ct + d) are the projective transformations of a line, the group PGL(2, ℝ). They change lengths, and they change ratios of lengths.

The one invariant. For four points on a line, the cross-ratio (A, B; C, D) = (AC·BD)/(BC·AD), with signed lengths, is unchanged by every projective map. It does for PGL(2) what length does for sliding and angle does for turning, and it is the only such number: any quantity that perspective preserves is a function of it.

The ruler. Take the near mark, the sign, and the horizon, at 0, 10, and ∞ metres. For a fourth height y, the cross-ratio in the picture equals the cross-ratio on the road, and solving gives the distance. The scale the ruler carries is just this formula, cut into wood. A ruler laid on proportionally, as if the picture were drawn to scale, is right at the sign and wrong everywhere else, without bound toward the horizon.

On a plane. A projective map of the plane is fixed by where it sends four points in general position, so one tile, pinned at its corners, calibrates the whole floor. The map that carries the pinned tile back to a unit square carries every other point to its true place in tiles. Along any line of the floor the old one-dimensional ruler still works.

Harmonic sets. When the cross-ratio is −1 the four points are called harmonic, and D is the harmonic conjugate of C with respect to A and B. The complete quadrilateral of Plate III constructs it with a straightedge alone. The proof is a single sentence: project the figure so that the line KN goes to infinity, and the construction becomes a parallelogram, whose diagonals bisect each other.

Further. Albrecht Dürer's and Piero della Francesca's treatises on perspective; Desargues's Brouillon project (1639), where the harmonic construction first appears in this form; and the opening chapters of Coxeter's Projective Geometry.

Drawn for the Wending House, 2026. Drag anything that looks like brass.The first edition of this page