Projective Geometry
Cross-Ratio
(A, B; C, D) = ((C−A)(D−B))/((C−B)(D−A)) — preserved by every projective transformation
The cross-ratio of four collinear points A, B, C, D (or four concurrent lines) is the value (A, B; C, D) = ((C−A)(D−B))/((C−B)(D−A)) — the unique projective invariant of four points on a line. Every projective transformation of ℝℙ¹ (Möbius transformations on ℂℙ¹) preserves the cross-ratio. Discovered in classical geometry (Pappus, ~340 CE); central to 19th-century projective geometry (Möbius, Cayley, Klein). It identifies which configurations of points are projectively equivalent: two quadruples on different lines are projectively equivalent iff their cross-ratios match. Generalizes to higher dimensions and complex variables; the cross-ratio is the "projective coordinate" giving the canonical parameter on ℝℙ¹.
- Formula(A,B;C,D) = ((C−A)(D−B))/((C−B)(D−A))
- InvarianceUnder projective transformations
- First studiedPappus ~340 CE
- Modern theoryMöbius, Cayley, Klein 1800s
- Möbius mapsPreserve cross-ratio on ℂℙ¹
- Permutation orbitSix values per quadruple
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Why cross-ratio matters
Felix Klein's Erlangen program (1872) classified geometries by the group of transformations they preserve, and the cross-ratio is the smallest non-trivial invariant of the projective group on the line. Any time you have four collinear points and a transformation that might warp or perspectivize them, the cross-ratio is the first thing to compute — it tells you what survives.
- Projective geometry. Cross-ratio is the only numerical invariant of four points on a line under projective change of coordinates — every other quantity (distance, midpoint, ratio of distances) gets distorted by perspective.
- Perspective drawing. Renaissance painters used the harmonic relationship implicitly when constructing receding tile floors; the cross-ratio formalizes why a checkerboard projects to a pattern that still looks "regular" in a precise sense.
- Computer vision homography. Camera calibration, single-view metrology, vanishing-point detection, and the projective reconstruction stage of structure-from-motion all rest on the fact that image cross-ratios equal world cross-ratios.
- Hyperbolic geometry. The hyperbolic distance in the Klein and Poincaré disc models is a logarithm of a cross-ratio with two boundary points — the cross-ratio is the bridge between projective and non-Euclidean metrics.
- Möbius transformations. On the Riemann sphere ℂℙ¹, Möbius maps are exactly the holomorphic bijections, and they are characterized as the maps preserving the complex cross-ratio.
- Algebraic geometry. The j-invariant of an elliptic curve y² = x(x − 1)(x − λ) is a symmetric function of the six cross-ratio values — the moduli space of four ordered points on ℙ¹ modulo projective equivalence is itself ℙ¹.
- Conic-section recognition. Steiner's theorem identifies a conic as the locus of intersection of corresponding lines from two pencils in projective correspondence — cross-ratios glue the construction together.
- Synthetic projective proofs. Pappus's hexagon theorem, Desargues's theorem, and Pascal's mystic hexagram all admit short proofs that ride on cross-ratio identities — the modern descendant of Pappus's original 4th-century construction.
- Photogrammetry and aerial mapping. Reconstructing a building's facade from a tilted photograph uses cross-ratios along facade edges to recover unknown distances without surveying every point — the same principle used in single-view metrology since the 1990s.
Common misconceptions
- "Cross-ratio preserves distance." It does not. Lengths, ratios of lengths, and even orderings can change under projection; only the specific quartic combination defining the cross-ratio survives.
- "Cross-ratio is always real." True only on the real projective line. On ℂℙ¹ the cross-ratio of four complex points is generally a complex number, and its argument encodes whether the four points are concyclic (cross-ratio is real iff they lie on a circle or line).
- "It depends on the parametrization." The numeric value of (A, B; C, D) computed with affine coordinates is independent of the choice of affine chart on ℝℙ¹: the fractional linear maps relating two charts cancel out exactly.
- "All 24 permutations give different values." They produce only six distinct values forming a group isomorphic to S₃, with two special cases (harmonic λ = −1 and equianharmonic) where the orbit is smaller.
- "It works on any four points in the plane." The classical cross-ratio requires the four points to be collinear (or on a common conic, with a generalized definition). For four points in general position you need a different invariant — there is no projective invariant of four arbitrary planar points.
- "It's just a curiosity from old geometry." Cross-ratio underlies modern projective reconstruction in computer vision, the j-invariant in arithmetic geometry, the hyperbolic distance formula, and the moduli of marked points on the line — it is alive throughout current research.
Frequently asked questions
Why is the cross-ratio invariant under projection?
Projective transformations of the line ℝℙ¹ are exactly fractional linear maps t ↦ (at + b)/(ct + d) with ad − bc ≠ 0. A direct calculation shows that substituting four such expressions into the cross-ratio formula leaves the value unchanged: the (ad − bc) factors in numerator and denominator cancel. Geometrically, projecting a line from one point onto another line preserves incidence and the cross-ratio of any four collinear points, even though distances and ratios of distances both change.
What is the projective line ℝℙ¹?
ℝℙ¹ is the set of one-dimensional linear subspaces of ℝ², equivalently the real line ℝ together with one extra point at infinity, equivalently a topological circle. It is parametrized by homogeneous coordinates [x : y] modulo nonzero scalar, which collapses to the affine coordinate t = x/y wherever y ≠ 0 and to ∞ when y = 0. The cross-ratio is well-defined on ℝℙ¹ even when one of the four points is the point at infinity (interpret the corresponding factor as a limit).
How many distinct values can a cross-ratio take when permuting points?
Six. The 4! = 24 permutations of A, B, C, D produce values that fall into orbits under a group isomorphic to S₃, so a generic cross-ratio λ has six distinct images: λ, 1/λ, 1 − λ, 1/(1 − λ), λ/(λ − 1), and (λ − 1)/λ. Special configurations have fewer: the harmonic value λ = −1 has three, and the equianharmonic values (sixth roots of unity) collapse to two.
What is the harmonic cross-ratio?
Four points are in harmonic position when their cross-ratio equals −1. Equivalently, C and D divide the segment AB internally and externally in the same ratio. Harmonic conjugates appear classically as the two diagonals of a complete quadrilateral cutting any given line in a harmonic quadruple, and they are the projective analog of midpoint plus point at infinity. Harmonic conjugation is a self-inverse involution of the line.
How is it used in computer vision (camera calibration)?
A pinhole camera maps the world projectively onto its image plane, so cross-ratios of collinear world features survive in the image. Surveyors use this to recover real-world positions from photographs without metric calibration: take any line in the scene with three known points and a fourth unknown, the image cross-ratio equals the world cross-ratio, and one equation solves for the missing position. The same identity drives single-view metrology, vanishing-point detection, and projective reconstruction in multi-view geometry.
What's the cross-ratio of four concurrent lines?
The cross-ratio of four lines through a common point P is defined by intersecting them with any transversal not through P and taking the cross-ratio of the four intersection points. The result is independent of the choice of transversal. This dual viewpoint is what makes projective invariance so powerful: a pencil of four lines can be cut in many ways, and every cut sees the same numerical fingerprint.