Duality and cross-ratio

Try

On a projective line, ordinary distances and ratios of two differences usually change under a fractional linear transformation. A ratio built from four ordered points survives. With the convention CR(a,b;c,d)=((c−a)(d−b))/((c−b)(d−a)), the ordered quadruple (0,1;3,4) has cross-ratio 9/8.

The ordering is part of the data. Exchanging two entries may change the value. A theorem about cross-ratio must state its convention before applying a memorized formula.

Predict

Predict whether cross-ratio can be evaluated by treating infinity as an ordinary real number. Use a limiting or homogeneous formula instead. For example CR(a,b;c,∞)=(c−a)/(c−b).

Predict whether a projective point and a projective line are fundamentally asymmetric in their coordinate descriptions. Both are nonzero triples up to scaling, with incidence expressed by one bilinear equation. This is the source of duality.

Experiment

Compare ordered quadruples.

Ordered entries	Cross-ratio
(0,1;3,4)	9/8
(1,0;3,4)	8/9
(−1,1;0,∞)	−1

Compute the first row directly. In the second row, swapping the first two entries exchanges numerator and denominator. In the third, take the limit as the last coordinate tends to infinity. The value −1 is called harmonic for this ordering; that name does not mean every permutation also gives −1.

Understand

Let T(t)=(at+b)/(ct+d), with ad−bc≠0. When the denominators are nonzero, T(u)−T(v)=(ad−bc)(u−v)/((cu+d)(cv+d)). Substitute this identity into the four differences defining cross-ratio. Each determinant factor and each denominator factor cancels, proving invariance. Cases involving infinity follow by the homogeneous extension or a limit where defined. The theorem assumes four distinct projective points so the relevant denominator does not vanish.

For duality in RP², represent a point by a nonzero column p and a line by a nonzero coefficient column ℓ. Incidence is ℓᵀp=0. Interchanging the roles of the two columns leaves the scalar equation unchanged. Joining two points corresponds to intersecting the two dual lines. Thus an incidence theorem proved from self-dual projective axioms has a dual theorem with points and lines exchanged.

Apply

Repair “Cross-ratio preserves the distance between the first two points.” It preserves one scalar associated with an ordered quadruple, not a metric. A projective map can stretch one region and compress another while leaving every cross-ratio unchanged.

For optional depth, three distinct projective-line points determine a unique projective transformation sending them to any other ordered triple of distinct points. One can send the first triple to 0,1,∞ and invert the analogous map for the second. A fourth point is then constrained by cross-ratio. This explains why four points reveal an invariant where three do not.

The duality discussion concerns incidence, not Euclidean perpendicularity. Identifying vectors with covectors through a dot product is a chosen coordinate device, not an intrinsic metric on the projective plane.

Continue

Keep representatives, point ordering and exceptional denominators explicit in each calculation. Continue to projective conics to use homogeneous quadratic forms and their associated polar lines.