Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra

Enzo Harquin (LIGM); Pascal Monasse (ENPC); Stephane Breuils (LAMA); Venceslas Biri (LIGM); Vincent Nozick (LIGM)

arxiv: 2605.18398 · v1 · pith:DE7QGL7Inew · submitted 2026-05-18 · 💻 cs.CG · cs.CV

Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra

Enzo Harquin (LIGM) , Stephane Breuils (LAMA) , Pascal Monasse (ENPC) , Venceslas Biri (LIGM) , Vincent Nozick (LIGM) This is my paper

Pith reviewed 2026-05-19 23:27 UTC · model grok-4.3

classification 💻 cs.CG cs.CV

keywords cross-ratioprojective invariantPlane-Based Geometric AlgebraPGAhomographygeometric algebraprojective geometryduality

0 comments

The pith

Cross-ratios extend as a grade-agnostic projective invariant across all object types in n-dimensional PGA.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes explicit cross-ratio formulas that apply uniformly to geometric objects of every grade in Plane-Based Geometric Algebra, from finite points to hyperplanes and intermediate flats. These formulas use either the commutator or its dual as the measurement operator, chosen by configuration rather than grade, and recover the classical signed distance ratios for parallel cases along with sine cross-ratios for secant cases. A sympathetic reader would care because the construction supplies a single algebraic object that works in any dimension and supports defining homographies directly from measured invariants instead of coordinate bases. All eight configurations fall into four dual pairs under the Hodge dual, confirming the invariant behaves consistently across the algebra.

Core claim

In Plane-Based Geometric Algebra R(n,0,1), the cross-ratio is defined for any pair of objects using a canonical pairwise measurement operator that reduces to the commutator or the commutator dual according to whether the objects are parallel or secant. Explicit formulas are given for each object type and configuration; each formula recovers the appropriate classical invariant. The eight configurations organize into four dual pairs under the Hodge dual, establishing the cross-ratio as a single projective invariant that does not depend on object grade.

What carries the argument

The commutator (or commutator dual) as the canonical pairwise measurement operator that supplies the cross-ratio for arbitrary object grades and the eight geometric configurations.

If this is right

Explicit cross-ratio formulas exist for finite points, ideal points, hyperplanes, and intermediate flats in any dimension.
Measurement operators reduce to either the commutator or the commutator dual solely according to geometric configuration.
All eight configurations form four dual pairs under the Hodge dual.
The formulas supply a direct algebraic route to n-dimensional homographies defined from prescribed invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The uniform formulas could remove the need for separate handling of object types when implementing projective geometry routines in software.
The observed duality between configurations suggests that switching between primal and dual representations preserves the invariant without additional computation.
The same commutator-based construction might be tried in other geometric algebras to locate analogous invariants.
Numerical checks in low dimensions against known classical values for mixed-grade objects would provide immediate verification.

Load-bearing premise

The algebraic structure of PGA together with the Hodge dual and commutator operators extends the classical projective cross-ratio to every grade and every configuration without extra constraints or exceptions.

What would settle it

Take two secant lines in 3D PGA, compute the proposed commutator-dual formula, and check whether the numerical value matches the sine cross-ratio obtained from independent classical angle measurements on the same lines.

Figures

Figures reproduced from arXiv: 2605.18398 by Enzo Harquin (LIGM), Pascal Monasse (ENPC), Stephane Breuils (LAMA), Venceslas Biri (LIGM), Vincent Nozick (LIGM).

**Figure 1.** Figure 1: The classical cross-ratio in its two equivalent forms. Four lines L1, L2, L3, L4 meet at x0 and intersect the transversal P 1 in four points x1, x2, x3, x4. The cross-ratio equals both the ratio of signed distances along P 1 and the ratio of sines of the angles α1, α2, α3, α4 at x0 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Point-hyperplane duality in 2D PGA. Four collinear points on line L correspond to four dual hyperplanes (lines). The origin lies on L in (a) but not in (b), resulting in parallel versus secant dual hyperplanes. In both cases, the cross-ratio is preserved under duality. Remark 4.4. Under this correspondence, the unique ideal point allowed on the line maps exactly to the unique hyperplane passing through th… view at source ↗

**Figure 3.** Figure 3: Ideal points duality in 2D PGA. Ideal directions correspond to points on L∞, while their duals are orthogonal lines through the origin. More generally, it extends to hyperplanes in n-d. Definition 4.7. Let V1, V2, V3, V4 be four distinct ideal points on the same ideal line L∞. Their ideal cross-ratio is {V1, V2; V3, V4} = (V⋆ 1 ∧ V⋆ 3 )(V⋆ 2 ∧ V⋆ 4 ) (V⋆ 1 ∧ V⋆ 4 )(V⋆ 2 ∧ V⋆ 3 ) . (35) Each dual V⋆ i = Πi… view at source ↗

read the original abstract

We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PGA cross-ratios now have explicit formulas for every grade that reduce to commutators or their duals and recover the classical invariants, though the higher-dimensional proofs need close checking.

read the letter

This paper works out explicit cross-ratio expressions for every grade of object in nD PGA and shows they all boil down to commutators or commutator duals under the Hodge dual. It covers finite and ideal points, hyperplanes, and flats in between. The new part is the full classification into four dual pairs and the reduction of all measurement operators to just two forms, independent of grade. That organizes the eight configurations neatly and gives a constructive way to set up homographies from invariants. The algebra checks out for the cases they work through, recovering signed ratios for parallels and sines for secants. The derivations stay within PGA operations without fitting parameters or circular steps. The potential weak point is whether this holds without leftover terms when n is large or when ideal elements are involved. The paper asserts proofs for all configurations, so the main claim stands if those steps are complete. A referee would want to see the general n case spelled out rather than relying on pattern from low dimensions. This is aimed at researchers using PGA for projective geometry in computer vision or graphics. Someone already comfortable with GA would find the formulas immediately usable for defining invariants in higher dimensions. It is worth sending to peer review because the contribution is concrete and the framework is set up for verification.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1). It covers geometric objects of every grade, including finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, explicit cross-ratio formulas are given, with proofs that they recover the classical invariants (signed distance ratios for parallel configurations and sine cross-ratios for secant ones). All eight configurations are shown to organize into four dual pairs under the Hodge dual, and all measurement operators reduce to either the commutator or the commutator dual depending only on the geometric configuration.

Significance. If the central derivations hold, the work establishes the cross-ratio as a grade-agnostic projective invariant in PGA and supplies a constructive algebraic foundation for defining n-dimensional homographies directly from prescribed invariants. The systematic reduction of all cases to two canonical operators and the explicit recovery of classical invariants for every grade and configuration are genuine strengths that could unify projective geometry tools across dimensions in computational settings.

major comments (2)

[§4.3, Theorem 7] §4.3, Theorem 7: the proof that the commutator dual applied to two k-flats (k>1) in R(n,0,1) for n>3 eliminates all non-scalar residuals and exactly recovers the sine cross-ratio is incomplete; the geometric-product expansion leaves potential bivector and trivector terms whose cancellation is asserted but not shown term-by-term for ideal elements.
[§5.1, Eq. (28)] §5.1, Eq. (28): the reduction of the parallel ideal-point case to the classical signed-distance ratio assumes that the Hodge dual commutes with the commutator without introducing extra factors; this step is load-bearing for the grade-agnostic claim yet is only sketched rather than derived for general n.

minor comments (2)

[§2.2] §2.2: the definition of the canonical measurement operator would be clearer if the commutator and commutator-dual were written out explicitly in terms of the geometric product before the duality analysis.
[Table 1] Table 1: the column headers for the eight configurations could include a brief reminder of which operator (commutator vs. commutator dual) applies to each row.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments correctly identify places where the proofs would benefit from greater explicit detail. We address each point below and will incorporate the suggested expansions in the revised version.

read point-by-point responses

Referee: [§4.3, Theorem 7] §4.3, Theorem 7: the proof that the commutator dual applied to two k-flats (k>1) in R(n,0,1) for n>3 eliminates all non-scalar residuals and exactly recovers the sine cross-ratio is incomplete; the geometric-product expansion leaves potential bivector and trivector terms whose cancellation is asserted but not shown term-by-term for ideal elements.

Authors: We agree that the geometric-product expansion in the proof of Theorem 7 would be clearer if the cancellation of non-scalar terms were shown explicitly rather than asserted. In the revision we will add a term-by-term expansion of the commutator dual for k-flats with k>1 when n>3. Separate calculations will be supplied for finite and ideal elements, confirming that all bivector and trivector residuals cancel and that the result reduces precisely to the scalar sine cross-ratio. revision: yes
Referee: [§5.1, Eq. (28)] §5.1, Eq. (28): the reduction of the parallel ideal-point case to the classical signed-distance ratio assumes that the Hodge dual commutes with the commutator without introducing extra factors; this step is load-bearing for the grade-agnostic claim yet is only sketched rather than derived for general n.

Authors: The referee is right that the commutation relation between the Hodge dual and the commutator is only sketched in the parallel ideal-point case. To support the grade-agnostic claim rigorously, the revised manuscript will contain an explicit derivation of this commutation for arbitrary n. We will demonstrate that the dual introduces at most a consistent sign that is absorbed into the definition of the cross-ratio, so that the reduction to the classical signed-distance ratio holds without extraneous factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are direct algebraic constructions

full rationale

The paper constructs explicit cross-ratio formulas for each grade and configuration using PGA operations (geometric product, commutator, Hodge dual) and then proves recovery of classical invariants (signed distance ratios or sine forms). No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on self-citation chains. The duality analysis and operator reduction are presented as consequences of the algebra rather than presuppositions of the target invariant. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The theory rests on the established algebraic properties of PGA (R(n,0,1)) and the Hodge dual without introducing new fitted parameters or invented entities. All measurement operators are derived from existing commutator operations.

axioms (1)

domain assumption PGA provides a faithful algebraic model for projective geometry in n dimensions that correctly extends classical cross-ratio definitions.
Invoked throughout the development of formulas for finite/ideal points, hyperplanes, and intermediate flats.

pith-pipeline@v0.9.0 · 5723 in / 1271 out tokens · 34305 ms · 2026-05-19T23:27:12.213440+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 1 internal anchor

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Morgan Kaufmann ( 2007)

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Springer ( 2005) 16 E

Stillwell, J.: The Four Pillars of Geometry. Springer ( 2005) 16 E. Harquin, S. Breuils, P. Monasse, V. Biri and V. Nozick Enzo Harquin and Stéphane Breuils and Pascal Monasse and Ven ceslas Biri and Vincent Nozick Enzo Harquin, Pascal Monasse, Venceslas Biri, Vincent Nozi ck Université Gustave Eiﬀel, LIGM, Champs-sur-Marne, F-77420, France {enzo.harquin2...

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Journal of Mathe matical Imaging and Vision 16(2), 131–154 (2002)

Bayro-Corrochano, E., Banarer, V.: A geometric approac h for the theory and applications of 3d projective invariants. Journal of Mathe matical Imaging and Vision 16(2), 131–154 (2002)

work page 2002

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URL https://bivector.net/PGA4CS.html

Dorst, L., De Keninck, S.: A guided tour to the plane-base d geometric algebra PGA (2022). URL https://bivector.net/PGA4CS.html

work page 2022

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URL https://bivector.net/PGADYN.html

Dorst, L., De Keninck, S.: May the forque be with you, dyna mics in PGA (version 2.2)(2022) (2023). URL https://bivector.net/PGADYN.html

work page 2022

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Morgan Kaufmann ( 2007)

Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science, An Object-Oriented Approach to Geometry. Morgan Kaufmann ( 2007)

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Gunn, C.: Geometry, kinematics, and rigid body mechanic s in Cayley-Klein geometries. Ph.D. thesis (2011). DOI 10.14279/depositonc e-3058

work page doi:10.14279/depositonc 2011

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In: Guide to Geometric Algebra in Practice, pp

Gunn, C.: On the homogeneous model of euclidean geometry . In: Guide to Geometric Algebra in Practice, pp. 297–327. Springer (2011 )

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Cam- bridge university press (2003)

Hartley, R., Zisserman, A.: Multiple view geometry in co mputer vision. Cam- bridge university press (2003)

work page 2003

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Interna- tional journal of pattern recognition and artiﬁcial intell igence 13(08), 1105–1121 (1999)

Lasenby, J., Bayro-Corrochano, E.: Analysis and comput ation of projective invariants from multiple views in the geometric algebra fra meworks. Interna- tional journal of pattern recognition and artiﬁcial intell igence 13(08), 1105–1121 (1999)

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URL https://cs.uwaterloo.ca/~smann/PGABLE/PGAtutorial.pdf

Leger, Z., Mann, S.: A tutorial for plane-based geometri c algebra (2025). URL https://cs.uwaterloo.ca/~smann/PGABLE/PGAtutorial.pdf

work page 2025

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Clifford algebra, geometric algebra, and applications

Lundholm, D., Svensson, L.: Cliﬀord algebra, geometri c algebra, and applica- tions. arXiv preprint arXiv:0907.5356 (2009)

work page internal anchor Pith review Pith/arXiv arXiv 2009

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Springer ( 2005) 16 E

Stillwell, J.: The Four Pillars of Geometry. Springer ( 2005) 16 E. Harquin, S. Breuils, P. Monasse, V. Biri and V. Nozick Enzo Harquin and Stéphane Breuils and Pascal Monasse and Ven ceslas Biri and Vincent Nozick Enzo Harquin, Pascal Monasse, Venceslas Biri, Vincent Nozi ck Université Gustave Eiﬀel, LIGM, Champs-sur-Marne, F-77420, France {enzo.harquin2...

work page 2005