Pith Number

pith:DE7QGL7I

pith:2026:DE7QGL7IWWIQEPW2OWIXK5RZPO

not attested not anchored not stored refs resolved

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)

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

arxiv:2605.18398 v1 · 2026-05-18 · cs.CG · cs.CV

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{DE7QGL7IWWIQEPW2OWIXK5RZPO}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The cross-ratio is established as a grade-agnostic projective invariant within PGA, with explicit formulas for each object type and configuration that recover the appropriate classical invariant (signed distance ratios for parallel cases and sine cross-ratios for secant cases).

C2weakest assumption

That the algebraic structure of PGA together with the Hodge dual and commutator operators faithfully extends the classical projective cross-ratio to all grades and all eight configurations without additional constraints or exceptions.

C3one line summary

Generalizes projective cross-ratios to every grade of geometric object in n-dimensional PGA with explicit formulas, duality pairs, and recovery of classical invariants.

References

11 extracted · 11 resolved · 1 Pith anchors

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

[2] URL https://bivector.net/PGA4CS.html 2022

[3] URL https://bivector.net/PGADYN.html 2022

[4] Morgan Kaufmann ( 2007) 2007

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

Receipt and verification

First computed	2026-05-20T00:05:58.787718Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

193f032fe8b591023eda75917576397b90541bef605712b4cf21ac59f689b1e0

Aliases

arxiv: 2605.18398 · arxiv_version: 2605.18398v1 · doi: 10.48550/arxiv.2605.18398 · pith_short_12: DE7QGL7IWWIQ · pith_short_16: DE7QGL7IWWIQEPW2 · pith_short_8: DE7QGL7I

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DE7QGL7IWWIQEPW2OWIXK5RZPO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 193f032fe8b591023eda75917576397b90541bef605712b4cf21ac59f689b1e0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "44390fda96fce6ba5afb88c569204748c082df43ac075f98efa34d21f2d26e63",
    "cross_cats_sorted": [
      "cs.CV"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.CG",
    "submitted_at": "2026-05-18T13:43:15Z",
    "title_canon_sha256": "58e42373613d7a9693652e0c8c4783459c10cd8b1935f13d783acc45b382a192"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.18398",
    "kind": "arxiv",
    "version": 1
  }
}