Pith Number

pith:FJYH3ABL

pith:2026:FJYH3ABL2D2SRDR7RHWFBU5HXF

not attested not anchored not stored refs resolved

Generalized Apollonius Circles As Equioptic Curves Of Circles In Constants Curvature Geometries

G\'eza Csima

An extended Apollonius circle coincides with the equioptic curves of two circles in spaces of constant curvature.

arxiv:2605.15457 v1 · 2026-05-14 · math.MG

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\usepackage{pith}
\pithnumber{FJYH3ABL2D2SRDR7RHWFBU5HXF}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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

There exists an Apollonius circle of the centers of two circles that coincides with their equioptic curves, as in Euclidean geometry.

C2weakest assumption

The extended definition of the Apollonius circle, while producing the classical curve in Euclidean geometry, remains well-defined and geometrically meaningful when the ambient space is replaced by a space of constant positive or negative curvature.

C3one line summary

A generalized definition of Apollonius circles is introduced that remains unchanged in Euclidean geometry yet permits a proof that the circle of centers coincides with the equioptic curve of two circles in hyperbolic and spherical geometries.

References

8 extracted · 8 resolved · 0 Pith anchors

[1] – Szirmai, J.: Isoptic curves of conic sections in constant curvature geometries.Mathematical Communications19(2014) 277–290 2014

[2] – Szirmai, J.: Isoptic curves of generalized conic sections in the hyperbolic plane.Ukrainian Mathematical Journal,71/12(2020), 1929- 1944, doi: 10.1007/s11253-020-01756-3 2020 · doi:10.1007/s11253-020-01756-3

[3] J.: The ”Circle” of Apollonius in Hyperbolic Geometry.Forum Geometricorum., V ol.18 2018

[4] – Szirmai, J.: Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries.Results in Mathematics,80, 190 (2025) 2025 · doi:10.1007/s00025-025-02503-5

[5] – Szirmai, J.: Symmetries in the 8 homogeneous 3-geometries 2010

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:59.555676Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2a707d802bd0f5288e3f89ec50d3a7b973a310d22390fbc415acb56c939c484c

Aliases

arxiv: 2605.15457 · arxiv_version: 2605.15457v1 · doi: 10.48550/arxiv.2605.15457 · pith_short_12: FJYH3ABL2D2S · pith_short_16: FJYH3ABL2D2SRDR7 · pith_short_8: FJYH3ABL

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/FJYH3ABL2D2SRDR7RHWFBU5HXF \
  | 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: 2a707d802bd0f5288e3f89ec50d3a7b973a310d22390fbc415acb56c939c484c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0c2ee6209dc9e65fc4888c9ea14995ff9979bd91af877e6500c303dffc4960e5",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.MG",
    "submitted_at": "2026-05-14T22:40:06Z",
    "title_canon_sha256": "478fd7a6be6cb9a4b541f50c99d5d4a6064b09c99854a14ade93323c9a89d3b7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15457",
    "kind": "arxiv",
    "version": 1
  }
}