{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:MMJC7CWU6QLJMMTHVB7VVQBVQQ","short_pith_number":"pith:MMJC7CWU","schema_version":"1.0","canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","source":{"kind":"arxiv","id":"1302.1813","version":1},"attestation_state":"computed","paper":{"title":"Polarit\\'es d\\'efinies par un triangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Beno\\^it Kloeckner (IF)","submitted_at":"2013-02-07T17:49:30Z","abstract_excerpt":"A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.1813","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","cross_cats_sorted":[],"title_canon_sha256":"dc3feed24c7dba6acfbe83b058078e91687e8ad9df76ba32dbf77633aa4a7b1f","abstract_canon_sha256":"470b410c86a4b5eaf1e69f711fdf903f17cc134ddc186edaad91a1b362a3fd54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:14.546730Z","signature_b64":"Pk7rFM79ttFEtSd6ukh5dloF+6MipCkO9HxBN/DPvak3CSWWn4P8hp18B85peQAzNTeXOaxyMsZEdlDtY+dpBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","last_reissued_at":"2026-05-18T03:34:14.545130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:14.545130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polarit\\'es d\\'efinies par un triangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Beno\\^it Kloeckner (IF)","submitted_at":"2013-02-07T17:49:30Z","abstract_excerpt":"A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1813","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.1813","created_at":"2026-05-18T03:34:14.546083+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.1813v1","created_at":"2026-05-18T03:34:14.546083+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.1813","created_at":"2026-05-18T03:34:14.546083+00:00"},{"alias_kind":"pith_short_12","alias_value":"MMJC7CWU6QLJ","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MMJC7CWU6QLJMMTH","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MMJC7CWU","created_at":"2026-05-18T12:27:52.871228+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ","json":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ.json","graph_json":"https://pith.science/api/pith-number/MMJC7CWU6QLJMMTHVB7VVQBVQQ/graph.json","events_json":"https://pith.science/api/pith-number/MMJC7CWU6QLJMMTHVB7VVQBVQQ/events.json","paper":"https://pith.science/paper/MMJC7CWU"},"agent_actions":{"view_html":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ","download_json":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ.json","view_paper":"https://pith.science/paper/MMJC7CWU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.1813&json=true","fetch_graph":"https://pith.science/api/pith-number/MMJC7CWU6QLJMMTHVB7VVQBVQQ/graph.json","fetch_events":"https://pith.science/api/pith-number/MMJC7CWU6QLJMMTHVB7VVQBVQQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/action/storage_attestation","attest_author":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/action/author_attestation","sign_citation":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/action/citation_signature","submit_replication":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/action/replication_record"}},"created_at":"2026-05-18T03:34:14.546083+00:00","updated_at":"2026-05-18T03:34:14.546083+00:00"}