{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-point-line-plane","semantic_family_id":"sf-point-line-plane","type":"concept","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"點、線、面","en":"Points, lines, and planes"},
  "summary":{"zh-Hant":"點、線與面提供描述位置、方向、共線、共面與入射關係的基礎幾何語言。"},
  "statement":{"zh-Hant":"在指定的幾何模型中，點表示位置；線與面是承載點並具有相應維度的幾何對象，其精確性質由該模型的公理與結構決定。"},
  "formula":{"tex":"P","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math>","semantic_ast":{"type":"symbol","name":"P","glyph":"P"},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"}},
  "assumptions":[{"id":"a1","zh-Hant":"已明示所使用的歐幾里得、仿射或其他幾何語境。"}],
  "symbols":[{"token":"P","role":"point","role_zh":"代表點","scope":"representative_formula"}],
  "dependencies":[],
  "explanation":{"paragraphs_zh":["點不是畫在紙上的墨點；圖上的大小只是視覺標記。線與面也不是由線條粗細或紙張邊界定義。","在仿射模型中，點與平移向量的角色分離，直線與平面可由適當維度的仿射子空間表示。","共線、共面與相交等陳述都依賴明示的入射或 ambient-space 關係。"],"common_misconception_zh":"常見誤解是把示意圖的外觀當成定義，或在未指定幾何模型時宣稱所有線、面性質都普遍成立。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"座標例子只驗證指定模型中的實例，不構成幾何公理或一般維度命題的證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"選定 affine-space 模型並分別綁定點、直線、平面、維度與入射聲明。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-point-line-plane.md 整理；待人工審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-17"}
}