{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-geometric-transformations",
  "semantic_family_id": "sf-geometric-transformations",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "平面座標的基本等距與均勻縮放"
  },
  "summary": {
    "zh-Hant": "平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。"
  },
  "statement": {
    "zh-Hant": "在等單位正交實平面，平移 H(x,y)=(x+tx,y+ty)、原點逆時針0／90／180／270度旋轉及關於x軸、y軸、y=x、y=−x的鏡射均為等距映射。均勻縮放 H_c(x,y)=(cx,cy)：c=0 是退化常值映射，|c|=1 是全平面等距，其他非零c是正倍率 |c| 的非等距相似映射。"
  },
  "formula": {
    "tex": "H(P)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>H</mi><mo>(</mo><mi>P</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "H",
        "glyph": "H"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "P",
          "glyph": "P"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3639680f02743ff5376e63aded303b48d753dde059c4b3de45a70ff0757bfda2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "定義域與陪域為整個實平面，所有座標、平移與縮放參數的題庫切片為安全整數。"
    },
    {
      "id": "a2",
      "zh-Hant": "旋轉中心固定原點、方向固定逆時針，turns為0至3的四分之一圈次數；鏡射軸須明示。"
    },
    {
      "id": "a3",
      "zh-Hant": "均勻縮放分類是全平面性質，不從某一個重合點或不變點的距離推論全域等距。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "explicit_real_plane_map",
      "role_zh": "明示的實平面座標映射",
      "scope": "representative_formula"
    },
    {
      "token": "P",
      "role": "coordinate_point",
      "role_zh": "輸入座標點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "映射的整個實平面論域。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "固定原點、軸及座標方向。"
    },
    {
      "id": "mko-distance-formula",
      "title_zh": "等單位正交座標的歐氏距離",
      "reason_zh": "全域距離不變的語義。"
    },
    {
      "id": "mko-similarity",
      "title_zh": "正倍率的標記三角形相似切片",
      "reason_zh": "分清座標係數與正長度倍率。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "90度逆時針送 (x,y) 到 (−y,x)，180度到 (−x,−y)，270度到 (y,−x)，0度是恆等。原點在每一種旋轉下不變，不靠不同角度重複算原點充量。",
      "x軸鏡射為 (x,−y)，y軸為 (−x,y)，y=x為 (y,x)，y=−x為 (−y,−x)。原點中心反演 (−x,−y) 是半轉，不混稱成關於某條軸的鏡射。",
      "均勻縮放使每個距離乘 |c|。c=0不能保持不同兩點的距離，c=−1卻能保持；負係數不是一律不等距。單點恰好不變不代表全平面映射等距。"
    ],
    "common_misconception_zh": "旋轉方向／中心不明、鏡射軸混用，把負係數當負長度倍率，或從一點不變推論全域等距。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示整數座標切片可重算；一般等距／相似映射及任意旋轉角的形式橋尚未建立。",
    "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": "在固定版 EuclideanSpace R (Fin 2) 上橋接明示等距、軸鏡射及均勻縮放，不使用普通乘積預設度量。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-geometric-transformations.md 整理；2026-10-06 脈衝 R07（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}