{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-geometric-transformations"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-geometric-transformations-01","title":{"zh-Hant":"幾何變換：證明基本等距並分類均勻縮放","en":"Geometric transformations: proving the basic isometries and classifying uniform scaling"},"summary":{"zh-Hant":"在等單位正交實平面中證明：平移、原點逆時針 90° 旋轉 R90(x,y)=(−y,x) 以及 MKO 列出的其他旋轉與軸鏡射都是等距映射；半轉 (−x,−y) 不是所列四條軸中任何一條的鏡射；均勻縮放 Hc(x,y)=(cx,cy) 使每個距離乘 |c|。據此分類 c=0、c=−1、c=−2、c=2，並以 P=(3,1)、Q=(−1,4) 做數值核對。","en":"On the equal-unit orthonormal real plane, prove that translations, the counterclockwise quarter-turn R90(x,y)=(−y,x) about the origin, and the other rotations and axis reflections listed in the MKO are isometries; that the half-turn (−x,−y) is not a reflection in any of the listed axes; and that uniform scaling Hc(x,y)=(cx,cy) multiplies every distance by |c|. Use this to classify c=0, c=−1, c=−2 and c=2, with a numerical check on P=(3,1), Q=(−1,4)."},"mko_ids":["mko-geometric-transformations"]}],"title":"概念推導導讀"}