积极 AI Logo 积极 AI

旋转叠加镜像：Turn-Flip（转折翻转）几何变换演示

旋转叠加镜像：Turn-Flip（转折翻转）几何变换演示
几何画板及绘图内容加载中，请稍等……
2024-11-19 23:57
多边形角向量变换几何作图
所属专题 Congruence Transformations
该范例通过“旋转 + 反射”的组合，展示一次 Turn-Flip（转折翻转）对多边形顶点与对应位置的作用规律。核心数学概念包括几何变换（旋转、镜面反射、平移、滑移等）、刚体运动的复合、以及由变换产生的对应点（preimage 与像点）关系。作图要点：由构造摘要可见，使用 Point、Polygon、Vector、Line 建立基准几何；通过 Rotate 与 Mirror 分别生成旋转与反射后的点/对象，并结合 Translate、glide 等建立复合变换效果；再用 Segment、Angle、Intersect 辅助确定连线、角度与交点位置，最终得到平移/反射/旋转后的多边形对应图（如 A’、B’、C’ 等）及其可视化结果。适合中学数学与几何教学（几何变换、坐标/向量辅助建模），也适合作为竞赛与数学可视化学习的入门案例。

GeoGebra 命令及参数

  1. Point(Circle[A, 1.2851459061133876]) → B
  2. Polygon(A, B, C, D, E, F) → preimage, a_{1}, b_1, c_1, d, e, f
  3. Vector(G, H) → Vector
  4. Line(I, J) → lineOfReflection
  5. Translate(A, Vector) → A'_{5}
  6. Translate(B, Vector) → B'_{5}
  7. Translate(C, Vector) → C'_{5}
  8. Translate(D, Vector) → D'_{5}
  9. Translate(E, Vector) → E'_{5}
  10. Translate(F, Vector) → F'_{5}
  11. Polygon(A'_{5}, B'_{5}, C'_{5}, D'_{5}, E'_{5}, F'_{5}) → translated, a', b', c', d', e', f'
  12. Mirror(A, lineOfReflection) → A'_{1}
  13. Mirror(B, lineOfReflection) → B'_{1}
  14. Mirror(C, lineOfReflection) → C'_{1}
  15. Mirror(D, lineOfReflection) → D'_{1}
  16. Mirror(E, lineOfReflection) → E'_{1}
  17. Mirror(F, lineOfReflection) → F'_{1}
  18. Polygon(A'_{1}, B'_{1}, C'_{1}, D'_{1}, E'_{1}, F'_{1}) → reflected, a'_{1}, b'_{1}, c'_{1}, d'_{1}, e'_{1}, f'_{1}
  19. Rotate(A, dummyRotationAngle, Center) → A'_{2}
  20. Rotate(B, dummyRotationAngle, Center) → B'_{2}
  21. Rotate(C, dummyRotationAngle, Center) → C'_{2}
  22. Rotate(D, dummyRotationAngle, Center) → D'_{2}
  23. Rotate(E, dummyRotationAngle, Center) → E'_{2}
  24. Rotate(F, dummyRotationAngle, Center) → F'_{2}
  25. Polygon(A'_{2}, B'_{2}, C'_{2}, D'_{2}, E'_{2}, F'_{2}) → rotated, a'_{2}, b'_{2}, c'_{2}, d'_{2}, e'_{2}, f'_{2}
  26. Line(K, lineOfReflection) → h
  27. Point(h) → L
  28. Vector(K, L) → Vector2
  29. Mirror(A, lineOfReflection) → A'_{3}
  30. Mirror(B, lineOfReflection) → B'_{3}
  31. Mirror(C, lineOfReflection) → C'_{3}
  32. Mirror(D, lineOfReflection) → D'_{3}
  33. Mirror(E, lineOfReflection) → E'_{3}
  34. Mirror(F, lineOfReflection) → F'_{3}
  35. Polygon(A'_{3}, B'_{3}, C'_{3}, D'_{3}, E'_{3}, F'_{3}) → intermediary, a'_{3}, b'_{3}, c'_{3}, d'_{3}, e'_{3}, f'_{3}
  36. Translate(A'_{3}, Vector2) → M
  37. Translate(B'_{3}, Vector2) → N
  38. Translate(C'_{3}, Vector2) → O
  39. Translate(D'_{3}, Vector2) → P
  40. Translate(E'_{3}, Vector2) → Q
  41. Translate(F'_{3}, Vector2) → R
  42. Polygon(M, N, O, P, Q, R) → image_{1}, m, n, o, p, q, r
  43. Line(S, T) → a
  44. Rotate(preimageList, angleAnimator, rotationCenter) → RFanimRotList
  45. Rotate(A, rotationAngle) → A'_{4}
  46. Rotate(C, rotationAngle) → C'_{4}
  47. Rotate(D, rotationAngle) → D'_{4}
  48. Rotate(E, rotationAngle) → E'_{4}
  49. Rotate(F, rotationAngle) → F'_{4}
  50. Rotate(A, rotationAngle, rotationCenter) → A'_{6}
  51. Rotate(C, rotationAngle, rotationCenter) → C'_{6}
  52. Rotate(D, rotationAngle, rotationCenter) → D'_{6}
  53. Rotate(E, rotationAngle, rotationCenter) → E'_{6}
  54. Rotate(F, rotationAngle, rotationCenter) → F'_{6}
  55. Mirror(A'_{6}, a) → W
  56. Mirror(C'_{6}, a) → F_1
  57. Mirror(D'_{6}, a) → G_1
  58. Mirror(E'_{6}, a) → H_1
  59. Mirror(F'_{6}, a) → I_1
  60. Rotate(preimageList, rotationAngle, rotationCenter) → RFintermList
  61. Mirror(RFintermList, a) → RFfinalList
  62. Mirror(preimageList, a) → FRintermList
  63. Rotate(FRintermList, rotationAngle, rotationCenter) → FRfinalList
  64. Rotate(FRintermList, angleAnimator, rotationCenter) → FRanimRotList
  65. Polygon(FRanimFlipList) → FRanimFlip
  66. Polygon(FRanimRotList) → FRanimRot
  67. Polygon(FRintermList) → FRinterm
  68. Polygon(RFanimRotList) → RFanimRot
  69. Polygon(RFintermList) → RFinterm
  70. Polygon(RFanimFlipList) → RFanimFlip
  71. Polygon(RFfinalList) → RFfinal
  72. Polygon(FRfinalList) → image
  73. Segment(rotationCenter, FRanimRotList(2)) → FRrotationSegment
  74. Segment(rotationCenter, RFanimRotList(2)) → RFrotSegment
  75. Line(rotationCenter, V) → b
  76. Rotate(V, rotationAngle / 2, rotationCenter) → V'
  77. Angle(V, rotationCenter, V') → linesAngle
  78. Line(rotationCenter, V') → c
  79. Rotate(b, pivotAngle1_{RefRot}, rotationCenter) → b'_{4}
  80. Intersect(b'_{4}, a) → Center_{ab}
  81. Rotate(a, pivotAngle2_{RefRot}, Center_{ab}) → a'_{4}
  82. Mirror(F, a'_{4}) → F'
  83. Mirror(E, a'_{4}) → E'
  84. Mirror(D, a'_{4}) → D'
  85. Mirror(C, a'_{4}) → C'
  86. Mirror(B, a'_{4}) → B'
  87. Mirror(A, a'_{4}) → A'
  88. Polygon(A', B', C', D', E', F') → preimage', g, i, j, k, l, s
  89. Rotate(b'_{4}, pivotAngle2_{RefRot}, Center_{ab}) → t
  90. Mirror(F', t) → F''
  91. Mirror(E', t) → E''
  92. Mirror(D', t) → D''
  93. Mirror(C', t) → C''
  94. Mirror(B', t) → B''
  95. Mirror(A', t) → A''
  96. Polygon(A'', B'', C'', D'', E'', F'') → preimage'', a'', b'', c'', d'', e'', f''
  97. Rotate(c, pivotAngle1_{RefRot}, rotationCenter) → c'_{4}
  98. Intersect(c, a) → Center_{ac}
  99. Rotate(a, pivotAngle1_{RotRef}, Center_{ac}) → a'_{5}
  100. Rotate(c, pivotAngle1_{RotRef}, Center_{ac}) → c'_{5}
  101. Intersect(b, c'_{5}) → Center_{bc}
  102. Rotate(b, pivotAngle2_{RotRef}, Center_{bc}) → b'_{5}
  103. Rotate(c'_{5}, pivotAngle2_{RotRef}, Center_{bc}) → f_1
  104. Mirror(A, b'_{5}) → A'_{7}
  105. Mirror(B, b'_{5}) → B'_{4}
  106. Mirror(C, b'_{5}) → C'_{7}
  107. Mirror(D, b'_{5}) → D'_{7}
  108. Mirror(E, b'_{5}) → E'_{7}
  109. Mirror(F, b'_{5}) → F'_{7}
  110. Polygon(A'_{7}, B'_{4}, C'_{7}, D'_{7}, E'_{7}, F'_{7}) → preimage'_{1}, a'_{7}, g_1, c'_{7}, d'_{7}, e'_{7}, f'_{7}
  111. Mirror(A'_{7}, f_1) → Z
  112. Mirror(B'_{4}, f_1) → B_1
  113. Mirror(C'_{7}, f_1) → C_1
  114. Mirror(D'_{7}, f_1) → D_1
  115. Mirror(E'_{7}, f_1) → E_1
  116. Mirror(F'_{7}, f_1) → J_1
  117. Polygon(Z, B_1, C_1, D_1, E_1, J_1) → poly1, z_1, b_{2}, c_{2}, d_1, e_1, j_1

AI 绘图提示语参考

画一个六边形（由六个点A到F组成），以及一条反射线（由点I和J确定）。首先绘制原始六边形和反射线。然后将六边形关于反射线进行反射，得到另一个六边形。接着将反射后的六边形绕某个中心旋转一定角度，得到最终图形。整个过程展示了旋转与反射的组合变换，强调变换前后图形的对应关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。