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转滑变换（Turn-Slide）的几何构造与验证

转滑变换（Turn-Slide）的几何构造与验证
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2024-12-05 03:36
多边形角向量变换几何作图
所属专题 Congruence Transformations
该范例通过把一个多边形先做旋转再做平移（或等效地先做平移再做旋转），演示“转滑（Turn-Slide）”对应图形在平面中的位置变化与对应关系。核心数学概念包括刚体变换、复合变换（旋转+平移/滑移）、反射与直线（可作为对照）、对应点位置的一致性，以及由构造得到的角度与线段对应。作图要点：从“构造摘要”可见使用了 GeoGebra 命令 Point、Polygon、Vector、Line、Translate、Mirror、Rotate、Segment、Angle、OrthogonalLine、Circle、Intersect；对象上构造并追踪多个关键点（如 A、B、C、D、E、F 及其变换后的对应点 A'、B' 等）、以及向量与辅助直线（lineOfReflection 等），随后用 Translate、Rotate 实现转滑复合变换，并通过 Angle、Segment、Intersect 等检验几何关系。适合中学阶段的几何变换教学（七/八年级到高中几何复习），也适用于课堂演示与学生探究活动，用于理解等距变换与复合变换的图形保持性质。

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_1, 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. Rotate(A, dummyRotationAngle, Center) → A'_{2}
  19. Rotate(B, dummyRotationAngle, Center) → B'_{2}
  20. Rotate(C, dummyRotationAngle, Center) → C'_{2}
  21. Rotate(D, dummyRotationAngle, Center) → D'_{2}
  22. Rotate(E, dummyRotationAngle, Center) → E'_{2}
  23. Rotate(F, dummyRotationAngle, Center) → F'_{2}
  24. 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}
  25. Line(K, lineOfReflection) → h
  26. Point(h) → L
  27. Vector(K, L) → Vector2
  28. Mirror(A, lineOfReflection) → A'_{3}
  29. Mirror(B, lineOfReflection) → B'_{3}
  30. Mirror(C, lineOfReflection) → C'_{3}
  31. Mirror(D, lineOfReflection) → D'_{3}
  32. Mirror(E, lineOfReflection) → E'_{3}
  33. Mirror(F, lineOfReflection) → F'_{3}
  34. 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}
  35. Translate(A'_{3}, Vector2) → M
  36. Translate(B'_{3}, Vector2) → N
  37. Translate(C'_{3}, Vector2) → O
  38. Translate(D'_{3}, Vector2) → P
  39. Translate(E'_{3}, Vector2) → Q
  40. Translate(F'_{3}, Vector2) → R
  41. Polygon(M, N, O, P, Q, R) → image_{1}, m, n, o, p, q, r
  42. Rotate(preimageList, angleAnimator, rotationCenter) → RTanimRotList
  43. Rotate(A, rotationAngle) → A'_{4}
  44. Rotate(C, rotationAngle) → C'_{4}
  45. Rotate(D, rotationAngle) → D'_{4}
  46. Rotate(E, rotationAngle) → E'_{4}
  47. Rotate(F, rotationAngle) → F'_{4}
  48. Rotate(A, rotationAngle, rotationCenter) → A'_{6}
  49. Rotate(C, rotationAngle, rotationCenter) → C'_{6}
  50. Rotate(D, rotationAngle, rotationCenter) → D'_{6}
  51. Rotate(E, rotationAngle, rotationCenter) → E'_{6}
  52. Rotate(F, rotationAngle, rotationCenter) → F'_{6}
  53. Rotate(preimageList, rotationAngle, rotationCenter) → RTintermList
  54. Polygon(RTanimRotList) → RTanimRot
  55. Polygon(RTintermList) → RTinterm
  56. Segment(rotationCenter, RTanimRotList(2)) → RFrotSegment
  57. Line(rotationCenter, V) → a
  58. Rotate(V, rotationAngle / 2, rotationCenter) → V'
  59. Angle(V, rotationCenter, V') → linesAngle
  60. Line(rotationCenter, V') → b
  61. Vector(S, T) → translVector
  62. OrthogonalLine(W, translVector, xOyPlane) → c
  63. Circle(W, abs(translVector) / 2) → k
  64. OrthogonalLine(W, c, xOyPlane) → i
  65. Circle(W, abs(translVector) / 2) → s
  66. Intersect(i, s, 2) → Z
  67. Line(Z, c) → d
  68. Translate(RTinterm, translVector) → image
  69. Mirror(A, lineOfReflection) → A'
  70. Mirror(B, lineOfReflection) → B'
  71. Mirror(C, lineOfReflection) → C'
  72. Mirror(D, lineOfReflection) → D'
  73. Mirror(E, lineOfReflection) → E'
  74. Mirror(F, lineOfReflection) → F'
  75. Polygon(A', B', C', D', E', F') → preimage', l, t, a_{2}, f_1, g_1, h_1
  76. Polygon(RTanimTranslList) → RTanimTransl
  77. Polygon(TRanimTranslList) → TRanimTransl
  78. Polygon(TRintermList) → TRinterm
  79. Rotate(TRintermList, angleAnimator, rotationCenter) → TRanimRotList
  80. Polygon(TRanimRotList) → TRanimRot
  81. Segment(rotationCenter, TRanimRotList(1)) → rotSegment
  82. Rotate(TRinterm, rotationAngle, rotationCenter) → TRinterm'
  83. Translate(A, translVector) → A'_{7}
  84. Translate(C, translVector) → C'_{7}
  85. Translate(D, translVector) → D'_{7}
  86. Translate(E, translVector) → E'_{7}
  87. Translate(F, translVector) → F'_{7}
  88. Mirror(A, c) → A'_{8}
  89. Mirror(B, c) → B'_{4}
  90. Mirror(C, c) → C'_{8}
  91. Mirror(D, c) → D'_{8}
  92. Mirror(E, c) → E'_{8}
  93. Mirror(F, c) → F'_{8}
  94. Polygon(A'_{8}, B'_{4}, C'_{8}, D'_{8}, E'_{8}, F'_{8}) → preimage'_{1}, a'_{8}, b'_{4}, c'_{8}, d'_{8}, e'_{8}, f'_{8}
  95. Mirror(preimage'_{1}, d) → poly1
  96. Mirror(poly1, a) → poly1'
  97. Mirror(poly1', b) → poly1''
  98. Mirror(A, a) → A'_{9}
  99. Mirror(B, a) → B'_{6}
  100. Mirror(C, a) → C'_{9}
  101. Mirror(D, a) → D'_{9}
  102. Mirror(E, a) → E'_{9}
  103. Mirror(F, a) → F'_{9}
  104. Polygon(A'_{9}, B'_{6}, C'_{9}, D'_{9}, E'_{9}, F'_{9}) → preimage'_{2}, a'_{9}, b'_{6}, c'_{9}, d'_{9}, e'_{9}, f'_{9}
  105. Mirror(A'_{9}, b) → B_1
  106. Mirror(B'_{6}, b) → C_1
  107. Mirror(C'_{9}, b) → D_1
  108. Mirror(D'_{9}, b) → E_1
  109. Mirror(E'_{9}, b) → F_1
  110. Mirror(F'_{9}, b) → G_1
  111. Polygon(B_1, C_1, D_1, E_1, F_1, G_1) → poly2, g, j, d_{2}, e_1, i_1, j_1
  112. Mirror(B_1, c) → H_1
  113. Mirror(C_1, c) → I_1
  114. Mirror(D_1, c) → J_1
  115. Mirror(E_1, c) → K_1
  116. Mirror(F_1, c) → L_1
  117. Mirror(G_1, c) → M_1
  118. Polygon(H_1, I_1, J_1, K_1, L_1, M_1) → poly2', k_1, l_1, m_1, n_1, p_1, q_1
  119. Mirror(poly2', d) → poly2''

AI 绘图提示语参考

绘制一个多边形作为原始图形，再画一条直线作为反射轴，并画一个向量表示平移方向。先沿该向量平移多边形得到平移后的图形，再关于反射轴作镜像得到镜像图形。接着以某点为旋转中心，将原始多边形旋转一个角度得到旋转图形。然后先关于反射轴镜像原始多边形，再沿另一个向量平移镜像结果，得到最终的复合变换图形。最后用虚线或不同颜色标出对应点、对应边和对应角，以展示旋转加平移的复合刚体运动效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。