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几何变换与圆弧构造：旋转/镜像生成多边形角度结构

几何变换与圆弧构造：旋转/镜像生成多边形角度结构
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2024-01-24 02:43
圆多边形角变换几何作图
所属专题 Exploration: One Special Constant
本例展示如何通过旋转、镜像、圆与圆弧等操作，在平面几何中构造出由若干角度与边段关系支撑的图形结构。核心数学概念包括几何变换（Rotate、Mirror、Dilate、Translate）、圆与圆弧（Circle、CircleArc）、直线与交点（Line、Intersect、OrthogonalLine）、以及角度度量（Angle）与线段约束（Segment）等。作图要点为：利用 Rotate 对关键点/向量进行旋转，再用 Mirror 进行对称复制，同时借助 Circle 与 CircleArc 生成圆弧边界；通过 Line、Ray 与 OrthogonalLine 建立辅助直线与垂线关系，并使用 Intersect 计算关键交点（如 C、O、H、D 等对象）；最后用 Angle 标注并校验角度关系，并可结合 Vector、Dilate、Translate 调整位置与尺度。该范例适合中学几何教学中的作图活动、竞赛几何训练与几何变换复习，也适合作为 GeoGebra 几何建模与可视化的示例。

GeoGebra 命令及参数

  1. Rotate(N, If[1 < a ≤ 2, (-(a - 1)), -1], M) → N'
  2. CircleArc(M, N', N) → c
  3. Mirror(c, yAxis) → c'
  4. Rotate(C, If[1 < a ≤ 2, (-(a - 1)), -1], N') → C'
  5. Rotate(C', -2, M) → C''_{1}
  6. Rotate(C''_{1}, -2, M) → O
  7. Line(O, C''_{1}) → t
  8. Line(M, t) → g_1
  9. Intersect(c', g_1) → H,
  10. Line(C', C''_{1}) → h
  11. Line(M, h) → i
  12. Intersect(c, i, 1) → D
  13. Angle(D, M, H) → β
  14. Rotate(D, If[5 < a ≤ 6, ((a - 5) * β), β], M) → D'
  15. Mirror(D', M) → D''
  16. Segment(D', D'') → s_1
  17. Circle(H, s_1 / 7) → t_1
  18. Intersect(t_1, s_1, 1) → R
  19. Mirror(H, R) → H'
  20. Mirror(R, H') → R'
  21. Mirror(H', R') → H''
  22. Mirror(C', yAxis) → C''
  23. Line(M, C'') → n_2
  24. OrthogonalLine(C'', n_2) → p_2
  25. Line(M, p_2) → q_2
  26. Point(q_2) → U
  27. Segment(((-r), 0), (r, 0)) → f
  28. Rotate(B, If[7.5 < a ≤ 8.5, ((-(a - 7.5)) * 2), -2], (0, 0)) → B'_{2}
  29. Line((0, 0), B'_{2}) → b
  30. OrthogonalLine(B'_{2}, b) → f_1
  31. Circle((0, 0), r) → d_1
  32. Vector(I, J) → v
  33. Rotate((r, 0), 1, (0, 0)) → K
  34. Rotate((r, 0), 3, (0, 0)) → L
  35. Mirror(K, y = 0) → K'
  36. Dilate(M, If[0 < a ≤ 1, 1 - a, 0], N) → M'
  37. Vector(M, M') → w
  38. Translate(f, w) → f'_{2}
  39. Mirror(N', yAxis) → N''
  40. Mirror(A, yAxis) → A'
  41. Circle(N', r - c) → d
  42. Segment(N', C') → g
  43. Mirror(N', yAxis) → N''_{1}
  44. Segment(N''_{1}, C'') → g'
  45. Rotate(C', -1, M) → C''_{2}
  46. Angle(B, M, D) → α
  47. Rotate(B, If[2 < a ≤ 3, ((a - 2) * α), α], M) → B'
  48. Mirror(B', M) → B''
  49. Segment(B'', B') → j
  50. Dilate(M, If[3 < a ≤ 4, 4 - a, 0], C''_{2}) → M'_{1}
  51. Vector(M, M'_{1}) → u
  52. Translate(B'', u) → B'''
  53. Translate(B', u) → B''_{1}
  54. Segment(B''', B''_{1}) → j'
  55. Rotate(C''_{2}, If[4 < a ≤ 5, a - 4, 1], M) → E
  56. Rotate(C''_{2}, If[4 < a ≤ 5, (-(a - 4)), -1], M) → F
  57. CircleArc(M, C''_{2}, E) → e
  58. CircleArc(M, F, C''_{2}) → k
  59. Circle(E, j / 2 - e) → p
  60. Line(M, E) → l
  61. OrthogonalLine(E, l) → m
  62. Rotate(M, ((-pi)) / 2, E) → M'_{2}
  63. Ray(E, M'_{2}) → n
  64. Intersect(p, n, 1) → G
  65. Rotate(M, pi / 2, E) → M'_{3}
  66. Segment(E, G) → q
  67. Line(M, C''_{2}) → s
  68. Mirror(E, s) → E'
  69. Mirror(G, s) → G'
  70. Segment(E', G') → q'
  71. Rotate(C''_{1}, -1, M) → P
  72. Segment(D', D'') → h_1
  73. Dilate(M, If[6 < a ≤ 7, 7 - a, 0], P) → M'_{4}
  74. Vector(M, M'_{4}) → u_1
  75. Translate(D', u_1) → D''_{1}
  76. Translate(D'', u_1) → D'''
  77. Segment(D''_{1}, D''') → i_1
  78. Rotate(P, If[7 < a ≤ 8, a - 7, 1], M) → P'
  79. Line(M, P) → j_1
  80. Mirror(P', j_1) → P''
  81. CircleArc(M, P, P') → c_1
  82. CircleArc(M, P'', P) → e_1
  83. Circle(P', h_1 / 2 - c_1) → k_1
  84. Circle(P'', h_1 / 2 - c_1) → p_1
  85. Line(M, P') → l_1
  86. OrthogonalLine(P', l_1) → m_1
  87. Rotate(M, ((-pi)) / 2, P') → M'_{5}
  88. Ray(P', M'_{5}) → n_1
  89. Intersect(k_1, n_1) → Q,
  90. Segment(P', Q) → q_1
  91. Mirror(Q, j_1) → Q'
  92. Segment(P'', Q') → r_1
  93. Mirror(R', H'') → R''
  94. Mirror(H'', R'') → H'''
  95. Mirror(R'', H''') → R'''
  96. Dilate(H, If[8 < a ≤ 9, 9 - a, 0], R) → H'_{1}
  97. Segment(H, H'_{1}) → a_1
  98. Rotate(H, If[9 < a ≤ 10, ((a - 9) * pi), pi], R) → H'_{2}
  99. Segment(R, H'_{2}) → b_1
  100. Rotate(R, If[10 < a ≤ 11, ((a - 10) * pi), pi], H') → R'_{1}
  101. Segment(H', R'_{1}) → f_2
  102. Rotate(H', If[11 < a ≤ 12, ((a - 11) * pi), pi], R') → H''_{1}
  103. Segment(R', H''_{1}) → g_2
  104. Rotate(R', If[12 < a ≤ 13, ((a - 12) * pi), pi], H'') → R''_{1}
  105. Segment(H'', R''_{1}) → h_2
  106. Rotate(H'', If[13 < a ≤ 14, ((a - 13) * pi), pi], R'') → H'''_{1}
  107. Segment(R'', H'''_{1}) → i_2
  108. Rotate(R'', If[14 < a ≤ 15, ((a - 14) * pi), pi], H''') → R'''_{1}
  109. Segment(H''', R'''_{1}) → j_2
  110. Line(M, C'') → k_2
  111. Angle(U, M, H'') → γ
  112. Dilate(H'', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → H'''_{2}
  113. Vector(H'', H'''_{2}) → v_1
  114. Dilate(R', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → R''_{2}
  115. Vector(R', R''_{2}) → w_1
  116. Translate(R', w_1) → R''_{3}
  117. Translate(H''_{1}, w_1) → S
  118. Segment(R''_{3}, S) → l_2
  119. Rotate(R''_{3}, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → T
  120. Rotate(S, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → S'
  121. Segment(T, S') → m_2
  122. Rotate(C'', If[16.5 < a ≤ 17.5, ((a - 16.5) * 0.28), 0.28], M) → C'''
  123. CircleArc(M, C'', C''') → c_2
  124. Circle(C''', (0.14 * s_1) - c_2) → d_2
  125. Line(C''', M) → r_2
  126. OrthogonalLine(C''', r_2) → s_2
  127. Rotate(M, ((-pi)) / 2, C''') → M'_{6}
  128. Ray(C''', M'_{6}) → t_2
  129. Intersect(d_2, t_2, 1) → V
  130. Segment(C''', V) → a_2
  131. If(8 < a ≤ 15, 0.75, 0.25) → o_1

AI 绘图提示语参考

在平面内构造一个以旋转、镜像与圆弧生成的几何配置：先给定中心点M及一条基准线，通过围绕M的旋转得到新点N′，再以M、N′、N确定一段圆弧c，随后把该圆弧关于y轴镜像得到c′；再把以相关圆弧产生的点与镜像点继续用以M为中心的旋转得到更多圆弧交互结构，并引入多条辅助直线以与圆弧求交，得到关键交点H与D；计算从D指向M与H的夹角并用该角量生成另一点D′，再用关于M与y轴的镜像与连线得到对称段与圆（如以H为圆心、由对称段确定半径），最后补充若干与圆、垂线、旋转构成的角度标注（如β与α）来完成整体结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。