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几何构造驱动的最值优化示例

几何构造驱动的最值优化示例
几何画板及绘图内容加载中，请稍等……
2020-09-15 11:43
圆多边形角最值优化几何作图
该范例通过几何作图与变换来演示一个“以几何形状参数为变量的优化”过程：在由线、圆与多边形共同约束的构型中，相关量（如距离、角或约束函数）随构造变化而达到最优。其核心数学概念包括：几何约束下的优化思想（optimization）、距离度量、角度刻画，以及利用交点与旋转生成对称或等价构型。作图要点可从“构造摘要”提炼：使用 Line 画直线、OrthogonalLine 作垂线、Point 构造关键点、Segment 与 Polygon 形成线段与多边形边界、Circle 建圆并用 Intersect 求交点；再用 Distance 量化度量（如点到点/点到线的距离或线段长度）、Rotate 实现旋转得到新位置；最后借助 Angle 提取角度关系。适合高中到大学的几何建模与可视化学习，也适合作为课堂演示“几何—函数量—优化”联系的教学案例。

GeoGebra 命令及参数

  1. Line(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. Point(Circle[A, 24]) → C
  4. Segment(A, C) → h
  5. Point(Circle[A, 16]) → D
  6. Segment(A, D) → i
  7. Circle(A, D) → c
  8. Intersect(c, g, 2) → E
  9. OrthogonalLine(E, g) → j
  10. OrthogonalLine(C, f) → k
  11. Segment(E, A) → l
  12. Segment(A, C) → m
  13. Intersect(j, k) → F
  14. Segment(C, F) → n
  15. Segment(F, E) → p
  16. Circle(A, Cutout) → d
  17. Circle(E, Cutout) → e
  18. Circle(F, Cutout) → q
  19. Circle(C, Cutout) → r
  20. Intersect(e, l, 1) → G
  21. Intersect(q, n, 1) → H
  22. Segment(G, H) → s
  23. Intersect(r, n, 1) → I
  24. Intersect(d, l, 1) → J
  25. Segment(I, J) → t
  26. Intersect(d, h, 1) → K
  27. Intersect(e, p, 1) → L
  28. Segment(K, L) → a
  29. Intersect(q, p, 1) → M
  30. Intersect(r, h, 1) → N
  31. Segment(M, N) → b
  32. Intersect(s, a) → O
  33. Intersect(s, b) → P
  34. Intersect(t, b) → Q
  35. Intersect(t, a) → R
  36. Segment(G, O) → f_1
  37. Segment(O, L) → g_1
  38. Segment(P, H) → i_1
  39. Segment(P, M) → j_1
  40. Segment(Q, I) → l_1
  41. Segment(R, J) → p_1
  42. Segment(R, K) → q_1
  43. Segment(Q, N) → r_1
  44. Polygon(G, O, L, E) → q1, g_2, o, l_2, e_1
  45. Polygon(J, R, K, A) → q2, j_2, r_2, k_1, a_1
  46. Polygon(N, Q, I, C) → q3, n_1, q_2, i_2, c_1
  47. Polygon(M, F, H, P) → q4, m_1, f_2, h_1, p_2
  48. Polygon(G, O, R, J, L, M, P, H, I, Q, N, K) → poly1, g_3, k_2, r_3, j_3, o_1, l_3, m_2, p_3, h_2, i_3, q_3, n_2
  49. Distance(J, G) → distanceJG
  50. Distance(L, M) → distanceLM
  51. Segment(G, J) → s_1
  52. Segment(L, M) → t_1
  53. Segment(G, E) → b_1
  54. Segment(J, A) → d_1
  55. Segment(G, J) → s_2
  56. Segment(L, M) → t_2
  57. Line(T, U) → a_2
  58. Rotate(T, -120°, U) → T'
  59. Angle(T', U, T) → α
  60. Line(U, T') → b_2
  61. Circle(U, j_3) → c_2
  62. Circle(U, l_3) → d_2
  63. Intersect(d_2, b_2, 2) → V
  64. Intersect(c_2, b_2, 2) → W
  65. Line(W, a_2) → e_2
  66. Intersect(d_2, a_2, 1) → Z
  67. Line(Z, b_2) → f_3
  68. Circle(U, g_1) → c_3
  69. OrthogonalLine(U, a_2) → h_3
  70. Line(W, h_3) → k_3
  71. Line(Z, h_3) → m_3
  72. Intersect(e_2, f_3) → A_1
  73. Line(A_1, h_3) → n_3
  74. Intersect(c_3, h_3, 2) → B_1
  75. Line(B_1, b_2) → s_3
  76. Line(B_1, a_2) → t_3
  77. Intersect(m_3, t_3) → C_1
  78. Line(C_1, s_3) → a_3
  79. Intersect(k_3, s_3) → D_1
  80. Line(D_1, t_3) → b_3
  81. Intersect(n_3, a_3) → E_1
  82. Intersect(f_3, t_3) → F_1
  83. Intersect(e_2, s_3) → G_1
  84. Segment(Z, U) → d_3
  85. Segment(U, B_1) → e_3
  86. Segment(U, W) → f_4
  87. Segment(W, D_1) → g_4
  88. Segment(D_1, G_1) → h_4
  89. Segment(G_1, B_1) → i_4
  90. Segment(B_1, F_1) → j_4
  91. Segment(F_1, C_1) → k_4
  92. Segment(C_1, Z) → l_4
  93. Segment(C_1, E_1) → m_4
  94. Segment(E_1, A_1) → n_4
  95. Segment(A_1, F_1) → p_4
  96. Segment(E_1, D_1) → q_4
  97. Segment(G_1, A_1) → r_4
  98. Polygon(F_1, C_1, E_1, A_1) → q5, s_4, t_4, a_4, b_4
  99. Polygon(E_1, D_1, G_1, A_1) → q6, c_4, d_4, e_4, f_5
  100. Polygon(A_1, G_1, B_1, F_1) → q7, g_5, h_5, i_5, j_5
  101. Polygon(B_1, D_1, W, U) → q8, k_5, l_5, w, u
  102. Polygon(C_1, Z, U, B_1) → q9, m_5, z_1, u_1, n_5

AI 绘图提示语参考

绘制一个由线段、圆和多边形构成的几何图形。首先画两条平行线f和g，然后在圆上取点C和D，连接AC和AD形成线段。接着画圆c并找到其与g的交点E，通过旋转和垂直线构造一系列交点F、G、H、I、J、K、L、M、N，最后连接这些点形成多个多边形，包括四边形GOLE、四边形JRAK、四边形NQIC和四边形MFHP。
以上提示语为根据原始绘图命令整理而来，仅供参考。