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模7：六边形截切下的菱形与梯形

模7：六边形截切下的菱形与梯形
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2020-09-15 19:57
多边形立体几何变换几何作图
该范例演示在立体几何中对六边形进行“三次截切”，观察截面图形如何对应出菱形与梯形等四边形结构，并通过旋转与平移建立稳定的构型关系。核心数学概念包括立体几何中的截面、正多边形相关的对称性、四边形分类（菱形、梯形）、以及利用几何变换把不同截面统一到同一构型框架中。作图要点：在 GeoGebra 中使用 Circle 生成圆与构造基准（如圆 r、中心点 O），借助 Point、Line、Segment 标定关键点与边界；通过 Rotate 与 Translate 实现对称/相对位置的变换，使用 Midpoint 取中点以保证截面几何约束；并用 Cube 建立空间实体，再以 Polygon 连接截面的多边形顶点（如 caraABCD、caraADHE、caraABFE、caraBCGF、caraCDHG、caraEFGH 等面），最终由面上的“arista*”边对应形成菱形/梯形的可视关系。适合中学高年级到大学基础几何课程，用于立体几何截面、几何变换与多边形性质的教学演示或探究作图练习。

GeoGebra 命令及参数

  1. Circle(O, r, xOyPlane) → c
  2. Point(c) → A1
  3. Rotate(A1, π / 2, O) → A2
  4. Midpoint(A1, A2) → B5
  5. Cube(A1, A2, zAxis) → a, A3, caraBCGF, caraCDHG, caraEFGH, aristaAD_1, aristaAB, aristaBC_1, aristaCD, aristaAE_1, aristaDH, arista, A4, aristaEF, aristaBF, aristaFG_1, aristaCG_1, aristaGH_1, A5, A6, A7, A8, caraABCD, caraADHE, caraABFE
  6. Midpoint(A6, A2) → B2
  7. Polygon(B5, B2, A2) → t3, a2, s_2, t_2
  8. Midpoint(A4, A8) → B4
  9. Midpoint(A1, A7) → centro
  10. Midpoint(A3, A7) → B3
  11. Midpoint(A1, A5) → B1
  12. Polygon(B5, B1, A1) → t1, a1, n_2, p_2
  13. Midpoint(A3, A4) → B7
  14. Polygon(A3, B3, B7) → t4, i_3, j_3, k_3
  15. Midpoint(A7, A8) → BC
  16. Polygon(BC, B4, A8) → t5, l_3, m_3, n_3
  17. Midpoint(A4, A1) → B8
  18. Polygon(A3, A2, A1, B8, B7) → p2, j_4, k_4, l_4, m_4, n_4
  19. Polygon(centro, BC, B4, B8) → c8, s_3, t_3, a_3, b_3
  20. Midpoint(A8, A5) → BD
  21. Polygon(B8, A1, B1, BD, A8, B4) → p3, c_3, d_3, e_3, f_4, g_4, h_4
  22. Polygon(BD, A8, BC) → t2, q_2, r_2, a8
  23. Polygon(B1, BD, BC, B5) → c5, j_2, k_2, l_2, m_2
  24. Midpoint(A6, A7) → BB
  25. Polygon(B8, BB, B3, B7) → c7, p_3, q_3, r_3, i_4
  26. Polygon(A2, A3, B3, BB, B2) → p1, e_2, a3, f_3, g_3, h_3
  27. Polygon(B5, B2, BB, centro) → c6, a_2, b_2, c_2, d_2
  28. Midpoint(A2, A3) → B6
  29. Vector(A5, A8) → u4
  30. Polygon(A2, A3, A8, A5) → c4, k_1, l_1, m_1, n_1
  31. Midpoint(A5, A6) → BA
  32. Polygon(A4, A1, A6, A7) → c3, g_1, h_1, i_1, j_1
  33. Polygon(A1, A3, A7, A5) → c2, q, s, e, f_1
  34. Vector(B1, B3) → u1
  35. Vector(B1, B3) → u3
  36. Sequence(Max[0, Min[1, t - s]], s, 0, 10) → sl
  37. Line(A5, A3) → k
  38. Segment(A7, A1) → h
  39. Midpoint(A1, A6) → O1
  40. Midpoint(A2, A7) → O2
  41. Midpoint(A3, A8) → O3
  42. Midpoint(A8, A1) → O4
  43. Midpoint(A5, A7) → O5
  44. Vector(B8, BB) → u2
  45. Polygon(A4, A2, A6, A8) → c1, l, m, n, p
  46. Polygon(A4, BD, A6, B6) → c3_1, e_1, f_2, g_2, h_2
  47. Translate(c1, u1) → c1'
  48. Polygon(A7, BD, A1, B6) → c2_1, f, g, i, j
  49. Translate(c3, u3) → c3'
  50. Translate(c2, u2) → c2'
  51. Translate(c1', Vector[((-sl(1)) * u1)]) → c1''
  52. Translate(c2', Vector[((-sl(2)) * u2)]) → c2''
  53. Translate(c3', Vector[((-sl(3)) * u3)]) → c3''
  54. Line(B1, B2) → t_1
  55. Translate(c4, u4) → c4'
  56. Translate(c4', Vector[((-sl(4)) * u4)]) → c4''
  57. Vector(A1, A8) → v
  58. Line(O1, O3) → bc
  59. Line(O5, O) → bb
  60. Line(O2, O4) → bd
  61. Vector(A2, A8) → v3
  62. Vector(BC, B5) → v1_1
  63. Vector(A6, A1) → v2_1
  64. Polygon(BC, B3, B6, B5, B1, BD) → po1, p_1, b3, b6, b5, b1, q_1
  65. Polygon(BC, B4, B8, B5, B2, BB) → po2, r_1, b4, b8, s_1, b2, a_1
  66. Polygon(BB, B3, B7, B8, B1, BA) → po3, b_1, c_1, b7, d_1, i_2, ba
  67. Translate(po1, u1) → po1'
  68. Translate(po2, u2) → po2'
  69. Translate(po3, u1) → po3'
  70. Translate(po1', Vector[((-sl(1)) * u1)]) → po1''
  71. Translate(po2', Vector[((-sl(2)) * u2)]) → po2''
  72. Translate(po3', Vector[((-sl(3)) * u3)]) → po3''
  73. Translate(ll1, u5) → ll2
  74. Rotate(ll2, (sl(6) * π), bd) → ll3
  75. Rotate(ll3, (sl(7) * π / 2), bb) → ll4
  76. Translate(ll2, Vector[((-sl(10)) * v)]) → ll5
  77. Translate(ll4, Vector[(sl(10) * v)]) → ll6

AI 绘图提示语参考

绘制一个六棱柱，并从三个不同的六边形切面切割出菱形和梯形。首先画出六棱柱的底面和侧面，然后通过旋转和中点构造出菱形和梯形的顶点。最后连接这些顶点形成菱形和梯形。
以上提示语为根据原始绘图命令整理而来，仅供参考。