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正六边形底面立体：平行于y轴的截面也是正六边形

正六边形底面立体：平行于y轴的截面也是正六边形
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2020-01-27 19:23
多边形立体几何微积分积分几何作图
①本范例演示一个具有正六边形底面的立体，其平行于y轴的截面均为正六边形；②涉及的核心数学概念包括立体几何、多边形性质、积分计算；③作图要点——从下方『构造摘要』提炼用到了Segment（线段）、Midpoint（中点）、Intersect（交点）、Line（直线）、Sphere（球体）、Textfield（文本框）、Rotate（旋转）、Polygon（多边形）、CurveCartesian（笛卡尔曲线）、Surface（曲面）等GeoGebra命令/对象；④适合高中及以上学生学习立体几何、积分应用以及空间想象能力培养。

GeoGebra 命令及参数

  1. Segment(H, I) → s_1
  2. Midpoint(s_1) → A_1
  3. Segment(J, K) → q_1
  4. Midpoint(q_1) → Z
  5. Midpoint(s_1) → E
  6. Midpoint(q_1) → F
  7. Intersect(upper, h, (0, 8.66025)) → B
  8. Intersect(lower, h, (0, -8.66025)) → C
  9. Segment(C, B) → q
  10. Intersect(upper, q, (0, 8.66025)) → G
  11. Midpoint(C, G) → T
  12. Line(B, D) → i
  13. Sphere(B, d) → e
  14. Segment(C, B) → p
  15. Textfield(upper) → InputBox1
  16. Textfield(lower) → InputBox2
  17. Textfield(a) → InputBox3
  18. Textfield(b) → InputBox4
  19. Segment(C, G) → k_1
  20. Line(C, G) → j
  21. Sphere(G, T) → k
  22. Line(T, T + (1, 0, 0)) → m
  23. Rotate(G, π / 3, F_1, m) → G'
  24. Rotate(G', π / 3, F_1, Line[F_1, F_1 + (1, 0, 0)]) → G''
  25. Line(F_1, F_1 + (1, 0, 0)) → n
  26. Rotate(G'', π / 3, F_1, n) → G'''
  27. Rotate(G''', π / 3, F_1, n) → G_1
  28. Polygon(C, G, G', G'', G''', G_1) → poly1, c_1, g_2, g', g'', g''', g_3
  29. Segment(G', G'') → l
  30. Segment(G'', G''') → r
  31. Segment(G''', G_1) → s
  32. Segment(G_1, C) → t
  33. Segment(G', G) → i_1
  34. Line(Z, Z + (1, 0, 0)) → f
  35. Line(D_1, D_1 + (1, 0, 0)) → g
  36. Rotate(J, π / 3, D_1, g) → J'
  37. Rotate(J', π / 3, D_1, g) → J''
  38. Rotate(J'', π / 3, D_1, g) → J'''
  39. Rotate(J''', π / 3, D_1, g) → E_1
  40. Segment(J, J') → f_1
  41. Segment(J', J'') → g_1
  42. Segment(J'', J''') → j_1
  43. Segment(J''', E_1) → n_1
  44. Segment(E_1, K) → p_1
  45. Line(H_1, H_1 - (1, 0, 0)) → l_1
  46. Rotate(I, π / 3, H_1, l_1) → I'
  47. Rotate(I', π / 3, H_1, l_1) → I''
  48. Rotate(I'', π / 3, H_1, l_1) → I'''
  49. Rotate(I''', π / 3, H_1, l_1) → I_1
  50. Segment(I, I') → m_1
  51. Segment(I', I'') → r_1
  52. Segment(I'', I''') → a_1
  53. Segment(I''', I_1) → b_1
  54. Segment(I_1, H) → d_1
  55. Polygon(H, I, I', I'', I''', I_1) → poly2, h_2, i_2, i', i'', i''', i_3
  56. Polygon(K, J, J', J'', J''', E_1) → poly3, k_2, j_2, j', j'', j''', e_1
  57. CurveCartesian(t, (0.5 * (lower(t) + upper(t))) - (upper(t) - lower(t)), (((-sqrt(3))) / 2 * (lower(t) - upper(t))), t, a, b) → v
  58. CurveCartesian(t, lower(t), ((-sqrt(3)) * (lower(t) - upper(t))), t, a, b) → c
  59. CurveCartesian(t, upper(t), ((-sqrt(3)) * (lower(t) - upper(t))), t, a, b) → w
  60. CurveCartesian(t, (0.5 * (lower(t) + upper(t))) + upper(t) - lower(t), (((-sqrt(3))) / 2 * (lower(t) - upper(t))), t, a, b) → o_1
  61. Surface(u, lower(u) + ((-v) * (upper(u) - lower(u)) / 2), ((v * sqrt(3)) * (upper(u) - lower(u)) / 2), u, a, b, v, 0, 1) → o
  62. Surface(u, lower(u) - (0.5 * (upper(u) - lower(u))) + ((v * 0.5) * (upper(u) - lower(u))), (sqrt(3) / 2 * (upper(u) - lower(u))) + ((v * sqrt(3) / 2) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u
  63. Surface(u, lower(u) + (v * (upper(u) - lower(u))), (sqrt(3) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u_1
  64. Surface(u, upper(u) + (0.5 * (upper(u) - lower(u))) + (((-v) * 0.5) * (upper(u) - lower(u))), (sqrt(3) / 2 * (upper(u) - lower(u))) + ((v * sqrt(3) / 2) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → v_1
  65. Surface(u, upper(u) + (v * (upper(u) - lower(u)) / 2), ((v * sqrt(3)) * (upper(u) - lower(u)) / 2), u, a, b, v, 0, 1) → w_1