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可修改固体：平行于y轴的正八边形截面

可修改固体：平行于y轴的正八边形截面
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2019-05-07 00:59
解析几何立体几何微积分几何作图
所属专题 GeoGebra 3D & AR: PreCalc & Calculus Resources Cross Sectional Specials (Calculus)
这个范例演示了一个可修改固体，其平行于y轴的截面是正八边形。涉及的核心数学概念包括立体几何、解析几何和微积分。作图要点使用了GeoGebra的Segment、Midpoint、Intersect、Line、Textfield、Rotate、Polygon、Surface、CurveCartesian等命令和对象。适合高中及以上学生学习和理解复杂几何体体积计算和截面分析。

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, (-10, 10.876772823989416)) → B
  8. Intersect(lower, h, (-10, -2.0796594266992683)) → C
  9. Segment(C, B) → q
  10. Intersect(upper, q, (-9.999999999999998, 10.876772823989416)) → G
  11. Midpoint(C, G) → T
  12. Line(B, D) → i
  13. Segment(C, B) → p
  14. Textfield(upper) → InputBox1
  15. Textfield(lower) → InputBox2
  16. Textfield(a) → InputBox3
  17. Textfield(b) → InputBox4
  18. Segment(C, G) → k_1
  19. Line(C, G) → j
  20. Line(T, T + (1, 0, 0)) → m
  21. Rotate(G, pi / 3, F_1, m) → G'
  22. Rotate(G', pi / 3, F_1, Line[F_1, F_1 + (1, 0, 0)]) → G''
  23. Line(F_1, F_1 + (1, 0, 0)) → n
  24. Rotate(G'', pi / 3, F_1, n) → G'''
  25. Rotate(G''', pi / 3, F_1, n) → G_1
  26. Line(Z, Z + (1, 0, 0)) → f
  27. Line(D_1, D_1 + (1, 0, 0)) → g
  28. Line(H_1, H_1 - (1, 0, 0)) → l_1
  29. Rotate(I, pi / 3, H_1, l_1) → I'
  30. Rotate(I', pi / 3, H_1, l_1) → I''
  31. Rotate(I'', pi / 3, H_1, l_1) → I'''
  32. Rotate(I''', pi / 3, H_1, l_1) → I_1
  33. Line(E_1, E_1 + (1, 0, 0)) → l
  34. Rotate(G, pi / 4, E_1, l) → G'_1
  35. Rotate(G'_1, pi / 4, E_1, l) → J_1
  36. Rotate(J_1, pi / 4, E_1, l) → K_1
  37. Rotate(K_1, pi / 4, E_1, l) → L_1
  38. Rotate(L_1, pi / 4, E_1, l) → M_1
  39. Rotate(M_1, pi / 4, E_1, l) → N_1
  40. Polygon(G, G'_1, J_1, K_1, L_1, M_1, N_1, C) → poly3, g_1, g'_1, j_1, k_2, l_2, m_1, n_1, c
  41. Segment(G'_1, J_1) → r
  42. Segment(J_1, K_1) → s
  43. Segment(K_1, L_1) → t
  44. Segment(L_1, M_1) → f_1
  45. Segment(M_1, N_1) → i_1
  46. Segment(N_1, C) → p_1
  47. Segment(G, G'_1) → r_1
  48. Rotate(J, pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → J'
  49. Rotate(J', pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → J''
  50. Rotate(J'', pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → J'''
  51. Rotate(J''', pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → P_1
  52. Rotate(P_1, pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → Q_1
  53. Rotate(Q_1, pi / 4, Line[O_1, O_1 + (1, 0, 0)]) → R_1
  54. Polygon(K, J, J', J'', J''', P_1, Q_1, R_1) → poly4, k_3, j_2, j', j'', j''', p_2, q_2, r_2
  55. Segment(J, J') → a_1
  56. Segment(J', J'') → b_1
  57. Segment(J'', J''') → d_1
  58. Segment(J''', P_1) → e_1
  59. Segment(P_1, Q_1) → f_2
  60. Segment(Q_1, R_1) → m_2
  61. Segment(R_1, K) → n_2
  62. Rotate(H, pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → H'
  63. Rotate(H', pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → H''
  64. Rotate(H'', pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → H'''
  65. Rotate(H''', pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → T_1
  66. Rotate(T_1, pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → U_1
  67. Rotate(U_1, pi / 4, Line[S_1, S_1 + (1, 0, 0)]) → V_1
  68. Polygon(H, I, V_1, U_1, T_1, H''', H'', H') → poly5, h_3, i_4, v_2, u_2, t_2, h''', h'', h'
  69. Segment(I, V_1) → s_2
  70. Segment(V_1, U_1) → a_2
  71. Segment(U_1, T_1) → b_2
  72. Segment(T_1, H''') → c_2
  73. Segment(H''', H'') → d_2
  74. Segment(H'', H') → e_2
  75. Segment(H', H) → f_3
  76. Surface(u, lower(u) - (v * (upper(u) - lower(u)) / sqrt(2)), (v * (upper(u) - lower(u)) / sqrt(2)), u, a, b, v, 0, 1) → v
  77. CurveCartesian(t, lower(t) - (upper(t) - lower(t)) / sqrt(2), (upper(t) - lower(t)) / sqrt(2), t, a, b) → o
  78. CurveCartesian(t, lower(t) - (upper(t) - lower(t)) / sqrt(2), (upper(t) - lower(t)) / sqrt(2) + upper(t) - lower(t), t, a, b) → u
  79. CurveCartesian(t, lower(t), (2 * (upper(t) - lower(t)) / sqrt(2)) + upper(t) - lower(t), t, a, b) → w
  80. CurveCartesian(t, upper(t), (2 * (upper(t) - lower(t)) / sqrt(2)) + upper(t) - lower(t), t, a, b) → o_1
  81. CurveCartesian(t, upper(t) + (upper(t) - lower(t)) / sqrt(2), (upper(t) - lower(t)) / sqrt(2) + upper(t) - lower(t), t, a, b) → u_1
  82. CurveCartesian(t, upper(t) + (upper(t) - lower(t)) / sqrt(2), (upper(t) - lower(t)) / sqrt(2), t, a, b) → v_1
  83. Surface(u, upper(u) + (v * (upper(u) - lower(u)) / sqrt(2)), (v * (upper(u) - lower(u)) / sqrt(2)), u, a, b, v, 0, 1) → w_1
  84. Surface(u, lower(u) - (upper(u) - lower(u)) / sqrt(2), (v * (upper(u) - lower(u))) + (upper(u) - lower(u)) / sqrt(2), u, a, b, v, 0, 1) → o_2
  85. Surface(u, lower(u) - (upper(u) - lower(u)) / sqrt(2) + (v * (upper(u) - lower(u)) / sqrt(2)), upper(u) - lower(u) + (upper(u) - lower(u)) / sqrt(2) + ((v * (upper(u) - lower(u)))) / sqrt(2), u, a, b, v, 0, 1) → w_2
  86. Surface(u, upper(u) + (upper(u) - lower(u)) / sqrt(2), (v * (upper(u) - lower(u))) + (upper(u) - lower(u)) / sqrt(2), u, a, b, v, 0, 1) → a_3
  87. Surface(u, lower(u) + (v * (upper(u) - lower(u))), upper(u) - lower(u) + (upper(u) - lower(u)) / sqrt(2) + (upper(u) - lower(u)) / sqrt(2), u, a, b, v, 0, 1) → b_3
  88. Surface(u, upper(u) + (v * (upper(u) - lower(u)) / sqrt(2)), ((1 + sqrt(2)) * (upper(u) - lower(u))) - (v * (upper(u) - lower(u)) / sqrt(2)), u, a, b, v, 0, 1) → c_3