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可修改固体：平行于y轴的正九边形截面

可修改固体：平行于y轴的正九边形截面
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2019-05-10 18:36
多边形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轴的截面是正九边形。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Intersect（相交）、Segment（线段）、Midpoint（中点）、Line（直线）、Sphere（球体）、Textfield（文本框）、Rotate（旋转）、Polygon（多边形）、CurveCartesian（笛卡尔曲线）和Surface（曲面）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及其截面性质。

GeoGebra 命令及参数

  1. Intersect(lower, h, (-10, -3)) → C
  2. Intersect(upper, h, (-10, 13.333333333333332)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 13.333333333333332)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Rotate(J, (2 * pi / 5), Z, c) → J'
  22. Rotate(J', (2 * pi / 5), Z, c) → J''
  23. Rotate(J'', (2 * pi / 5), Z, c) → J'''
  24. Rotate(H, (2 * pi / 5), A_1, r_1) → H'
  25. Rotate(H', (2 * pi / 5), A_1, r_1) → H''
  26. Rotate(H'', (2 * pi / 5), A_1, r_1) → H'''
  27. Rotate(G, (2 * pi / 5), B_1, l_1) → G'
  28. Rotate(G', (2 * pi / 5), B_1, l_1) → G''
  29. Rotate(G'', (2 * pi / 5), B_1, l_1) → G'''
  30. Segment(C, G) → f
  31. Line(C_1, C_1 + (1, 0, 0)) → g
  32. Rotate(G, (2 * pi / 9), C_1, g) → G'_1
  33. Rotate(G'_1, (2 * pi / 9), C_1, g) → D_1
  34. Rotate(D_1, (2 * pi / 9), C_1, g) → E_1
  35. Rotate(E_1, (2 * pi / 9), C_1, g) → F_1
  36. Rotate(F_1, (2 * pi / 9), C_1, g) → G_1
  37. Rotate(G_1, (2 * pi / 9), C_1, g) → H_1
  38. Rotate(H_1, (2 * pi / 9), C_1, g) → I_1
  39. Polygon(C, G, G'_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly1, c_1, g_1, g'_1, d_1, e_1, f_1, g_2, h_2, i_1
  40. Segment(G, G'_1) → j
  41. Segment(G'_1, D_1) → k
  42. Segment(D_1, E_1) → l
  43. Segment(E_1, F_1) → m
  44. Segment(F_1, G_1) → n
  45. Segment(G_1, H_1) → r
  46. Segment(H_1, I_1) → s
  47. Segment(I_1, C) → t
  48. Rotate(J, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → J'_1
  49. Rotate(J'_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → K_1
  50. Rotate(K_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → L_1
  51. Rotate(L_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → M_1
  52. Rotate(M_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → N_1
  53. Rotate(N_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → O_1
  54. Rotate(O_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → P_1
  55. Polygon(K, J, J'_1, K_1, L_1, M_1, N_1, O_1, P_1) → poly2, k_2, j_1, j'_1, k_3, l_2, m_1, n_1, o_1, p_1
  56. Segment(J, J'_1) → a_1
  57. Segment(J'_1, K_1) → b_1
  58. Segment(K_1, L_1) → f_2
  59. Segment(L_1, M_1) → i_2
  60. Segment(M_1, N_1) → j_2
  61. Segment(N_1, O_1) → m_2
  62. Segment(O_1, P_1) → n_2
  63. Segment(P_1, K) → p_2
  64. Rotate(H, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → H'_1
  65. Rotate(H'_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → R_1
  66. Rotate(R_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → S_1
  67. Rotate(S_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → T_1
  68. Rotate(T_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → U_1
  69. Rotate(U_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → V_1
  70. Rotate(V_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → W_1
  71. Polygon(H, H'_1, R_1, S_1, T_1, U_1, V_1, W_1, I) → poly3, h_3, h'_1, r_2, s_2, t_2, u_1, v_1, w_1, i_3
  72. Segment(I, W_1) → q_2
  73. Segment(W_1, V_1) → a_2
  74. Segment(V_1, U_1) → b_2
  75. Segment(U_1, T_1) → c_2
  76. Segment(T_1, S_1) → d_2
  77. Segment(S_1, R_1) → e_2
  78. Segment(R_1, H'_1) → f_3
  79. Segment(H'_1, H) → g_3
  80. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos((2 * pi / 9))), ((upper(t) - lower(t)) * sin((2 * pi / 9))), t, a, b) → o
  81. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos((2 * pi / 9))), ((upper(t) - lower(t)) * sin((2 * pi / 9))), t, a, b) → u
  82. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)))), t, a, b) → v
  83. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)))), t, a, b) → w
  84. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5)), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2)), t, a, b) → o_2
  85. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5)), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2)), t, a, b) → u_2
  86. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * (tan((7 * pi / 18)) + 1 / cos((7 * pi / 18)))), t, a, b) → v_2
  87. Surface(u, lower(u) - ((v * (upper(u) - lower(u))) * cos((2 * pi / 9))), ((v * (upper(u) - lower(u))) * sin((2 * pi / 9))), u, a, b, v, 0, 1) → w_2
  88. Surface(u, upper(u) + ((v * (upper(u) - lower(u))) * cos((2 * pi / 9))), ((v * (upper(u) - lower(u))) * sin((2 * pi / 9))), u, a, b, v, 0, 1) → a_3
  89. Surface(u, lower(u) - ((upper(u) - lower(u)) * cos((2 * pi / 9))) - (((upper(u) - lower(u)) * cos((4 * pi / 9))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 9))) + (((upper(u) - lower(u)) * sin((4 * pi / 9))) * v), u, a, b, v, 0, 1) → b_3
  90. Surface(u, upper(u) + ((upper(u) - lower(u)) * cos((2 * pi / 9))) + (((upper(u) - lower(u)) * cos((4 * pi / 9))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 9))) + (((upper(u) - lower(u)) * sin((4 * pi / 9))) * v), u, a, b, v, 0, 1) → c_3
  91. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))) + ((0.5 * (upper(u) - lower(u))) * v), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + (sqrt(3) / 2 * v))), u, a, b, v, 0, 1) → d_3
  92. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - (0.5 * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + (sqrt(3) / 2 * v))), u, a, b, v, 0, 1) → e_3
  93. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5 - (cos(pi / 9) * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2 + (v * sin(pi / 9)))), u, a, b, v, 0, 1) → l_3
  94. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5 - (cos(pi / 9) * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2 + (v * sin(pi / 9)))), u, a, b, v, 0, 1) → j_3