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可修改固体：平行于y轴的正十边形截面

可修改固体：平行于y轴的正十边形截面
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2019-05-14 08:38
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)
①本范例演示一个可修改固体，其平行于y轴的截面均为正十边形；②涉及核心数学概念包括立体几何、解析几何、微积分中的积分与函数；③作图要点提炼自GeoGebra命令：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）、Surface（曲面）；④适合高中及以上学生及微积分学习者，用于理解截面法与几何变换。

GeoGebra 命令及参数

  1. Intersect(lower, h, (-10, -11.367936667331891)) → C
  2. Intersect(upper, h, (-10, 1)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 1)) → 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. Rotate(J, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → J'_1
  40. Rotate(J'_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → K_1
  41. Rotate(K_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → L_1
  42. Rotate(L_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → M_1
  43. Rotate(M_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → N_1
  44. Rotate(N_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → O_1
  45. Rotate(O_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → P_1
  46. Rotate(H, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → H'_1
  47. Rotate(H'_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → R_1
  48. Rotate(R_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → S_1
  49. Rotate(S_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → T_1
  50. Rotate(T_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → U_1
  51. Rotate(U_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → V_1
  52. Rotate(V_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → W_1
  53. Line(Z_1, Z_1 + (1, 0, 0)) → j
  54. Rotate(J, pi / 5, Z_1, j) → J'_2
  55. Rotate(J'_2, pi / 5, Z_1, j) → A_2
  56. Rotate(A_2, pi / 5, Z_1, j) → B_2
  57. Rotate(B_2, pi / 5, Z_1, j) → C_2
  58. Rotate(C_2, pi / 5, Z_1, j) → D_2
  59. Rotate(D_2, pi / 5, Z_1, j) → E_2
  60. Rotate(E_2, pi / 5, Z_1, j) → F_2
  61. Rotate(F_2, pi / 5, Z_1, j) → G_2
  62. Polygon(K, J, J'_2, A_2, B_2, C_2, D_2, E_2, F_2, G_2) → poly2, k, g_3, j_1, j'_2, a_2, b_2, c_2, d_2, e_2, f_2
  63. Rotate(G, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → G'_2
  64. Rotate(G'_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → I_2
  65. Rotate(I_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → J_2
  66. Rotate(J_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → K_2
  67. Rotate(K_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → L_2
  68. Rotate(L_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → M_2
  69. Rotate(M_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → N_2
  70. Rotate(N_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → O_2
  71. Polygon(C, O_2, N_2, M_2, L_2, K_2, J_2, I_2, G'_2, G) → poly3, c_4, g_4, o_2, n_2, m_2, l_2, k_2, j_2, i_2, g'_2
  72. Segment(C, O_2) → l
  73. Segment(O_2, N_2) → m
  74. Segment(N_2, M_2) → n
  75. Segment(M_2, L_2) → r
  76. Segment(L_2, K_2) → s
  77. Segment(K_2, J_2) → t
  78. Segment(J_2, I_2) → m_1
  79. Segment(I_2, G'_2) → n_1
  80. Segment(G'_2, G) → p_1
  81. Segment(J, J'_2) → a_1
  82. Segment(J'_2, A_2) → b_1
  83. Segment(A_2, B_2) → p_2
  84. Segment(B_2, C_2) → q_2
  85. Segment(C_2, D_2) → r_2
  86. Segment(D_2, E_2) → s_2
  87. Segment(E_2, F_2) → t_2
  88. Segment(F_2, G_2) → f_3
  89. Segment(G_2, K) → h_3
  90. Rotate(H, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → H'_2
  91. Rotate(H'_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → Q_2
  92. Rotate(Q_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → R_2
  93. Rotate(R_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → S_2
  94. Rotate(S_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → T_2
  95. Rotate(T_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → U_2
  96. Rotate(U_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → V_2
  97. Rotate(V_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → W_2
  98. Polygon(I, H, H'_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2) → poly4, i_3, w_1, h_4, h'_2, q_3, r_3, s_3, t_3, u_2, v_2
  99. Segment(H, H'_2) → k_3
  100. Segment(H'_2, Q_2) → m_3
  101. Segment(Q_2, R_2) → n_3
  102. Segment(R_2, S_2) → p_3
  103. Segment(S_2, T_2) → f_4
  104. Segment(T_2, U_2) → i_4
  105. Segment(U_2, V_2) → j_4
  106. Segment(V_2, W_2) → k_4
  107. Segment(W_2, I) → l_4
  108. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * sin(pi / 5)), t, a, b) → o
  109. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * sin(pi / 5)), t, a, b) → u
  110. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos(pi / 5) + cos((2 * pi / 5)))), ((upper(t) - lower(t)) * (sin(pi / 5) + sin((2 * pi / 5)))), t, a, b) → v
  111. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos(pi / 5) + cos((2 * pi / 5)))), ((upper(t) - lower(t)) * (sin(pi / 5) + sin((2 * pi / 5)))), t, a, b) → w
  112. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * (sin(pi / 5) + (2 * sin((2 * pi / 5))))), t, a, b) → o_1
  113. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * (sin(pi / 5) + (2 * sin((2 * pi / 5))))), t, a, b) → u_1
  114. CurveCartesian(t, lower(t), ((upper(t) - lower(t)) * ((2 * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), t, a, b) → v_1
  115. CurveCartesian(t, upper(t), ((upper(t) - lower(t)) * ((2 * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), t, a, b) → o_3
  116. Surface(u, lower(u) - (((upper(u) - lower(u)) * cos(pi / 5)) * v), (((upper(u) - lower(u)) * sin(pi / 5)) * v), u, a, b, v, 0, 1) → u_3
  117. Surface(u, upper(u) + (((upper(u) - lower(u)) * cos(pi / 5)) * v), (((upper(u) - lower(u)) * sin(pi / 5)) * v), u, a, b, v, 0, 1) → v_3
  118. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos(pi / 5) + (v * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + (v * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → w_3
  119. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos(pi / 5) + (v * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + (v * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → a_4
  120. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos(pi / 5) + ((1 - v) * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + ((1 + v) * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → b_4
  121. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos(pi / 5) + ((1 - v) * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + ((1 + v) * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → d_4
  122. Surface(u, lower(u) - ((upper(u) - lower(u)) * ((1 - v) * cos(pi / 5))), ((upper(u) - lower(u)) * (((1 + v) * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → e_4
  123. Surface(u, upper(u) + ((upper(u) - lower(u)) * ((1 - v) * cos(pi / 5))), ((upper(u) - lower(u)) * (((1 + v) * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → m_4
  124. Surface(u, lower(u) + (v * (upper(u) - lower(u))), ((upper(u) - lower(u)) * tan((2 * pi / 5))), u, a, b, v, 0, 1) → n_4

AI 绘图提示语参考

绘制一个可调节的立体图形，其沿y轴方向的横截面为正十边形。首先在y轴下方和上方分别绘制两个正十边形，然后通过旋转和移动这些正十边形生成立体图形。使用滑动条调整正十边形的大小和位置，观察立体图形的变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。