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卡瓦利埃原理：棱柱与立方体截面面积与体积关系演示

卡瓦利埃原理：棱柱与立方体截面面积与体积关系演示
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2016-03-27 13:30
立体几何积分几何作图
所属专题 Geometry - Area & Volume
该范例演示卡瓦利埃原理，通过构造棱柱与立方体并比较其平行截面面积，说明等高且各对应截面积相等的立体图形体积相等。核心数学概念涵盖立体几何、体积不变性、横截面积、平行截面、卡瓦利埃原理及其与定积分体积公式的联系。作图要点：调用了 GeoGebra 命令 Cube、Prism、Plane、Intersect、Polygon、Segment、Line、Circle、PointIn，建立了顶点 A、B、C、D、E、F、G、H，以及面对象 caraABDC、caraBDHE 等和棱对象 aristaBD、aristaAB 等，并借助点 I、K、L、M、N、Q、R 与线段 b、c、d、e、j 实现截面动态展示。适合高中立体几何课堂教学，也可作为大学微积分引入体积积分的直观辅助工具。

GeoGebra 命令及参数

  1. Cube(B, A, xOyPlane) → a, C, caraACGF, caraCDHG, caraEFGH, aristaBD, aristaAB, aristaAC, aristaCD, aristaBE, aristaDH, aristaEH, D, aristaEF, aristaAF, aristaFG, aristaCG, aristaGH, E, F, G, H, caraABDC, caraBDHE, caraABEF
  2. Intersect(xAxis, yAxis) → I
  3. Plane(K, xOyPlane) → c
  4. PointIn(caraEFGH) → MOVER
  5. Segment(MOVER, A) → b
  6. Intersect(b, c) → L
  7. Line(L, xAxis) → d
  8. Circle(L, 5.7, c) → e
  9. Intersect(d, e) → M, N
  10. Plane(Q, c) → j
  11. Intersect(b, j) → R
  12. Line(R, d) → k
  13. Circle(R, 5.4, j) → p
  14. Intersect(k, p) → S, T
  15. Plane(W, j) → o
  16. Intersect(b, o) → Z
  17. Line(Z, d) → r
  18. Circle(Z, 5.1, o) → s
  19. Intersect(r, s) → A_1, B_1
  20. Plane(E_1, o) → u
  21. Intersect(b, u) → F_1
  22. Line(F_1, d) → d_1
  23. Circle(F_1, 4.8, u) → e_1
  24. Intersect(d_1, e_1) → G_1, H_1
  25. Polygon(F_1, H_1, 4, u) → polígono4, f_1, g_1, h_1, i_1, I_1, J_1
  26. Plane(K_1, u) → v
  27. Intersect(b, v) → L_1
  28. Line(L_1, d) → j_1
  29. Circle(L_1, 4.5, v) → k_1
  30. Intersect(j_1, k_1) → M_1, N_1
  31. Plane(Q_1, v) → w
  32. Intersect(b, w) → R_1
  33. Line(R_1, d) → q_1
  34. Circle(R_1, 4.2, w) → r_1
  35. Intersect(q_1, r_1) → S_1, T_1
  36. Plane(W_1, w) → o_1
  37. Intersect(b, o_1) → Z_1
  38. Line(Z_1, d) → c_2
  39. Circle(Z_1, 3.9, o_1) → d_2
  40. Intersect(c_2, d_2) → A_2, B_2
  41. Plane(E_2, o_1) → u_1
  42. Intersect(b, u_1) → F_2
  43. Line(F_2, d) → i_2
  44. Circle(F_2, 3.6, u_1) → k_2
  45. Intersect(i_2, k_2) → G_2, H_2
  46. Polygon(F_2, H_2, 4, u_1) → polígono8, j_2, l_2, m_2, n_2, I_2, J_2
  47. Plane(K_2, u_1) → v_1
  48. Intersect(b, v_1) → L_2
  49. Line(L_2, d) → p_2
  50. Circle(L_2, 3.3000000000000003, v_1) → q_2
  51. Intersect(p_2, q_2) → M_2, N_2
  52. Plane(Q_2, v_1) → w_1
  53. Intersect(b, w_1) → R_2
  54. Line(R_2, d) → b_3
  55. Circle(R_2, 3, w_1) → c_3
  56. Intersect(b_3, c_3) → S_2, T_2
  57. Plane(W_2, w_1) → o_2
  58. Intersect(b, o_2) → Z_2
  59. Line(A_3, b) → h_3
  60. Line(Z_2, d) → i_3
  61. Circle(Z_2, 2.7, o_2) → k_3
  62. Intersect(i_3, k_3) → B_3, C_3
  63. Plane(F_3, o_2) → u_2
  64. Intersect(b, u_2) → G_3
  65. Line(G_3, d) → p_3
  66. Circle(G_3, 2.4000000000000004, u_2) → q_3
  67. Intersect(p_3, q_3) → H_3, I_3
  68. Polygon(G_3, I_3, 4, u_2) → polígono12, r_3, s_3, t_3, a_4, J_3, K_3
  69. Plane(L_3, u_2) → v_2
  70. Intersect(b, v_2) → M_3
  71. Line(M_3, d) → b_4
  72. Circle(M_3, 2.1, v_2) → c_4
  73. Intersect(b_4, c_4) → N_3, O_3
  74. Plane(R_3, v_2) → w_2
  75. Intersect(b, w_2) → S_3
  76. Line(S_3, d) → h_4
  77. Circle(S_3, 1.7999999999999998, w_2) → k_4
  78. Intersect(h_4, k_4) → T_3, U_3
  79. Plane(Z_3, w_2) → o_3
  80. Intersect(b, o_3) → A_4
  81. Line(A_4, d) → n_4
  82. Circle(A_4, 1.5, o_3) → p_4
  83. Intersect(n_4, p_4) → B_4, C_4
  84. Plane(F_4, o_3) → u_3
  85. Intersect(b, u_3) → G_4
  86. Line(G_4, d) → a_5
  87. Circle(G_4, 1.2000000000000002, u_3) → c_5
  88. Intersect(a_5, c_5) → H_4, I_4
  89. Polygon(G_4, I_4, 4, u_3) → polígono16, b_5, d_5, e_5, f_5, J_4, K_4
  90. Plane(L_4, u_3) → v_3
  91. Intersect(b, v_3) → M_4
  92. Line(M_4, d) → g_5
  93. Circle(M_4, 0.9000000000000004, v_3) → h_5
  94. Intersect(g_5, h_5) → N_4, O_4
  95. Plane(R_4, v_3) → w_3
  96. Intersect(b, w_3) → S_4
  97. Line(S_4, d) → m_5
  98. Circle(S_4, 0.6000000000000005, w_3) → p_5
  99. Intersect(m_5, p_5) → T_4, U_4
  100. Plane(Z_4, w_3) → o_4
  101. Intersect(b, o_4) → A_5
  102. Line(A_5, d) → t_5
  103. Circle(A_5, 0.2999999999999998, o_4) → c_6
  104. Intersect(t_5, c_6) → B_5, C_5
  105. Polygon(B, A, C, D) → polígono20, b_7, a_7, c_7, d_7
  106. Prism(polígono20, 1.2) → u_4, V_2, arista1, arista2, arista3, arista4, arista5, arista6, arista7, arista8, D_3, E_3, P_3, cara1, cara2, cara3, cara4, cara5
  107. Prism(polígono4, 1.2) → u_5, D_2, arista33, arista34, arista35, arista36, arista37, arista38, arista39, arista40, O_2, P_2, U_2, cara21, cara22, cara23, cara24, cara25
  108. Prism(polígono8, 1.2) → g_6, J, arista9, arista10, arista11, arista12, arista13, arista14, arista15, arista16, O, P, U, cara6, cara7, cara8, cara9, cara10
  109. Prism(polígono12, 1.2) → k_6, V, arista17, arista18, arista19, arista20, arista21, arista22, arista23, arista24, C_1, D_1, O_1, cara11, cara12, cara13, cara14, cara15
  110. Prism(polígono16, 1.2) → o_6, P_1, arista25, arista26, arista27, arista28, arista29, arista30, arista31, arista32, U_1, V_1, C_2, cara16, cara17, cara18, cara19, cara20