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矩形平截头体的体积计算与动态演示

矩形平截头体的体积计算与动态演示
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2017-09-09 12:28
多边形立体几何导数极限几何作图
① 本范例通过GeoGebra三维视图动态演示一个矩形平截头体（即被平面截去顶部的四棱锥台）的体积计算过程，直观展示其立体结构与尺寸变化。② 核心数学概念包括立体几何中的棱台体积公式、相似三角形比例关系、导数与极限思想（当顶部截面趋近于底面时退化为棱柱，趋近于顶点时退化为棱锥），以及体积随高度变化的函数关系。③ 作图要点：利用GeoGebra命令如Midpoint（中点）、Translate（平移）、Line（直线）、OrthogonalPlane（正交平面）、Polygon（多边形）、Vector（向量）等构造底面与顶面矩形，通过Segment（线段）连接对应顶点形成棱台，并使用If（条件判断）控制动态参数；对象包括多个顶点（A, B, C, D等）、辅助点（H, I, J等）以及平面和向量，整体支持滑块调节高度与截面位置。④ 适合高中或大学低年级学生用于理解立体几何体积公式的推导，也可作为微积分中“截面法”求体积的直观教具，帮助建立从几何到分析的思维过渡。

GeoGebra 命令及参数

  1. Midpoint(B2', B3) → U
  2. Translate(Prism[H, B2', B2, I], Vector[(0, 0, 0)]) → q
  3. Midpoint(B, B1') → V
  4. Line(U, V) → p
  5. If(t2 < 0.5, (2 * t2), 1) → t2'
  6. If(t2 < 0.5, 0, (2 * (t2 - 0.5))) → t3'
  7. Midpoint(B3, B') → K
  8. Midpoint(B2, B1') → L
  9. Line(L, K) → i
  10. Segment(E, B3') → m
  11. Midpoint(m) → N
  12. OrthogonalPlane(N, m) → n_1
  13. Segment(D, B') → k
  14. Midpoint(D, B') → M
  15. OrthogonalPlane(M, k) → l
  16. Translate(Pyramid[A, D, B', C, B], Vector[(t1 * Vector[B', O])]) → e
  17. Translate(Pyramid[A1, F, B1', E, B1], Vector[(t1 * Vector[B1', O])]) → f
  18. Translate(Pyramid[A3, J, B3', I, B3], Vector[(t1 * Vector[B3', O])]) → g
  19. Translate(Prism[J, B3', B3, C], Vector[(0, 0, 0)]) → j
  20. Translate(Pyramid[A2, G, B2', H, B2], Vector[(t1 * Vector[B2', O])]) → h
  21. Translate(Rotate[Rotate[q, 180°, zAxis], (180° * t2'), p], Vector[(t3' * Vector[B', I])]) → q'
  22. Translate(Rotate[Prism[F, B1', B1, G], ((t2' * 180))°, i], Vector[(t3' * Vector[B2', J])]) → o
  23. Translate(Prism[B', B1', B2', B3', B], Vector[(0, 0 - 0)]) → r
  24. Polygon(B, B1, B2, B3) → q2, b_1, b1, b2, b3
  25. Segment(A, B) → s
  26. Segment(A1, B1) → t
  27. Segment(A2, B2) → f_1
  28. Segment(A3, B3) → g_1
  29. Segment(P - (0.5, 0), P - (1.5, 0)) → b
  30. Polygon(A, A1, A2, A3) → q1, a, a1, a2, a3
  31. Translate(b, Vector[(0, 0, 0), (0, 0, z(B))]) → b'
  32. Midpoint(b) → Q
  33. Translate(Q, Vector[(0, 0, 0), (0, 0, z(B))]) → Q'
  34. Vector(Q, Q') → u
  35. Vector(Q', Q) → v
  36. Segment(H - (0, 1, 0), H - (0, 2, 0)) → c
  37. Segment(I - (0, 1, 0), I - (0, 2, 0)) → d
  38. Vector(Midpoint[c], Midpoint[d]) → w
  39. Vector(Midpoint[d], Midpoint[c]) → u_1
  40. Segment(R - (0, 1, 0), R - (0, 2, 0)) → h_1
  41. Segment(S - (0, 1, 0), S - (0, 2, 0)) → i_1
  42. Vector(Midpoint[h_1], Midpoint[i_1]) → v_1
  43. Vector(Midpoint[i_1], Midpoint[h_1]) → w_1
  44. Translate(Polygon[A, B, B3, A3], Vector[(0, 0)]) → q3
  45. Translate(Polygon[A, A1, B1, B], Vector[(0, 0)]) → q4
  46. Translate(Polygon[A1, A2, B2, B1], Vector[(0, 0)]) → q5
  47. Translate(Polygon[A2, A3, B3, B2], Vector[(0, 0)]) → q6
  48. If(t0 ≠ 0 ∧ t0 ≠ 1 ∨ t1 ≠ 0 ∧ t1 ≠ 1 ∨ t2 ≠ 0 ∧ t2 ≠ 1, false, true) → show
  49. Segment(S + (0.5, 0), S + (1.5, 0)) → j_1
  50. Segment(S + (0.5, 0), S + (1.5, 0)) → k_1
  51. Segment(T + (0.5, 0), T + (1.5, 0)) → l_1
  52. Vector(Midpoint[k_1], Midpoint[l_1]) → a_1
  53. Vector(Midpoint[l_1], Midpoint[k_1]) → c_1
  54. Segment(C + (0.3, 0), C + (1.2, 0)) → p_1
  55. Segment(J + (0.3, 0), J + (1.2, 0)) → m_1
  56. Vector(Midpoint[m_1], Midpoint[p_1]) → d_1
  57. Vector(Midpoint[p_1], Midpoint[m_1]) → e_1
  58. Segment(A3 + (0.3, 0), A3 + (1.2, 0)) → q_1
  59. Segment(Midpoint[m_1], Midpoint[q_1]) → r_1
  60. Midpoint(r_1) → W
  61. Segment(W, Z) → s_1
  62. Segment(A3 - (0, 1, 0), A3 - (0, 2, 0)) → t_1
  63. Segment(Midpoint[d], Midpoint[t_1]) → v_2
  64. Midpoint(v_2) → A_1
  65. Segment(B_1, A_1) → f_2