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三等分长方体

三等分长方体
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2015-03-26 03:07
向量立体几何几何作图
①本范例演示如何通过几何构造将长方体三等分体积；②涉及核心数学概念包括立体几何、体积计算、向量运算、多边形与棱锥构造；③构造过程中运用了Pyramid（棱锥）、Plane（平面）、IntersectPath（路径相交）、PointIn（点在路径上）、Volume（体积）、Point（点）等GeoGebra命令与对象；④适合高中及以上学生用于学习立体几何体积分割、空间向量及几何作图。

GeoGebra 命令及参数

  1. Pyramid(E, B, D, C, A) → a, faceBDCE, edgeAE, edgeAB_1, edgeAD_1, edgeAC_1, faceABE, faceABD_1, faceACD_1, faceACE, edgeBE, edgeBD_1, edgeCD_1, edgeCE
  2. Pyramid(G, F, D, C, A) → b, faceCDFG, edgeAG, edgeAF_1, edgeAD_2, edgeAC, faceAFG, faceADF_1, faceACD, faceACG, edgeFG, edgeDF_1, edgeCD, edgeCG
  3. Pyramid(faceADF_1, F) → c, faceDFF, faceADF_2, faceAFF, edgeFF, edgeDF_1, edgeAF_1
  4. Pyramid(H, B, D, F, A) → d, faceBDFH, edgeAH, edgeAB, edgeAD, edgeAF, faceABH, faceABD, faceADF, faceAFH, edgeBH, edgeBD, edgeDF, edgeFH
  5. Plane(I, faceAFG) → k
  6. Plane(J, faceACE) → l
  7. Plane(K, faceCDFG) → m
  8. IntersectPath(k, b) → poly1, L, M, N, O, q, r, s, t
  9. IntersectPath(k, d) → poly2, poly3, c_1, d_1, e_1, f_1, P, Q, R, S, T, U, a_1, b_1
  10. IntersectPath(l, a) → poly4, poly6, i_1, j_1, p_1, q_1, V, W, Z, A_1, F_1, G_1, g_1, h_1
  11. IntersectPath(l, b) → poly5, poly7, m_1, n_1, r_1, s_1, B_1, C_1, D_1, E_1, H_1, I_1, k_1, l_1
  12. IntersectPath(m, a) → poly8, poly10, b_2, c_2, h_2, i_2, J_1, K_1, L_1, M_1, R_1, S_1, t_1, a_2
  13. IntersectPath(m, d) → poly9, poly11, f_2, g_2, j_2, k_2, N_1, O_1, P_1, Q_1, T_1, U_1, d_2, e_2
  14. PointIn(faceABH) → PointfaceABH
  15. PointIn(faceABE) → PointfaceABE
  16. PointIn(poly9) → Pointpoly9_1
  17. PointIn(poly8) → Pointpoly8_1
  18. PointIn(faceACG) → PointfaceACG
  19. PointIn(faceACE) → PointfaceACE
  20. PointIn(poly5) → Pointpoly5_1
  21. PointIn(poly4) → Pointpoly4_1
  22. PointIn(poly2) → Pointpoly2_1
  23. PointIn(poly1) → Pointpoly1_1
  24. PointIn(poly5) → Pointpoly5_2
  25. PointIn(poly4) → Pointpoly4_2
  26. PointIn(poly9) → Pointpoly9_2
  27. PointIn(poly8) → Pointpoly8_2
  28. Volume(d) → volumed
  29. Point(d) → Pointd_1
  30. Volume(b) → volumeb
  31. Point(b) → Pointb_1
  32. Volume(d) → o_1
  33. Point(d) → Pointd_2
  34. PointIn(poly2) → Pointpoly2_2
  35. PointIn(poly1) → Pointpoly1_2
  36. Volume(d) → l_2
  37. Point(d) → Pointd_3
  38. Volume(b) → m_2
  39. Point(b) → Pointb
  40. Volume(a) → volumea
  41. Point(a) → Pointa
  42. Volume(d) → n_2
  43. Point(d) → Pointd
  44. PointIn(poly2) → Pointpoly2_3
  45. PointIn(poly1) → Pointpoly1_3
  46. PointIn(poly2) → Pointpoly2_4
  47. PointIn(poly1) → Pointpoly1_4
  48. PointIn(poly5) → Pointpoly5
  49. PointIn(poly4) → Pointpoly4
  50. PointIn(poly9) → Pointpoly9_3
  51. PointIn(poly8) → Pointpoly8
  52. PointIn(poly9) → Pointpoly9
  53. PointIn(poly2) → Pointpoly2
  54. PointIn(poly1) → Pointpoly1

AI 绘图提示语参考

绘制一个长方体，将其体积三等分。先构造四个三棱锥，分别以长方体的不同顶点为顶点，底面为长方体的面或对角面。然后，在三个关键截面上各取一个平面，这些平面分别穿过三棱锥的特定面，并与三棱锥相交，形成多个交线路径和多边形截面。最后，在生成的各个面和多边形内部随机放置点，并计算各三棱锥的体积，以验证三等分效果。整体图形展示空间几何中的体积分割与截面构造。
以上提示语为根据原始绘图命令整理而来，仅供参考。