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立方体的两部分（三）——三维分割与几何构造

立方体的两部分（三）——三维分割与几何构造
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2019-04-28 02:58
向量立体几何变换最值优化几何作图
所属专题 TWO PARTS OF A CUBE
该范例演示了将一个立方体分割为两部分的三维几何图形，重点展示空间中的切割平面、棱与面的关系。核心数学概念包括立体几何中的立方体性质、中点、向量、平移、旋转以及棱柱（Prism）的构造。作图要点：使用GeoGebra命令Point、Cube、Midpoint、Line、Vector、Translate、Rotate、Prism，并涉及多个对象如顶点A-H、面faceABCD等、棱edgeAD等以及辅助点L、M、N，通过中点与向量操作确定分割位置，再以旋转和平移构造出两部分棱柱。适合高中或大学阶段的立体几何教学，也可用于空间想象与几何作图练习。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → c, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(xAxis) → L
  6. Point(xAxis) → M
  7. Point(Circle[M, Distance[L, M], Segment[L, M]]) → N
  8. Cube(L, M, N) → k, O, faceNOSR, facePQRS, edgeLO, edgeLM, edgeMN, edgeNO, edgeLP, edgeOS, edgePS, edgePQ, P, edgeMQ, edgeQR, edgeNR, edgeRS, Q, R, S, faceLMNO, faceLOSP, faceLMQP, faceMNRQ
  9. Midpoint(H, B) → S_2
  10. Midpoint(S, M) → S_1
  11. Line(S_1, S_2) → p_1
  12. Line(S_2, yAxis) → f
  13. Line(S_1, yAxis) → g
  14. Line(S_2, zAxis) → h
  15. Line(S_1, zAxis) → i
  16. Point(p_1) → I
  17. Point(p_1) → J
  18. Vector(I, J) → u
  19. Translate(i, Vector[(p * u)]) → i'
  20. Translate(g, Vector[(p * u)]) → g'
  21. Rotate(A, α, p_1) → A'
  22. Rotate(B, α, p_1) → B'
  23. Rotate(C, α, p_1) → C'
  24. Rotate(D, α, p_1) → D'
  25. Rotate(E, α, p_1) → E'
  26. Rotate(F, α, p_1) → F'
  27. Rotate(G, α, p_1) → G'
  28. Rotate(H, α, p_1) → H'
  29. Rotate(A', β, f) → A''
  30. Rotate(B', β, f) → B''
  31. Rotate(C', β, f) → C''
  32. Rotate(D', β, f) → D''
  33. Rotate(E', β, f) → E''
  34. Rotate(F', β, f) → F''
  35. Rotate(G', β, f) → G''
  36. Rotate(H', β, f) → H''
  37. Translate(k, Vector[(p * u)]) → k'
  38. Translate(L, Vector[(p * u)]) → L'
  39. Translate(M, Vector[(p * u)]) → M'
  40. Translate(N, Vector[(p * u)]) → N'
  41. Translate(O, Vector[(p * u)]) → O'
  42. Translate(P, Vector[(p * u)]) → P'
  43. Translate(Q, Vector[(p * u)]) → Q'
  44. Translate(R, Vector[(p * u)]) → R'
  45. Translate(S, Vector[(p * u)]) → S'
  46. Rotate(c, α, p_1) → c'
  47. Rotate(c', β, f) → c''
  48. Rotate(G'', γ, h) → G'''
  49. Rotate(H'', γ, h) → H'''
  50. Rotate(E'', γ, h) → E'''
  51. Rotate(F'', γ, h) → F'''
  52. Rotate(D'', γ, h) → D'''
  53. Rotate(C'', γ, h) → C'''
  54. Rotate(A'', γ, h) → A'''
  55. Rotate(B'', γ, h) → B'''
  56. Rotate(c'', γ, h) → c'''
  57. Rotate(k', δ, p_1) → k''
  58. Rotate(L', δ, p_1) → L''
  59. Rotate(M', δ, p_1) → M''
  60. Rotate(N', δ, p_1) → N''
  61. Rotate(O', δ, p_1) → O''
  62. Rotate(P', δ, p_1) → P''
  63. Rotate(Q', δ, p_1) → Q''
  64. Rotate(R', δ, p_1) → R''
  65. Rotate(S', δ, p_1) → S''
  66. Rotate(k'', ε, g') → k'''
  67. Rotate(L'', ε, g') → L'''
  68. Rotate(M'', ε, g') → M'''
  69. Rotate(N'', ε, g') → N'''
  70. Rotate(O'', ε, g') → O'''
  71. Rotate(P'', ε, g') → P'''
  72. Rotate(Q'', ε, g') → Q'''
  73. Rotate(R'', ε, g') → R'''
  74. Rotate(S'', ε, g') → S'''
  75. Rotate(k''', ζ, i') → a
  76. Rotate(L''', ζ, i') → K
  77. Rotate(M''', ζ, i') → T
  78. Rotate(N''', ζ, i') → U
  79. Rotate(O''', ζ, i') → V
  80. Rotate(P''', ζ, i') → W
  81. Rotate(Q''', ζ, i') → Z
  82. Rotate(R''', ζ, i') → A_1
  83. Rotate(S''', ζ, i') → B_1
  84. Midpoint(H''', G''') → C_1
  85. Midpoint(E''', F''') → D_1
  86. Midpoint(D''', C''') → E_1
  87. Midpoint(A''', B''') → F_1
  88. Midpoint(A''', D''') → G_1
  89. Midpoint(E''', H''') → H_1
  90. Midpoint(F''', G''') → I_1
  91. Midpoint(B''', C''') → J_1
  92. Midpoint(C_1, D_1) → K_1
  93. Midpoint(C_1, E_1) → L_1
  94. Midpoint(G_1, H_1) → M_1
  95. Midpoint(D_1, F_1) → N_1
  96. Midpoint(J_1, I_1) → O_1
  97. Midpoint(G_1, J_1) → P_1
  98. Midpoint(V, T) → Q_1
  99. Midpoint(B_1, Z) → R_1
  100. Midpoint(A_1, T) → T_1
  101. Midpoint(B_1, U) → U_1
  102. Midpoint(B_1, K) → V_1
  103. Midpoint(Z, K) → W_1
  104. Midpoint(B_1, A_1) → Z_1
  105. Midpoint(A_1, Z) → A_2
  106. Midpoint(Z, W) → B_2
  107. Midpoint(V, U) → C_2
  108. Midpoint(K, T) → D_2
  109. Midpoint(T, U) → E_2
  110. Midpoint(V, K) → F_2
  111. Midpoint(B_1, W) → G_2
  112. Midpoint(Z, T) → H_2
  113. Midpoint(K, W) → I_2
  114. Midpoint(B_1, V) → J_2
  115. Midpoint(U, A_1) → K_2
  116. Midpoint(G''', C''') → L_2
  117. Midpoint(D''', H''') → M_2
  118. Midpoint(A''', E''') → N_2
  119. Midpoint(F''', B''') → O_2
  120. Prism(D''', A''', B''', M_2) → b, P_2, edgeB'''D''', edge1, edge2, edge3, edge4, edge6, edge5, Q_2, faceA'''B'''D''', face2, face4, face3, face1, edgeA'''D''', edgeA'''B'''
  121. Prism(M_2, L_2, P_2, H''') → d, R_2, edge13, edge7, edge12, edge14, edge8, edge15, edge9, T_2, face6, face7, face9, face8, face5, edge10, edge11
  122. Prism(V, T, U, J_2) → e, U_2, edgeUV, edge18, edge16, edge17, edge19, edge21, edge20, V_2, faceTUV, face12, face11, face13, face10, edgeTV, edgeTU
  123. Point(edge21) → W_2
  124. Prism(K, Q_1, T, I_2) → j, Z_2, edgeKT, edge23, edge26, edge25, edge27, edge29, edge28, A_3, face14, face17, face18, face16, face15, edge22, edge24
  125. Prism(I_2, V_2, A_3, W) → l, B_3, edge34, edge30, edge36, edge37, edge31, edge38, edge32, C_3, face20, face21, face23, face22, face19, edge33, edge35
  126. Prism(W, V_1, B_1, Z) → n, D_3, edge39, edgeWZ, edge45, edge44, edge41, edge46, edge42, E_3, face24, face26, face28, face27, face25, edge40, edge43
  127. Prism(V, B_1, V_1, C_2) → o, F_3, edge48, edge49, edge51, edge52, edge53, edge55, edge54, G_3, face29, face31, face33, face32, face30, edge47, edge50
  128. Prism(W, V_1, K, B_2) → q, H_3, edgeKW, edge59, edge60, edge57, edge61, edge63, edge62, I_3, face34, face38, face37, face36, face35, edge58, edge56
  129. Prism(K, B_1, V, D_2) → s, J_3, edgeKV, edge65, edge68, edge67, edge69, edge71, edge70, K_3, face39, face42, face43, face41, face40, edge64, edge66
  130. Prism(B_1, V_1, W, A_1) → t, L_3, edge72, edge75, edge79, edge74, edge76, edge80, edge77, M_3, face44, face48, face47, face46, face45, edge78, edge73
  131. Prism(W, V, B_1, B_2) → w, N_3, edge83, edge84, edge82, edge85, edge86, edge88, edge87, O_3, face49, face51, face52, face53, face50, edgeVW, edge81
  132. Prism(O_3, N_3, D_2, A_1) → a_1, P_3, edge93, edge89, edge96, edge94, edge90, edge97, edge91, Q_3, face55, face57, face58, face56, face54, edge95, edge92

AI 绘图提示语参考

在三维空间中，绘制一个立方体被切割成两部分的过程，展示切割平面、棱边和面的关系。首先构建两个立方体，然后找到关键顶点的中点，并绘制连接中点的直线。利用向量和中点确定切割方向，通过旋转和平移将立方体分割成两部分，并形成新的棱柱。最终展示切割后的两部分以及它们与原始立方体的关系，强调几何构造和空间几何性质。
以上提示语为根据原始绘图命令整理而来，仅供参考。