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立方体面积的平移几何演示（Cube Area）

立方体面积的平移几何演示（Cube Area）
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2023-10-01 20:56
向量立体几何变换几何作图
本范例通过在三维坐标系中对立方体进行平移与构造，展示与“立方体侧面积/截面面积”相关的几何关系。核心数学概念包括立体几何（立方体的面与面积）、向量平移（translation）、以及用交点与几何约束建立可视化结构。作图要点：利用 GeoGebra 的 Intersect（求交）来确定关键点位置，使用 Vector 与 Point 表示方向与顶点，结合 Cube 生成立方体，借助 Translate 实现几何平移，并用 Textfield 标注/输入参数或说明对象（如得到的点与关系）。对象层面包含多个点（A、B、C…等）与向量（u、v、w）以及由立方体与平移产生的面/截面标注元素。适合中学到大学入门阶段的立体几何教学、建模演示与可视化学习，用于帮助理解空间几何中的面积对应关系。

GeoGebra 命令及参数

  1. Intersect(zAxis, xOyPlane) → A
  2. Vector(A, B) → u
  3. Vector(A, C) → v
  4. Vector(A, D) → w
  5. Vector(A, E) → a
  6. Vector(A, S) → c
  7. Vector(A, O) → d
  8. Vector(A, K) → e
  9. Vector(A, G) → f
  10. Vector(A, T) → g
  11. Vector(A, P) → h
  12. Vector(A, L) → i
  13. Vector(A, H) → j
  14. Vector(A, R) → k
  15. Vector(A, N) → l
  16. Vector(A, J) → m
  17. Vector(A, F) → n
  18. Vector(A, U) → p
  19. Vector(A, Q) → q
  20. Vector(A, M) → r
  21. Vector(A, I) → s
  22. Point(xAxis) → V
  23. Point(xAxis) → W
  24. Point(Circle[W, Distance[V, W], Segment[V, W]]) → Z
  25. Cube(V, W, Z) → o, A_1, face5, face6, edge1, edgeVW_{1}, edgeWZ, edge4, edge2, edge6, edge8, edge7, B_1, edge3, edge9, edge5, edge10, C_1, D_1, E_1, face1, face3, face2, face4
  26. Intersect(zAxis, xAxis) → F_1
  27. Point(Circle[F_1, Distance[W, F_1], Segment[W, F_1]]) → G_1
  28. Cube(W, F_1, G_1) → t, H_1, face11, face12, edge12, edge11, edge14, edge16, edge13, edge18, edge20, edge19, I_1, edge15, edge21, edge17, edge22, J_1, K_1, L_1, face7, face9, face8, face10
  29. Point(xAxis) → M_1
  30. Point(Circle[M_1, Distance[F_1, M_1], Segment[F_1, M_1]]) → N_1
  31. Cube(F_1, M_1, N_1) → a_1, O_1, face17, face18, edge24, edge23, edge26, edge28, edge25, edge30, edge32, edge31, P_1, edge27, edge33, edge29, edge34, Q_1, R_1, S_1, face13, face15, face14, face16
  32. Point(xAxis) → T_1
  33. Point(Circle[T_1, Distance[M_1, T_1], Segment[M_1, T_1]]) → U_1
  34. Cube(M_1, T_1, U_1) → b_1, V_1, face23, face24, edge36, edge35, edge38, edge40, edge37, edge42, edge44, edge43, W_1, edge39, edge45, edge41, edge46, Z_1, A_2, B_2, face19, face21, face20, face22
  35. Point(Circle[V, Distance[W, V], Segment[W, V]]) → C_2
  36. Cube(W, V, C_2) → c_1, D_2, face29, face30, edge49, edgeVW, edge47, edge51, edge50, edge53, edge55, edge54, E_2, edge48, edge56, edge52, edge57, F_2, G_2, H_2, face25, face28, face26, face27
  37. Point(Circle[W, Distance[F_1, W], Segment[F_1, W]]) → I_2
  38. Cube(F_1, W, I_2) → d_1, J_2, face35, face36, edge61, edge58, edge59, edge63, edge62, edge65, edge67, edge66, K_2, edge60, edge68, edge64, edge69, L_2, M_2, N_2, face31, face34, face32, face33
  39. Point(Circle[F_1, Distance[M_1, F_1], Segment[M_1, F_1]]) → O_2
  40. Cube(M_1, F_1, O_2) → e_1, P_2, face41, face42, edge73, edge70, edge71, edge75, edge74, edge77, edge79, edge78, Q_2, edge72, edge80, edge76, edge81, R_2, S_2, T_2, face37, face40, face38, face39
  41. Point(Circle[M_1, Distance[T_1, M_1], Segment[T_1, M_1]]) → U_2
  42. Cube(T_1, M_1, U_2) → f_1, V_2, face47, face48, edge85, edge82, edge83, edge87, edge86, edge89, edge91, edge90, W_2, edge84, edge92, edge88, edge93, Z_2, A_3, B_3, face43, face46, face44, face45
  43. Point(Circle[U_2, Distance[V_2, U_2], Segment[V_2, U_2]]) → C_3
  44. Cube(V_2, U_2, C_3) → g_1, D_3, face53, face54, edge97, edge94, edge95, edge99, edge98, edge101, edge103, edge102, E_3, edge96, edge104, edge100, edge105, F_3, G_3, H_3, face49, face52, face50, face51
  45. Point(Circle[O_2, Distance[U_2, O_2], Segment[U_2, O_2]]) → I_3
  46. Cube(U_2, O_2, I_3) → h_1, J_3, face59, face60, edge109, edge106, edge107, edge111, edge110, edge113, edge115, edge114, K_3, edge108, edge116, edge112, edge117, L_3, M_3, N_3, face55, face58, face56, face57
  47. Point(Circle[I_2, Distance[O_2, I_2], Segment[O_2, I_2]]) → O_3
  48. Cube(O_2, I_2, O_3) → i_1, P_3, face65, face66, edge121, edge118, edge119, edge123, edge122, edge125, edge127, edge126, Q_3, edge120, edge128, edge124, edge129, R_3, S_3, T_3, face61, face64, face62, face63
  49. Point(Circle[C_2, Distance[I_2, C_2], Segment[I_2, C_2]]) → U_3
  50. Cube(I_2, C_2, U_3) → j_1, V_3, face71, face72, edge133, edge130, edge131, edge135, edge134, edge137, edge139, edge138, W_3, edge132, edge140, edge136, edge141, Z_3, A_4, B_4, face67, face70, face68, face69
  51. Point(Circle[U_1, Distance[N_1, U_1], Segment[N_1, U_1]]) → C_4
  52. Cube(N_1, U_1, C_4) → k_1, D_4, face77, face78, edge143, edge142, edge145, edge147, edge144, edge149, edge151, edge150, E_4, edge146, edge152, edge148, edge153, F_4, G_4, H_4, face73, face75, face74, face76
  53. Point(Circle[N_1, Distance[G_1, N_1], Segment[G_1, N_1]]) → I_4
  54. Cube(G_1, N_1, I_4) → l_1, J_4, face83, face84, edge155, edge154, edge157, edge159, edge156, edge161, edge163, edge162, K_4, edge158, edge164, edge160, edge165, L_4, M_4, N_4, face79, face81, face80, face82
  55. Point(Circle[G_1, Distance[Z, G_1], Segment[Z, G_1]]) → O_4
  56. Cube(Z, G_1, O_4) → m_1, P_4, face89, face90, edge167, edge166, edge169, edge171, edge168, edge173, edge175, edge174, Q_4, edge170, edge176, edge172, edge177, R_4, S_4, T_4, face85, face87, face86, face88
  57. Point(Circle[Z, Distance[A_1, Z], Segment[A_1, Z]]) → U_4
  58. Cube(A_1, Z, U_4) → n_1, V_4, face95, face96, edge181, edge178, edge179, edge183, edge182, edge185, edge187, edge186, W_4, edge180, edge188, edge184, edge189, Z_4, A_5, B_5, face91, face94, face92, face93
  59. Translate(face18, g) → face18'
  60. Translate(face16, g) → face16'
  61. Translate(face14, g) → face14'
  62. Translate(face15, g) → face15'
  63. Translate(face17, g) → face17'
  64. Translate(face13, g) → face13'
  65. Translate(face36, k) → face36'
  66. Translate(face35, k) → face35'
  67. Translate(face34, k) → face34'
  68. Translate(face33, k) → face33'
  69. Translate(face32, k) → face32'
  70. Translate(face31, k) → face31'
  71. Translate(face42, p) → face42'
  72. Translate(face41, p) → face41'
  73. Translate(face40, p) → face40'
  74. Translate(face37, p) → face37'
  75. Translate(face39, p) → face39'
  76. Translate(face41, p) → face41'_{1}
  77. Translate(face12, c) → face12'
  78. Translate(face10, c) → face10'
  79. Translate(face8, c) → face8'
  80. Translate(face11, c) → face11'
  81. Translate(face9, c) → face9'
  82. Translate(face7, c) → face7'
  83. Translate(face24, h) → face24'
  84. Translate(face22, h) → face22'
  85. Translate(face20, h) → face20'
  86. Translate(face23, h) → face23'
  87. Translate(face21, h) → face21'
  88. Translate(face19, h) → face19'
  89. Translate(face30, l) → face30'
  90. Translate(face29, l) → face29'
  91. Translate(face28, l) → face28'
  92. Translate(face27, l) → face27'
  93. Translate(face26, l) → face26'
  94. Translate(face25, l) → face25'
  95. Translate(face60, q) → face60'
  96. Translate(face59, q) → face59'
  97. Translate(face58, q) → face58'
  98. Translate(face56, q) → face56'
  99. Translate(face57, q) → face57'
  100. Translate(face55, q) → face55'
  101. Translate(face90, d) → face90'
  102. Translate(face86, d) → face86'
  103. Translate(face88, d) → face88'
  104. Translate(face89, d) → face89'
  105. Translate(face87, d) → face87'
  106. Translate(face85, d) → face85'
  107. Translate(face47, q) → face47'
  108. Translate(face46, q) → face46'
  109. Translate(face48, q) → face48'
  110. Translate(face44, q) → face44'
  111. Translate(face45, q) → face45'
  112. Translate(face43, q) → face43'
  113. Translate(face65, l) → face65'
  114. Translate(face66, l) → face66'
  115. Translate(face64, l) → face64'
  116. Translate(face63, l) → face63'
  117. Translate(face62, l) → face62'
  118. Translate(face61, l) → face61'
  119. Translate(face84, h) → face84'
  120. Translate(face82, h) → face82'
  121. Translate(face80, h) → face80'
  122. Translate(face83, h) → face83'
  123. Translate(face81, h) → face81'
  124. Translate(face79, h) → face79'
  125. Translate(face6, d) → face6'
  126. Translate(face2, d) → face2'
  127. Translate(face4, d) → face4'
  128. Translate(face3, d) → face3'
  129. Translate(face5, d) → face5'
  130. Translate(face1, d) → face1'
  131. Translate(face78, i) → face78'
  132. Translate(face76, i) → face76'
  133. Translate(face74, i) → face74'
  134. Translate(face54, r) → face54'
  135. Translate(face53, r) → face53'
  136. Translate(face52, r) → face52'
  137. Translate(face70, m) → face70'
  138. Translate(face71, m) → face71'
  139. Translate(face72, m) → face72'
  140. Translate(face92, e) → face92'
  141. Translate(face93, e) → face93'
  142. Translate(face96, e) → face96'
  143. Translate(face51, r) → face51'
  144. Translate(face50, r) → face50'
  145. Translate(face75, i) → face75'
  146. Translate(face77, i) → face77'
  147. Translate(face94, e) → face94'
  148. Translate(face95, e) → face95'
  149. Translate(face69, m) → face69'
  150. Translate(face68, m) → face68'
  151. Translate(face67, m) → face67'
  152. Translate(face91, e) → face91'
  153. Translate(face73, i) → face73'
  154. Translate(face49, r) → face49'
  155. Translate(face38, p) → face38'
  156. Textfield(side) → InputBox1
  157. Textfield(side_{1}) → InputBox2
  158. Textfield(side_{2}) → InputBox1_{2}
  159. Textfield(side_{2}) → InputBox1_{3}
  160. Textfield(side_{2}) → InputBox1_{1}

AI 绘图提示语参考

画一个立方体，然后通过空间几何变换将其平移，展示立方体表面积与空间中面积关系。首先构建一个立方体，然后利用向量确定方向，通过计算交点找到关键位置，将立方体沿向量进行平移，形成新的立方体。最终图形包含原始和移动后的立方体，以及它们之间的交线和面，用于说明立方体表面积在空间变换中的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。