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开放中部立方体的表面积最大化（V2）

开放中部立方体的表面积最大化（V2）
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2021-01-04 23:12
多边形立体几何最值优化几何作图
该范例展示“去掉中间立方体（开放中部）”后，如何通过改变立方体分割与棱长关系来研究剩余立体的表面积最大化问题。核心数学概念包括：立体几何的曲面/表面积计算、由正方形面拼接形成的棱柱几何、线性方程与向量（用于描述边界与方向关系）、以及用滑块/参数连续变化来观察目标函数的极值趋势。作图要点：利用构造摘要中的 Polygon 与 Prism 命令创建正方形截面多边形（如 A_2、B_2、C_2、D_2 与 poly1_2，并在另一组中对应 A_3、B_3、C_3、D_3、poly1_3），再用 Prism 从截面拉伸生成立体；面对象 faceABFE_2、faceBCGF_2、faceCDHG_2、faceADHE_2、faceEFGH_2 用于显式管理外露面；边对象 edgeAE_2、edgeBF_2、edgeCG_2、edgeDH_2、edgeEF_2、edgeFG_2、edgeGH_2、edgeEH_2 用于标定关键长度；由 a/b/c/d、e 等参数点（如 a_2、b_2、c_2、d_2、e_2、e_3 等）建立可调几何约束，从而驱动表面积随参数的变化并可进一步配合连续性观察与分布统计可视化。适合高中到大学的立体几何进阶、最值优化入门与用 GeoGebra 做建模/探索的课堂活动或自学用途。

GeoGebra 命令及参数

  1. Polygon(A_2, B_2, C_2, D_2) → poly1_2, a_2, b_2, c_2, d_2
  2. Prism(poly1_2, E_2) → e_2, F_2, edgeBF_2, edgeCG_2, edgeDH_2, edgeEF_2, edgeFG_2, edgeGH_2, edgeEH_2, G_2, H_2, faceABFE_2, faceBCGF_2, faceCDHG_2, faceADHE_2, faceEFGH_2, edgeAE_2
  3. Polygon(A_3, B_3, C_3, D_3) → poly1_3, a_3, b_3, c_3, d_3
  4. Prism(poly1_3, E_3) → e_3, F_3, edgeBF_3, edgeCG_3, edgeDH_3, edgeEF_3, edgeFG_3, edgeGH_3, edgeEH_3, G_3, H_3, faceABFE_3, faceBCGF_3, faceCDHG_3, faceADHE_3, faceEFGH_3, edgeAE_3
  5. Polygon(A_4, B_4, C_4, D_4) → poly1_4, a_4, b_4, c_4, d_4
  6. Prism(poly1_4, E_4) → e_4, F_4, edgeBF_4, edgeCG_4, edgeDH_4, edgeEF_4, edgeFG_4, edgeGH_4, edgeEH_4, G_4, H_4, faceABFE_4, faceBCGF_4, faceCDHG_4, faceADHE_4, faceEFGH_4, edgeAE_4
  7. Polygon(A_5, B_5, C_5, D_5) → poly1_5, a_5, b_5, c_5, d_5
  8. Prism(poly1_5, E_5) → e_5, F_5, edgeBF_5, edgeCG_5, edgeDH_5, edgeEF_5, edgeFG_5, edgeGH_5, edgeEH_5, G_5, H_5, faceABFE_5, faceBCGF_5, faceCDHG_5, faceADHE_5, faceEFGH_5, edgeAE_5
  9. Polygon(A_6, B_6, C_6, D_6) → poly1_6, a_6, b_6, c_6, d_6
  10. Prism(poly1_6, E_6) → e_6, F_6, edgeBF_6, edgeCG_6, edgeDH_6, edgeEF_6, edgeFG_6, edgeGH_6, edgeEH_6, G_6, H_6, faceABFE_6, faceBCGF_6, faceCDHG_6, faceADHE_6, faceEFGH_6, edgeAE_6
  11. Polygon(A_9, B_9, C_9, D_9) → poly1_9, a_9, b_9, c_9, d_9
  12. Prism(poly1_9, E_9) → e_9, F_9, edgeBF_9, edgeCG_9, edgeDH_9, edgeEF_9, edgeFG_9, edgeGH_9, edgeEH_9, G_9, H_9, faceABFE_9, faceBCGF_9, faceCDHG_9, faceADHE_9, faceEFGH_9, edgeAE_9
  13. Polygon(A_{10}, B_{10}, C_{10}, D_{10}) → poly1_{10}, a_{10}, b_{10}, c_{10}, d_{10}
  14. Prism(poly1_{10}, E_{10}) → e_{10}, F_{10}, edgeBF_{10}, edgeCG_{10}, edgeDH_{10}, edgeEF_{10}, edgeFG_{10}, edgeGH_{10}, edgeEH_{10}, G_{10}, H_{10}, faceABFE_{10}, faceBCGF_{10}, faceCDHG_{10}, faceADHE_{10}, faceEFGH_{10}, edgeAE_{10}
  15. Polygon(A_{11}, B_{11}, C_{11}, D_{11}) → poly1_{11}, a_{11}, b_{11}, c_{11}, d_{11}
  16. Prism(poly1_{11}, E_{11}) → e_{11}, F_{11}, edgeBF_{11}, edgeCG_{11}, edgeDH_{11}, edgeEF_{11}, edgeFG_{11}, edgeGH_{11}, edgeEH_{11}, G_{11}, H_{11}, faceABFE_{11}, faceBCGF_{11}, faceCDHG_{11}, faceADHE_{11}, faceEFGH_{11}, edgeAE_{11}
  17. Polygon(A_1, B_1, C_1, D_1) → poly1_1, a_1, b_1, c_1, d_1
  18. Prism(poly1_1, E_1) → e_1, F_1, edgeBF_1, edgeCG_1, edgeDH_1, edgeEF_1, edgeFG_1, edgeGH_1, edgeEH_1, G_1, H_1, faceABFE_1, faceBCGF_1, faceCDHG_1, faceADHE_1, faceEFGH_1, edgeAE_1
  19. Polygon(E, B, C, D) → poly1, a, b, c, d
  20. Polygon(A_{13}, B_{13}, C_{13}, D_{13}) → poly1_{13}, a_{13}, b_{13}, c_{13}, d_{13}
  21. Prism(poly1_{13}, E_{13}) → e_{13}, F_{13}, edgeBF_{13}, edgeCG_{13}, edgeDH_{13}, edgeEF_{13}, edgeFG_{13}, edgeGH_{13}, edgeEH_{13}, G_{13}, H_{13}, faceABFE_{13}, faceBCGF_{13}, faceCDHG_{13}, faceADHE_{13}, faceEFGH_{13}, edgeAE_{13}
  22. Polygon(A_{14}, B_{14}, C_{14}, D_{14}) → poly1_{14}, a_{14}, b_{14}, c_{14}, d_{14}
  23. Prism(poly1_{14}, E_{14}) → e_{14}, F_{14}, edgeBF_{14}, edgeCG_{14}, edgeDH_{14}, edgeEF_{14}, edgeFG_{14}, edgeGH_{14}, edgeEH_{14}, G_{14}, H_{14}, faceABFE_{14}, faceBCGF_{14}, faceCDHG_{14}, faceADHE_{14}, faceEFGH_{14}, edgeAE_{14}
  24. Polygon(A_{15}, B_{15}, C_{15}, D_{15}) → poly1_{15}, a_{15}, b_{15}, c_{15}, d_{15}
  25. Prism(poly1_{15}, E_{15}) → e_{15}, F_{15}, edgeBF_{15}, edgeCG_{15}, edgeDH_{15}, edgeEF_{15}, edgeFG_{15}, edgeGH_{15}, edgeEH_{15}, G_{15}, H_{15}, faceABFE_{15}, faceBCGF_{15}, faceCDHG_{15}, faceADHE_{15}, faceEFGH_{15}, edgeAE_{15}
  26. Polygon(A_{16}, B_{16}, C_{16}, D_{16}) → poly1_{16}, a_{16}, b_{16}, c_{16}, d_{16}
  27. Prism(poly1_{16}, E_{16}) → e_{16}, F_{16}, edgeBF_{16}, edgeCG_{16}, edgeDH_{16}, edgeEF_{16}, edgeFG_{16}, edgeGH_{16}, edgeEH_{16}, G_{16}, H_{16}, faceABFE_{16}, faceBCGF_{16}, faceCDHG_{16}, faceADHE_{16}, faceEFGH_{16}, edgeAE_{16}
  28. Polygon(A_{17}, B_{17}, C_{17}, D_{17}) → poly1_{17}, a_{17}, b_{17}, c_{17}, d_{17}
  29. Prism(poly1_{17}, E_{17}) → e_{17}, F_{17}, edgeBF_{17}, edgeCG_{17}, edgeDH_{17}, edgeEF_{17}, edgeFG_{17}, edgeGH_{17}, edgeEH_{17}, G_{17}, H_{17}, faceABFE_{17}, faceBCGF_{17}, faceCDHG_{17}, faceADHE_{17}, faceEFGH_{17}, edgeAE_{17}
  30. Polygon(A_{18}, B_{18}, C_{18}, D_{18}) → poly1_{18}, a_{18}, b_{18}, c_{18}, d_{18}
  31. Prism(poly1_{18}, E_{18}) → e_{18}, F_{18}, edgeBF_{18}, edgeCG_{18}, edgeDH_{18}, edgeEF_{18}, edgeFG_{18}, edgeGH_{18}, edgeEH_{18}, G_{18}, H_{18}, faceABFE_{18}, faceBCGF_{18}, faceCDHG_{18}, faceADHE_{18}, faceEFGH_{18}, edgeAE_{18}
  32. Polygon(A_{19}, B_{19}, C_{19}, D_{19}) → poly1_{19}, a_{19}, b_{19}, c_{19}, d_{19}
  33. Prism(poly1_{19}, E_{19}) → e_{19}, F_{19}, edgeBF_{19}, edgeCG_{19}, edgeDH_{19}, edgeEF_{19}, edgeFG_{19}, edgeGH_{19}, edgeEH_{19}, G_{19}, H_{19}, faceABFE_{19}, faceBCGF_{19}, faceCDHG_{19}, faceADHE_{19}, faceEFGH_{19}, edgeAE_{19}
  34. Polygon(A_{21}, B_{21}, C_{21}, D_{21}) → poly1_{21}, a_{21}, b_{21}, c_{21}, d_{21}
  35. Prism(poly1_{21}, E_{21}) → e_{21}, F_{21}, edgeBF_{21}, edgeCG_{21}, edgeDH_{21}, edgeEF_{21}, edgeFG_{21}, edgeGH_{21}, edgeEH_{21}, G_{21}, H_{21}, faceABFE_{21}, faceBCGF_{21}, faceCDHG_{21}, faceADHE_{21}, faceEFGH_{21}, edgeAE_{21}
  36. Polygon(A_{22}, B_{22}, C_{22}, D_{22}) → poly1_{22}, a_{22}, b_{22}, c_{22}, d_{22}
  37. Prism(poly1_{22}, E_{22}) → e_{22}, F_{22}, edgeBF_{22}, edgeCG_{22}, edgeDH_{22}, edgeEF_{22}, edgeFG_{22}, edgeGH_{22}, edgeEH_{22}, G_{22}, H_{22}, faceABFE_{22}, faceBCGF_{22}, faceCDHG_{22}, faceADHE_{22}, faceEFGH_{22}, edgeAE_{22}
  38. Polygon(A_{24}, B_{24}, C_{24}, D_{24}) → poly1_{24}, a_{24}, b_{24}, c_{24}, d_{24}
  39. Prism(poly1_{24}, E_{24}) → e_{24}, F_{24}, edgeBF_{24}, edgeCG_{24}, edgeDH_{24}, edgeEF_{24}, edgeFG_{24}, edgeGH_{24}, edgeEH_{24}, G_{24}, H_{24}, faceABFE_{24}, faceBCGF_{24}, faceCDHG_{24}, faceADHE_{24}, faceEFGH_{24}, edgeAE_{24}
  40. Polygon(E_{25}, B_{25}, C_{25}, D_{25}) → poly1_{25}, a_{25}, b_{25}, c_{25}, d_{25}
  41. Polygon(A_{25}, B_{26}, C_{26}, D_{26}) → poly1_{26}, a_{26}, b_{26}, c_{26}, d_{26}
  42. Prism(poly1_{26}, E_{26}) → e_{25}, F_{25}, edgeBF_{25}, edgeCG_{25}, edgeDH_{25}, edgeEF_{25}, edgeFG_{25}, edgeGH_{25}, edgeEH_{25}, G_{25}, H_{25}, faceABFE_{25}, faceBCGF_{25}, faceCDHG_{25}, faceADHE_{25}, faceEFGH_{25}, edgeAE_{25}
  43. Polygon(A_{26}, B_{27}, C_{27}, D_{27}) → poly1_{27}, a_{27}, b_{27}, c_{27}, d_{27}
  44. Prism(poly1_{27}, E_{27}) → e_{26}, F_{26}, edgeBF_{26}, edgeCG_{26}, edgeDH_{26}, edgeEF_{26}, edgeFG_{26}, edgeGH_{26}, edgeEH_{26}, G_{26}, H_{26}, faceABFE_{26}, faceBCGF_{26}, faceCDHG_{26}, faceADHE_{26}, faceEFGH_{26}, edgeAE_{26}
  45. Polygon(A_{27}, B_{28}, C_{28}, D_{28}) → poly1_{28}, a_{28}, b_{28}, c_{28}, d_{28}
  46. Prism(poly1_{28}, E_{28}) → e_{27}, F_{27}, edgeBF_{27}, edgeCG_{27}, edgeDH_{27}, edgeEF_{27}, edgeFG_{27}, edgeGH_{27}, edgeEH_{27}, G_{27}, H_{27}, faceABFE_{27}, faceBCGF_{27}, faceCDHG_{27}, faceADHE_{27}, faceEFGH_{27}, edgeAE_{27}
  47. Polygon(A_{28}, B_{29}, C_{29}, D_{29}) → poly1_{29}, a_{29}, b_{29}, c_{29}, d_{29}
  48. Prism(poly1_{29}, E_{29}) → e_{28}, F_{28}, edgeBF_{28}, edgeCG_{28}, edgeDH_{28}, edgeEF_{28}, edgeFG_{28}, edgeGH_{28}, edgeEH_{28}, G_{28}, H_{28}, faceABFE_{28}, faceBCGF_{28}, faceCDHG_{28}, faceADHE_{28}, faceEFGH_{28}, edgeAE_{28}
  49. Polygon(A_{29}, B_{30}, C_{30}, D_{30}) → poly1_{30}, a_{30}, b_{30}, c_{30}, d_{30}
  50. Prism(poly1_{30}, E_{30}) → e_{29}, F_{29}, edgeBF_{29}, edgeCG_{29}, edgeDH_{29}, edgeEF_{29}, edgeFG_{29}, edgeGH_{29}, edgeEH_{29}, G_{29}, H_{29}, faceABFE_{29}, faceBCGF_{29}, faceCDHG_{29}, faceADHE_{29}, faceEFGH_{29}, edgeAE_{29}
  51. Polygon(A_{30}, B_{31}, C_{31}, D_{31}) → poly1_{31}, a_{31}, b_{31}, c_{31}, d_{31}
  52. Prism(poly1_{31}, E_{31}) → e_{30}, F_{30}, edgeBF_{30}, edgeCG_{30}, edgeDH_{30}, edgeEF_{30}, edgeFG_{30}, edgeGH_{30}, edgeEH_{30}, G_{30}, H_{30}, faceABFE_{30}, faceBCGF_{30}, faceCDHG_{30}, faceADHE_{30}, faceEFGH_{30}, edgeAE_{30}
  53. Polygon(A_{31}, B_{32}, C_{32}, D_{32}) → poly1_{32}, a_{32}, b_{32}, c_{32}, d_{32}
  54. Prism(poly1_{32}, E_{32}) → e_{31}, F_{31}, edgeBF_{31}, edgeCG_{31}, edgeDH_{31}, edgeEF_{31}, edgeFG_{31}, edgeGH_{31}, edgeEH_{31}, G_{31}, H_{31}, faceABFE_{31}, faceBCGF_{31}, faceCDHG_{31}, faceADHE_{31}, faceEFGH_{31}, edgeAE_{31}
  55. Polygon(A_{32}, B_{33}, C_{33}, D_{33}) → poly1_{33}, a_{33}, b_{33}, c_{33}, d_{33}
  56. Prism(poly1_{33}, E_{33}) → e_{32}, F_{32}, edgeBF_{32}, edgeCG_{32}, edgeDH_{32}, edgeEF_{32}, edgeFG_{32}, edgeGH_{32}, edgeEH_{32}, G_{32}, H_{32}, faceABFE_{32}, faceBCGF_{32}, faceCDHG_{32}, faceADHE_{32}, faceEFGH_{32}, edgeAE_{32}
  57. Polygon(E_{38}, B_{38}, C_{38}, D_{38}) → poly1_{38}, a_{38}, b_{38}, c_{38}, d_{38}

AI 绘图提示语参考

在三维空间中绘制一个“开中间立方体”外表面用于优化研究的几何模型：先画若干个关于同一竖直方向的正方形截面，确定其四个角点并让它们共形成一组参数化的正方面；随后沿竖直方向把该正方形截面拉伸成棱柱，使上下对应的点连接生成立方体骨架。用显式的面对象标注哪些侧面被保留为外露区域（围成的若干矩形面），并与被移走的中部区域相对应；同时在关键棱上标出长度以体现几何参数随位置变化。整体可重复生成多套相似构型，通过调整参数持续观察外表面积的变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。