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开中间立方体：最大化表面积的几何与函数演示

开中间立方体：最大化表面积的几何与函数演示
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2021-01-04 23:35
函数立体几何最值优化几何作图函数图像
该范例演示“在一个立方体中挖去/打开中间部分（开中间立方体）”时，如何通过变形与求解使剩余表面积达到最大。核心数学概念包括：立体几何的表面积分解、立方体/棱柱的面与棱的关联、几何差与交集带来的面积表达、以及用二次函数/幂函数刻画表面积随参数变化的最值问题（常与坡度/斜率的分析相联系）。作图要点是先用 GeoGebra 的 Point 与 Cube 构造基准立方体，再通过 Translate 产生移动后的点/结构以形成“开口”边界；随后使用 Intersect 计算交线/交点并用 Segment 连接关键点，得到各面区域（如 face1~face8 等）与对应边（edge1~edge7 等），最终把表面积表达映射到由参数控制的面片集合。适合中学竞赛或高中拓展学习，用于立体几何建模、几何—代数联系与最值优化思维训练，也可作为课堂演示或自主探究作业。

GeoGebra 命令及参数

  1. Point(Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]) → A
  2. Cube((0, 0), (1, 0), A) → a, B, faceABFE, faceCDEF, edge2, edge1, edge4, edgeAB, edge3, edgeBF, edgeCF, edgeCD, C, edge5, edgeDE, edgeAE, edgeEF, D, E, F, face1, face3, face2, face4
  3. Point(Circle[(2, 0), Distance[(1, 0), (2, 0)], Segment[(1, 0), (2, 0)]]) → G
  4. Cube((1, 0), (2, 0), G) → b, H, faceGHLK, faceIJKL, edge7, edge6, edge9, edgeGH, edge8, edgeHL, edgeIL, edgeIJ, I, edge10, edgeJK, edgeGK, edgeKL, J, K, L, face5, face7, face6, face8
  5. Point(Circle[(3, 0), Distance[(2, 0), (3, 0)], Segment[(2, 0), (3, 0)]]) → M
  6. Cube((2, 0), (3, 0), M) → c, N, faceMNRQ, faceOPQR, edge12, edge11, edge14, edgeMN, edge13, edgeNR, edgeOR, edgeOP, O, edge15, edgePQ, edgeMQ, edgeQR, P, Q, R, face9, face11, face10, face12
  7. Translate(a, Vector[(0, 0, 1)]) → a'
  8. Translate(c, Vector[(0, 0, 1)]) → c'
  9. Translate(c, Vector[(0, 0, 2)]) → c'_{1}
  10. Intersect(edge1, zAxis) → S
  11. Point(Segment[(0, 0, 2), (0, -3, 2)]) → T
  12. Point(Circle[T + (1, 0, 0), Distance[T, T + (1, 0, 0)], Segment[T, T + (1, 0, 0)]]) → U
  13. Cube(T, T + (1, 0, 0), U) → e, V, face17, face18, edgeTV, edge16, edge17, edgeUV, edgeTW, edge20, edge21, edgeWZ, W, edge18, edge22, edge19, edge23, Z, A_1, B_1, face13, face15, face14, face16
  14. Point(Segment[(1, 0, 2), (1, -3, 2)]) → C_1
  15. Point(Circle[C_1 + (1, 0, 0), Distance[C_1, C_1 + (1, 0, 0)], Segment[C_1, C_1 + (1, 0, 0)]]) → D_1
  16. Cube(C_1, C_1 + (1, 0, 0), D_1) → f, E_1, face23, face24, edge25, edge24, edge27, edge29, edge26, edge31, edge33, edge32, F_1, edge28, edge34, edge30, edge35, G_1, H_1, I_1, face19, face21, face20, face22
  17. Point(Segment[(1, 0, 1), (1, -3, 1)]) → J_1
  18. Point(Circle[J_1 + (1, 0, 0), Distance[J_1, J_1 + (1, 0, 0)], Segment[J_1, J_1 + (1, 0, 0)]]) → K_1
  19. Cube(J_1, J_1 + (1, 0, 0), K_1) → g, L_1, face29, face30, edge37, edge36, edge39, edge41, edge38, edge43, edge45, edge44, M_1, edge40, edge46, edge42, edge47, N_1, O_1, P_1, face25, face27, face26, face28
  20. Point(Circle[(1, 1, 0), Distance[(0, 1, 0), (1, 1, 0)], Segment[(0, 1, 0), (1, 1, 0)]]) → Q_1
  21. Cube((0, 1, 0), (1, 1, 0), Q_1) → d, R_1, face35, face36, edge49, edge48, edge51, edge53, edge50, edge55, edge57, edge56, S_1, edge52, edge58, edge54, edge59, T_1, U_1, V_1, face31, face33, face32, face34
  22. Translate(d, Vector[(1, 0)]) → d'
  23. Translate(d, Vector[(2, 0)]) → d'_{1}
  24. Translate(d, Vector[(0, 1)]) → d'_{2}
  25. Translate(d, Vector[(1, 1)]) → d'_{3}
  26. Translate(d, Vector[(2, 1)]) → d'_{4}
  27. Translate(d, Vector[(0, 0, 1)]) → d'_{5}
  28. Translate(d, Vector[(0, 0, 2)]) → d'_{6}
  29. Translate(d, Vector[(1, 0, 1)]) → d'_{7}
  30. Translate(d, Vector[(2, 0, 1)]) → d'_{8}
  31. Translate(d, Vector[(2, 0, 2)]) → d'_{9}
  32. Translate(d, Vector[(1, 0, 2)]) → d'_{10}
  33. Translate(d, Vector[(0, 1, 1)]) → d'_{11}
  34. Translate(d, Vector[(0, 1, 2)]) → d'_{12}
  35. Translate(d, Vector[(1, 1, 1)]) → d'_{13}
  36. Translate(d, Vector[(1, 1, 2)]) → d'_{14}
  37. Translate(d, Vector[(2, 1, 1)]) → d'_{15}
  38. Translate(d, Vector[(2, 1, 2)]) → d'_{16}
  39. Segment(I, E) → h
  40. Segment((0, 0, 2), (0, 1, 2)) → i
  41. Segment((0, 1, 2), (0, 1, 3)) → j
  42. Segment((0, 1, 3), (1, 1, 3)) → k
  43. Segment((0, 1, 2), (1, 1, 2)) → l
  44. Segment((0, 0, 2), (1, 0, 2)) → m
  45. Segment((1, 1, 3), (2, 1, 3)) → n
  46. Segment((2, 0, 3), (2, 1, 3)) → p
  47. Segment((2, 0, 3), (2, 0, 2)) → q
  48. Segment((1, 1, 3), (1, 1, 2)) → r
  49. Segment((2, 1, 2), (1, 1, 2)) → s
  50. Segment((1, 0, 2), (1, 0, 1)) → t
  51. Segment((2, 0, 2), (2, 0, 1)) → f_1
  52. Segment((2, 0, 1), (2, 1, 1)) → g_1
  53. Segment((2, 1, 2), (2, 1, 1)) → h_1
  54. Segment((2, 1, 2), (2, 1, 3)) → i_1

AI 绘图提示语参考

画一个立方体，然后在其中心移除一部分，形成一个开放中间的立方体结构。展示移除部分后剩余表面的各个面和边，用线段连接边界点，并用不同颜色或样式区分各个面和边。先画出原始立方体，再画出移除部分，最后连接剩余部分的边界，形成开放中间的立方体，并可视化其表面结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。