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长方体体积计算演示

长方体体积计算演示
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2015-10-17 22:12
多边形数列立体几何矩阵
本示例演示了如何利用 GeoGebra 计算长方体的体积。通过定义长、宽、高的参数，并使用向量和矩阵变换，系统地构建了三维立方体模型。核心数学概念包括体积公式 V = a·b·c、向量运算、矩阵表示以及正弦参数化。作图中运用了 Sequence、Element、Point、Polygon、Segment 等命令，以及 α、β 旋转矩阵 R_x、R_y 等对象，实现了动态调整边长的交互式探索。适合高中或大学一年级几何与线性代数课程使用。

GeoGebra 命令及参数

  1. Sequence((Mat * Element[cube, k]), k, 1, 8) → cube'
  2. Element(Element[AE, 3], 1) → droite
  3. Sequence((Element[cube', k, 1, 1], Element[cube', k, 2, 1]), k, 1, 8) → listepoints
  4. Point(listepoints) → A
  5. Point(listepoints) → E
  6. Point(listepoints) → C_1
  7. Point(listepoints) → D
  8. Polygon(A, E, C_1, D) → poly1, a_1, b, c_1, d
  9. Point(listepoints) → B
  10. Point(listepoints) → F
  11. Point(listepoints) → G
  12. Point(listepoints) → H
  13. Polygon(B, F, G, H) → poly2, e, f, g, h
  14. Point(listepoints) → C
  15. Polygon(F, C, A, B) → poly3, f_1, i, a_2, b_1
  16. Element(AB, 3, 1) → avant
  17. Polygon(C_1, G, H, E) → poly4, c_2, g_1, h_1, e_1
  18. Element(AC, 3, 1) → dessus
  19. Polygon(F, C, C_1, G) → poly5, f_2, c_3, c_4, g_2
  20. Polygon(A, B, H, E) → poly6, a_3, b_2, h_2, e_2
  21. Segment(E, A) → c
  22. Segment(A, C) → j
  23. Segment(A, B) → k
  24. Segment(C, C_1) → l
  25. Segment(C_1, E) → m
  26. Segment(H, E) → n
  27. Segment(A, B) → p
  28. Segment(B, H) → q
  29. Segment(H, E) → r
  30. Segment(H, G) → s
  31. Segment(B, F) → t
  32. Segment(G, C_1) → d_1
  33. Segment(F, G) → i_1
  34. Segment(F, C) → j_1
  35. Sequence(Sequence[Sequence[Polygon[Translate[A, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[E, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[C_1, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[D, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste1
  36. Sequence(Sequence[Sequence[Polygon[Translate[B, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[F, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[G, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[H, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste2
  37. Sequence(Sequence[Sequence[Polygon[Translate[F, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[C, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[A, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[B, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste3
  38. Sequence(Sequence[Sequence[Polygon[Translate[C_1, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[G, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[H, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[E, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste4
  39. Sequence(Sequence[Sequence[Polygon[Translate[F, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[C, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[C_1, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[G, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste5
  40. Sequence(Sequence[Sequence[Polygon[Translate[A, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[B, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[H, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]], Translate[E, Vector[(k * Vector[A, E]) + (l * Vector[A, B]) + (i * Vector[A, D])]]], k, 0, u], l, 0, k_1], i, 0, l_1) → liste6

AI 绘图提示语参考

绘制一个长方体，其长、宽、高可调。先确定八个顶点，其中四个构成底面，四个构成顶面，并连接对应顶点形成侧棱。用六个面围成立体，每个面为四边形。然后，沿长方体的三条棱方向，按整数倍平移复制这些小面，形成网格状的长方体堆叠，展示体积分割。整体呈现三维透视效果，可旋转观察。
以上提示语为根据原始绘图命令整理而来，仅供参考。