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长方体体积的动态演示

长方体体积的动态演示
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2017-04-17 12:25
多边形立体几何几何作图滑块动画
① 本范例通过GeoGebra三维视图动态展示一个长方体，并实时计算其体积，直观呈现体积与长、宽、高的关系。② 核心数学概念是立体几何中的长方体体积公式 V = 长×宽×高，涉及棱、面、顶点等基本元素以及体积的度量。③ 作图要点：从构造摘要看，使用了Line、OrthogonalLine、Circle、Intersect、Rotate、Angle、Segment、Midpoint、Polygon、Translate、Sequence、Point等命令，构建了长方体的框架、旋转视角、辅助线段和标注点，并通过多边形poly1、poly2、poly3等围合成立体表面，最终用变量V表示体积。④ 适合初中或高中立体几何教学，用于帮助学生理解长方体体积公式的几何意义，也可作为教师演示体积计算与空间想象的交互工具。

GeoGebra 命令及参数

  1. Line(A, xAxis) → a
  2. OrthogonalLine(A, a) → q_4
  3. Circle(A, 2) → r
  4. Intersect(r, q_4, 2) → E
  5. Intersect(r, a, 2) → B
  6. OrthogonalLine(E, q_4) → s
  7. OrthogonalLine(B, a) → t
  8. Intersect(s, t) → F
  9. Rotate(F, 45°, E) → S'
  10. Angle(F, E, S') → α
  11. Segment(E, S') → i_1
  12. Segment(E, F) → j_1
  13. Midpoint(j_1) → T
  14. Circle(E, T) → k_1
  15. Intersect(k_1, i_1, 1) → H
  16. Segment(E, H) → p_1
  17. Line(H, j_1) → q_1
  18. Line(F, p_1) → r_1
  19. Intersect(q_1, r_1) → V
  20. Segment(A, E) → s_1
  21. Segment(F, B) → t_1
  22. Line(B, r_1) → a_2
  23. Line(V, s_1) → b_2
  24. Intersect(a_2, b_2) → W
  25. Polygon(A, B, F, E) → poly1, p_2, r_2, s_2, q_2
  26. Polygon(B, W, V, F) → poly2, r_3, w, v, s_3
  27. Polygon(E, F, V, H) → poly3, q_3, s_4, v_1, u
  28. Translate(A, B - A) → A'
  29. Translate(E, B - A) → E'
  30. Sequence(Sequence[Polygon[Translate[A, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[B, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[F, Vector[(k * Vector[A, B]) + (l * Vector[A, E])]], Translate[E, Vector[(k * Vector[A, B]) + (l * Vector[E, A])]]], k, 0, o], l, 0, 0) → ligne
  31. Sequence(Sequence[Polygon[Translate[E, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[F, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[V, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]], Translate[H, Vector[(l * Vector[A, B]) + (m * Vector[E, H])]]], l, 0, o], m, 0, q) → liste3
  32. Sequence(Polygon[Translate[B, (k * (W - B)) + (o * (B - A))], Translate[W, (k * (W - B)) + (o * (B - A))], Translate[V, (k * (W - B)) + (o * (B - A))], Translate[F, (k * (W - B)) + (o * (B - A))]], k, 0, q) → ligne2
  33. Sequence(Sequence[Polygon[Translate[A, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[B, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[F, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]], Translate[E, Vector[(k * Vector[B, A]) - (l * Vector[E, A])]]], k, 0, 0], l, 0, p) → liste
  34. Sequence(Sequence[Polygon[Translate[B, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[W, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[V, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]], Translate[F, Vector[((-l) * Vector[E, A]) + (m * Vector[E, H])]]], l, 0, p], m, 0, 0) → liste2
  35. Translate(E, (p * (E - A))) → E'_1
  36. Translate(F, (p * (E - A))) → F'
  37. Translate(V, (p * (E - A))) → V'
  38. Translate(H, (p * (E - A))) → H'
  39. Polygon(E'_1, F', V', H') → Haut, e'_1, f', v', h'
  40. Translate(A, ((o + 1) * (B - A))) → Sommet1
  41. Translate(A, ((p + 1) * (E - A))) → Sommet2
  42. OrthogonalLine(E'_1, s_1) → b
  43. OrthogonalLine(Sommet1, p_2) → c
  44. Intersect(b, c) → Sommet3
  45. Translate(A, ((q + 1) * (H - E))) → Sommet4
  46. OrthogonalLine(Sommet4, p_2) → d
  47. Segment(A, Sommet4) → e
  48. Segment(A, E'_1) → f
  49. Line(E'_1, e) → g
  50. Intersect(d, g) → C
  51. Line(Sommet3, g) → h
  52. Line(C, b) → i
  53. Intersect(h, i) → D
  54. Line(D, d) → j
  55. Line(Sommet1, h) → k
  56. Intersect(j, k) → G
  57. Segment(E'_1, Sommet3) → l
  58. Segment(Sommet3, Sommet1) → m
  59. Segment(Sommet3, D) → n
  60. Segment(D, G) → a_1
  61. Segment(D, C) → b_1
  62. Segment(C, E'_1) → c_1
  63. Segment(F', B) → d_1
  64. Segment(V', V) → e_1
  65. Segment(H', V') → f_1
  66. Segment(A, Sommet1) → g_1
  67. Point(g_1) → I
  68. Segment(I, G) → h_1
  69. Segment(F, E) → l_1
  70. Segment(F, B) → m_1
  71. Line(E, g_1) → n_1
  72. Intersect(n_1, m) → J
  73. Segment(F, J) → c_2
  74. Segment(F', V') → d_2
  75. Segment(B, W) → e_2
  76. Line(J, h_1) → f_2
  77. Intersect(a_1, f_2) → K
  78. Point(h_1) → L
  79. Segment(L, K) → g_2
  80. Segment(J, K) → h_2
  81. Line(H, h_1) → i_2
  82. Line(K, l) → j_2
  83. Intersect(i_2, j_2) → M
  84. Segment(M, K) → k_2
  85. Intersect(i_2, e_1) → N
  86. Segment(N, M) → l_2
  87. Intersect(r_1, j_2) → O
  88. Segment(F, O) → m_2
  89. Sequence(Segment[Translate[F, ((-k) * (A - B))], Translate[O, ((-k) * (A - B))]], k, 0, o) → seg4
  90. Line(N, b_1) → n_2
  91. Intersect(n_2, h_2) → P
  92. Segment(N, P) → t_2
  93. Line(V, t_2) → a_3
  94. Intersect(a_3, h_2) → Q
  95. Line(S', a_3) → b_3
  96. Intersect(b_3, h_2) → R
  97. Segment(V, Q) → c_3
  98. Polygon(F, V, N, M, K, J) → poly4, f_3, v_2, n_3, m_3, k_3, j_3
  99. Polygon(V, F, F', V') → poly5, v_3, f_4, f'_1, v'_1
  100. Segment(N, M) → d_3
  101. Segment(M, K) → e_3
  102. Sequence(Segment[Translate[F, ((-k) * (A - E))], Translate[E, ((-k) * (A - E))]], k, 0, p) → seg_1
  103. Sequence(Segment[Translate[F, (k * (B - A))], Translate[B, (k * (B - A))]], k, 0, o) → seg_2
  104. Sequence(Segment[Translate[F, ((-l) * (E - H)) + (o * (B - A))], Translate[B, ((-l) * (E - H)) + (o * (B - A))]], l, 1, q) → seg10
  105. Sequence(Segment[Translate[N, (l * (H - E))], Translate[P, (l * (H - E))]], l, -2, q - 3) → seg5
  106. Sequence(Segment[Translate[B, ((-k) * (B - F))], Translate[W, ((-k) * (B - F))]], k, 1, p) → seg_3

AI 绘图提示语参考

绘制一个长方体，包含长度、宽度和高度三个可调节的滑块。首先，创建一个点A作为长方体的一个顶点，然后通过A点和x轴创建一条线a。接着，从A点出发，创建一个半径为2的圆r，并与a和q_4的交点分别为B和E。通过旋转、角度、线段、中点等操作，构建长方体的框架，形成三个底面多边形poly1、poly2和poly3。最后，通过平移和旋转，形成长方体的立体结构，并显示其体积V。
以上提示语为根据原始绘图命令整理而来，仅供参考。