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体积的复制与变换探究

体积的复制与变换探究
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2017-09-17 03:58
多边形角数列变换几何作图
① 本范例通过GeoGebra动态演示了三维体积概念在二维平面上的投影与复制过程，利用旋转、平移和序列等命令构建多个多边形并计算其面积，从而直观展示体积与面积的关系。② 核心数学概念包括体积、角度、线性方程、三角形、微积分基础以及几何变换，涉及多边形面积计算、旋转对称性和序列生成。③ 作图要点：使用Line、OrthogonalLine、Circle、Intersect、Rotate、Angle、Segment、Midpoint、Polygon、Translate、Sequence、Point等命令，创建了多个点、线、圆和多边形对象（如A、B、E、F、S'、poly1、poly2、poly3等），通过旋转和平移构造出不同位置的图形，并利用角度和线段测量辅助分析。④ 适合中学高年级至大学低年级学生，用于几何与微积分的衔接教学，也可作为教师演示体积概念和几何变换的互动工具。

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