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动画演示几何变换

动画演示几何变换
几何画板及绘图内容加载中，请稍等……
2017-03-09 10:25
向量解析几何变换几何作图
所属专题 CCSS High School: Geometry (Congruence) Volume 2
①本范例通过动画演示了正方形、等边三角形及圆的几何变换过程；②涉及的核心数学概念包括几何作图、相似变换、向量运算、坐标几何；③作图要点提炼了 Polygon（多边形）、Dilate（缩放）、Rotate（旋转）、Circle（圆）、Intersect（交点）、Vector（向量）、Translate（平移）等 GeoGebra 命令与对象；④适合高中及以上学生学习几何变换、向量与坐标几何知识，也可用于数学教学演示。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  3. Segment(A, A') → f
  4. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  5. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  6. Segment(A'_1, A'') → f'
  7. Segment(A, B) → g
  8. Segment(B, C) → h
  9. Segment(C, D) → i
  10. Segment(D, A) → j
  11. Circle(C, (0.15 * j)) → k
  12. Circle(B, (0.15 * j)) → p
  13. Circle(A, (0.15 * j)) → q
  14. Circle(D, (0.15 * j)) → r
  15. Segment(A, C) → l
  16. Segment(D, B) → m
  17. Intersect(r, j, 1) → F
  18. Intersect(r, m, 1) → G
  19. Intersect(r, i, 1) → H
  20. Intersect(k, i, 1) → I
  21. Intersect(k, l, 1) → J
  22. Intersect(k, h, 1) → K
  23. Intersect(p, h, 1) → L
  24. Intersect(p, m, 1) → M
  25. Intersect(p, g, 1) → N
  26. Intersect(q, g, 1) → O
  27. Intersect(q, l, 1) → P
  28. Intersect(q, j, 1) → Q
  29. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  30. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  31. Segment(R, A''') → f''
  32. CircleSector(B, L, N) → s
  33. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  34. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  35. Segment(R', S) → f'''
  36. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  37. Vector(B, B') → u
  38. Translate(s, u) → s'
  39. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  40. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  41. Vector(C, C') → v
  42. Translate(s'', v) → s'''
  43. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  44. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  45. Vector(D, D') → w
  46. Translate(t, w) → t'
  47. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  48. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  49. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  50. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  51. Vector(B, B'_1) → v_1
  52. Vector(B, B'_3) → w_1
  53. Vector(B, B'_2) → i_1
  54. Point(f'') → V
  55. Segment(A, V) → n
  56. Angle(B, A, V) → α
  57. Rotate(B, α / 2, A) → B'_4
  58. Angle(B, A, B'_4) → β
  59. Ray(A, B'_4) → j_1
  60. Intersect(f', j_1) → W
  61. Segment(A, W) → l_1
  62. Intersect(q, l_1) → A_1,
  63. Dilate(A, If[13 < e ≤ 14, 14 - e, 0], W) → A'_3
  64. Segment(A, A'_3) → m_1
  65. Segment(A''', V) → n_1
  66. Circle(V, n_1) → d_1
  67. Intersect(d_1, n, 1) → B_1
  68. Intersect(q, n, 1) → E
  69. CircleSector(A, A_1, E) → c_1
  70. Dilate(A, If[11 < e ≤ 12, 12 - e, 0], B_1) → A'_2
  71. Segment(A, A'_2) → f_1
  72. Dilate(B_1, If[12 < e ≤ 13, 13 - e, 0], V) → U
  73. Segment(B_1, U) → g_1
  74. Rotate(c_1, If[14.5 < e ≤ 15.5, ((-(e - 14.5)) * β), (-β)], A) → e_1
  75. Dilate(A''', If[15.5 < e ≤ 16.5, 16.5 - e, 0], V) → T
  76. Dilate(B, If[15.5 < e ≤ 16.5, 16.5 - e, 0], W) → B'_5
  77. Segment(A''', T) → a_1
  78. Segment(B, B'_5) → b_1
  79. Rotate(A''', If[17 < e ≤ 18, ((2 * (e - 17)) * β), (2 * β)], V) → Z
  80. Rotate(T, If[17 < e ≤ 18, ((2 * (e - 17)) * β), (2 * β)], V) → T'
  81. Segment(Z, T') → h_1
  82. Line(B, f_1) → k_1
  83. Circle(B, W) → p_1
  84. Intersect(p_1, k_1, 2) → C_1
  85. Angle(C_1, B, W) → γ
  86. Rotate(B, If[18 < e ≤ 19, ((-(e - 18)) * γ), (-γ)], B) → B'_6
  87. Rotate(B'_5, If[18 < e ≤ 19, ((-(e - 18)) * γ), (-γ)], B) → D_1
  88. Segment(B'_6, D_1) → q_1
  89. Dilate(B, If[19 < e ≤ 20, 20 - e, 0], A) → B'_7
  90. Vector(B, B'_7) → u_1
  91. Translate(B'_6, u_1) → E_1
  92. Translate(D_1, u_1) → F_1
  93. Segment(E_1, F_1) → r_1