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动画演示多边形变换与相似性

动画演示多边形变换与相似性
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2016-06-18 06:00
多边形向量解析几何变换几何作图
①本范例通过动画演示多边形在变换过程中的相似性与几何属性；②涉及核心数学概念包括多边形构造、相似变换、向量运算、坐标几何等；③作图要点运用了 Polygon（多边形）、Dilate（缩放）、Segment（线段）、Rotate（旋转）、Circle（圆）、Intersect（交点）、CircleSector（扇形）、Vector（向量）、Translate（平移）等 GeoGebra 命令；④适合高中及以上数学学习者，用于理解几何变换与序列生成的动态过程。

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(l, m) → E
  18. Intersect(r, j, 1) → F
  19. Intersect(r, m, 1) → G
  20. Intersect(r, i, 1) → H
  21. Intersect(k, i, 1) → I
  22. Intersect(k, l, 1) → J
  23. Intersect(k, h, 1) → K
  24. Intersect(p, h, 1) → L
  25. Intersect(p, m, 1) → M
  26. Intersect(p, g, 1) → N
  27. Intersect(q, g, 1) → O
  28. Intersect(q, l, 1) → P
  29. Intersect(q, j, 1) → Q
  30. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  31. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  32. Segment(R, A''') → f''
  33. CircleSector(B, L, N) → s
  34. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  35. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  36. Segment(R', S) → f'''
  37. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  38. Vector(B, B') → u
  39. Translate(s, u) → s'
  40. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  41. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  42. Vector(C, C') → v
  43. Translate(s'', v) → s'''
  44. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  45. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  46. Vector(D, D') → w
  47. Translate(t, w) → t'
  48. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  49. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], E) → A'_2
  50. Segment(A, A'_2) → n
  51. Dilate(E, If[11 < e ≤ 12, 12 - e, 0], C) → E'
  52. Segment(E, E') → a_1
  53. Dilate(D, If[12 < e ≤ 13, 13 - e, 0], E) → D'_1
  54. Segment(D, D'_1) → b_1
  55. Dilate(E, If[13 < e ≤ 14, 14 - e, 0], B) → E'_1
  56. Segment(E, E'_1) → c_1
  57. Rotate(t, If[12 < e ≤ 13, (((e - 12) * 3) * π / 4), 2.356194490192345], D) → t'_1
  58. Dilate(D, If[17 < e ≤ 18, 18 - e, 0], E) → D'_2
  59. Vector(D, D'_2) → u_1
  60. Translate(t'_1, u_1) → d_1
  61. Rotate(E, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5707963267948966], E) → E'_2
  62. Rotate(E'_1, If[14 < e ≤ 15, ((e - 14) * π / 2), 1.5708], E) → T
  63. Segment(E'_2, T) → e_1
  64. Rotate(E'_2, If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5707963267948966], E) → U
  65. Rotate(T, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5708], E) → T'
  66. Segment(U, T') → f_1
  67. Rotate(U, If[17 < e ≤ 18, ((e - 17) * π / 2), 1.5707963267948966], E) → U'
  68. Rotate(T', If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5708], E) → T''
  69. Segment(U', T'') → g_1
  70. CircleSector(B, L, M) → h_1
  71. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  72. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  73. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  74. Vector(B, B'_1) → v_1
  75. Vector(B, B'_3) → w_1
  76. Vector(B, B'_2) → i_1
  77. Translate(h_1, v_1) → k_1
  78. Translate(h_1, i_1) → p_1
  79. Translate(h_1, w_1) → q_1
  80. Rotate(k_1, If[19 < e ≤ 20, ((e - 19) * π / 2), 1.5707963267948966], C) → r_1
  81. Rotate(p_1, If[19 < e ≤ 20, (((e - 19) * 3) * π / 2), 4.71238898038469], A) → t_1
  82. Rotate(h_1, If[20.5 < e ≤ 21.5, ((e - 20.5) * π / 4), 0.7853981633974483], B) → c_2
  83. Rotate(r_1, If[20.5 < e ≤ 21.5, ((e - 20.5) * π / 4), 0.7853981633974483], C) → d_2
  84. Rotate(q_1, If[19 < e ≤ 20, ((e - 19) * π), 3.141592653589793], D) → s_1
  85. Rotate(s_1, If[20.5 < e ≤ 21.5, ((e - 20.5) * π / 4), 0.7853981633974483], D) → e_2
  86. Rotate(t_1, If[20.5 < e ≤ 21.5, ((e - 20.5) * π / 4), 0.7853981633974483], A) → f_2