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正多边形构造与变换演示

正多边形构造与变换演示
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2017-01-02 04:05
多边形数列变换几何作图滑块动画
①本范例通过GeoGebra动态演示正多边形的生成过程，并展示其平移、旋转、缩放等变换操作。②核心数学概念包括正多边形的定义（各边相等、各角相等）、中心与半径的关系、旋转对称性，以及向量平移、位似缩放等几何变换。③作图要点：使用Sequence命令生成顶点序列，Distance计算边长，Center确定中心，Rotate实现旋转，Polygon构造多边形，Vector和Translate进行平移，Dilate进行缩放，Intersect求交点，Segment连接线段，Vertex提取顶点；涉及对象包括Sides、β、list1、h、c、A、B_1、list2、b、list4、list5、A'、list3、list6、poly1、d、e、a'、u、list7、f、g、ah、C、D、i、D'、v、list1'、c'、list2'、poly2、E、F、w、poly3、poly3'、G、j、poly4等。④适合中学几何教学，用于理解正多边形性质及动态变换，也可用于探究多边形内角和、对称轴等拓展内容。

GeoGebra 命令及参数

  1. Sequence((cos(((k * π) * β / 180)), sin(((k * π) * β / 180))), k, 0, Sides, 1) → list1
  2. Distance(Element[list1, 1], Element[list1, 2]) → h
  3. Center(c) → A
  4. Sequence(Polygon[Element[list1, k], A, Element[list1, k + 1]], k, 1, Sides, 1) → list2
  5. Sequence(Vertex[Element[list2, k], 3], k, 2, Sides, 1) → list4
  6. Sequence(Vertex[Element[list2, k], 3], k, 1, Sides - 1) → list5
  7. Rotate(A, (((-b) * 360) * ° / Sides), B_1) → A'
  8. Sequence(Rotate[Element[list1, k], ((((-(k - 1)) * b) * 360) * ° / Sides), Element[list1, k - 1]], k, 2, Sides, 1) → list3
  9. Sequence(Rotate[Element[list3, k], ((((-(k - 1)) * b) * 360) * ° / Sides), Element[list3, k - 1]], k, 2, Sides - 1, 1) → list6
  10. Polygon(Element[list1, 1], A', Element[list3, 1]) → poly1, d, e, a'
  11. Vector(Element[list1, 2], Element[list3, 1]) → u
  12. Sequence(Vector[Element[list1, k], Element[list3, k - 1]], k, 2, Sides, 1) → list7
  13. Segment((0, -1), (6.283185307179586, -1)) → f
  14. Intersect(c, d, 1) → D
  15. Segment(D, C) → i
  16. Dilate(D, If[0 < g ≤ 1, 1 - g, 0], C) → D'
  17. Vector(D, D') → v
  18. Translate(list1, v) → list1'
  19. Translate(c, v) → c'
  20. Translate(list2, v) → list2'
  21. Rotate(Element[list2', 1], (((-g) * 360 / Sides))°, Element[list1', 1]) → poly2
  22. Vertex(Element[list2', 2], 1) → E
  23. Vertex(poly2, 3) → F
  24. Vector(Vertex[Element[list2', 2], 1], Vertex[poly2, 3]) → w
  25. Translate(Element[list2', 2], w) → poly3
  26. Rotate(poly3, ((((-g) * 2) * 360 / Sides))°, F) → poly3'
  27. Vertex(poly3', 3) → G
  28. Vector(Vertex[Element[list2', 3], 1], Vertex[poly3', 3]) → j
  29. Translate(Element[list2', 3], j) → poly4
  30. Rotate(poly4, ((((-g) * 3) * 360 / Sides))°, G) → poly4'
  31. Vertex(poly4', 3) → H
  32. Vector(Element[list1', 4], H) → k
  33. Translate(Element[list2', 4], k) → poly5
  34. Rotate(poly5, ((((-g) * 4) * 360 / Sides))°, H) → poly5'
  35. Vertex(poly5', 3) → I
  36. Vector(Element[list1', 5], I) → l
  37. Translate(Element[list2', 5], l) → poly6
  38. Rotate(poly6, ((((-g) * 5) * 360 / Sides))°, I) → poly6'
  39. Vertex(poly6', 3) → J
  40. Vector(Element[list1', 6], J) → m
  41. Translate(Element[list2', 6], m) → poly7
  42. Rotate(poly7, ((((-g) * 6) * 360 / Sides))°, J) → poly7'
  43. Rotate(poly7, ((((-g) * 6) * 360 / Sides))°, J) → poly7'_1
  44. Vertex(poly7'_1, 3) → K
  45. Vector(Element[list1', 7], K) → n
  46. Translate(Element[list2', 7], n) → poly8
  47. Rotate(poly8, ((((-g) * 7) * 360 / Sides))°, K) → poly8'
  48. Vertex(poly8', 3) → L
  49. Vector(Element[list1', 8], L) → p
  50. Translate(Element[list2', 8], p) → poly9
  51. Rotate(poly9, ((((-g) * 8) * 360 / Sides))°, L) → poly9'
  52. Vertex(poly9', 3) → M
  53. Vector(Element[list1', 9], M) → q
  54. Translate(Element[list2', 9], q) → poly10
  55. Vertex(poly2, 2) → B
  56. Vertex(poly3', 2) → N
  57. Vertex(poly4', 2) → O
  58. Vertex(poly5', 2) → P
  59. Vertex(poly6', 2) → Q
  60. Vertex(poly7'_1, 2) → R
  61. Vertex(poly8', 2) → S
  62. Vertex(poly9', 2) → T
  63. Rotate(poly2, (((-o) * 180))°, B) → poly2'
  64. Rotate(poly3', (((-o) * 180))°, N) → poly3''
  65. Rotate(poly4', (((-o) * 180))°, O) → poly4''
  66. Rotate(poly5', (((-o) * 180))°, P) → poly5''
  67. Rotate(poly6', (((-o) * 180))°, Q) → poly6''
  68. Rotate(poly7'_1, (((-o) * 180))°, R) → poly11
  69. Rotate(poly8', (((-o) * 180))°, S) → poly8''
  70. Rotate(poly9', (((-o) * 180))°, T) → poly9''
  71. Dilate(B, 1 - r, C) → B'
  72. Dilate(N, 1 - r, F) → N'
  73. Dilate(O, 1 - r, G) → O'
  74. Dilate(P, 1 - r, H) → P'
  75. Dilate(Q, 1 - r, I) → Q'
  76. Dilate(R, 1 - r, J) → R'
  77. Dilate(S, 1 - r, K) → S'
  78. Dilate(T, 1 - r, L) → T'
  79. Vector(N, N') → s_1
  80. Vector(B, B') → t
  81. Vector(O, O') → u_1
  82. Vector(P, P') → v_1
  83. Vector(Q, Q') → w_1
  84. Vector(R, R') → a_1
  85. Vector(S, S') → b_1
  86. Vector(T, T') → c_1
  87. Translate(poly2', t) → poly2''
  88. Translate(poly3'', s_1) → poly3'''
  89. Translate(poly4'', u_1) → poly4'''
  90. Translate(poly5'', v_1) → poly5'''
  91. Translate(poly6'', w_1) → poly6'''
  92. Translate(poly11, a_1) → poly11'
  93. Translate(poly8'', b_1) → poly8'''
  94. Translate(poly9'', c_1) → poly9'''