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动画演示多边形与几何变换

动画演示多边形与几何变换
几何画板及绘图内容加载中，请稍等……
2017-02-14 20:13
多边形向量微积分几何作图
①本范例通过动画演示多边形、三角形等图形的旋转、平移与缩放变换；②涉及核心数学概念包括多边形构造、向量运算、角度计算、函数图像、微积分等；③构造过程中运用了 Rotate, Ray, Point, Segment, Circle, Intersect, Line, Angle, Min, CircleSector, Dilate, Vector, Translate, Polygon, OrthogonalLine 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高等数学应用。

GeoGebra 命令及参数

  1. Rotate(A, 108°, B) → A'
  2. Ray(B, A') → g
  3. Ray(B, A) → f
  4. Point(f) → C
  5. Segment(C, B) → n
  6. Circle(C, n) → d_1
  7. Rotate(B, (-(108°)), C) → B'
  8. Ray(C, B') → h
  9. Intersect(d_1, h, 1) → G
  10. Circle(G, n) → e_1
  11. Point(h) → D
  12. Rotate(C, (-(108°)), D) → C'
  13. Ray(D, C') → i
  14. Point(i) → E
  15. Rotate(D, (-(108°)), E) → D'
  16. Ray(E, D') → j
  17. Intersect(g, j) → F
  18. Segment(F, E) → k
  19. Segment(E, D) → l
  20. Line(G, l) → g_1
  21. Intersect(e_1, g_1, 1) → J
  22. Line(J, k) → h_1
  23. Intersect(h_1, g) → K
  24. Angle(A, B, A') → α
  25. Angle(B', C, B) → β
  26. Angle(C', D, C) → γ
  27. Angle(D', E, D) → δ
  28. Segment(D, C) → m
  29. Segment(B, F) → p
  30. Min(list) → a
  31. Circle(B, (0.2 * n)) → q
  32. Intersect(q, p) → H,
  33. Intersect(q, n) → I,
  34. CircleSector(B, I, H) → r
  35. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], C) → B'_1
  36. Vector(B, B'_1) → u
  37. Translate(r, u) → r'
  38. Rotate(r', If[1 < o ≤ 2, (((o - 1) * 2) * π / 5), (2 * π / 5)], C) → r''
  39. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], G) → C'_1
  40. Vector(C, C'_1) → v
  41. Translate(r'', v) → r'''
  42. Rotate(r''', If[3 < o ≤ 4, (((o - 3) * 2) * π / 5), (2 * π / 5)], G) → s
  43. Dilate(G, If[4 < o ≤ 5, 5 - o, 0], J) → D'_1
  44. Vector(G, D'_1) → w
  45. Translate(s, w) → s'
  46. Rotate(s', If[5 < o ≤ 6, (((o - 5) * 2) * π / 5), (2 * π / 5)], J) → s''
  47. Dilate(J, If[6 < o ≤ 7, 7 - o, 0], K) → E'
  48. Vector(J, E') → t
  49. Translate(s'', t) → s'''
  50. Circle(J, n) → k_1
  51. Rotate(s''', If[7 < o ≤ 8, (((o - 7) * 2) * π / 5), (2 * π / 5)], K) → c_1
  52. Segment(B, B'_1) → b
  53. Segment(C, C'_1) → c
  54. Segment(G, D'_1) → d
  55. Segment(J, E') → e
  56. Dilate(K, If[8 < o ≤ 9, 9 - o, 0], B) → K'
  57. Segment(K, K') → f_1
  58. Polygon(B, C, G, J, K) → poly1, b_1, c_2, g_2, j_1, k_2
  59. Rotate(B, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → B'_2
  60. Rotate(B'_1, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → L
  61. Segment(B'_2, L) → b'
  62. Rotate(B'_2, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G) → M
  63. Rotate(L, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G) → L'
  64. Segment(M, L') → b''
  65. Rotate(M, If[11 < o ≤ 12, ((-(o - 11)) * α), (-α)], J) → M'
  66. Rotate(L', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], J) → L''
  67. Segment(M', L'') → b'''
  68. Rotate(M', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → M''
  69. Rotate(L'', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → L'''
  70. Segment(M'', L''') → i_1
  71. Segment(J, B) → l_1
  72. Segment(C, J) → m_1
  73. Point(b'') → N
  74. Segment(K, C) → n_1
  75. Segment(N, C) → p_1
  76. Segment(N, K) → q_1
  77. OrthogonalLine(N, n_1) → r_1
  78. Intersect(r_1, n_1) → O
  79. OrthogonalLine(B, m_1) → s_1
  80. Intersect(s_1, m_1) → P
  81. Polygon(B, C, J) → poly2, j_2, b_2, c_3
  82. Segment(N, O) → t_1
  83. Segment(B, P) → a_1
  84. Dilate(B, If[15 < o ≤ 16, 16 - o, 0], J) → B'_3
  85. Segment(B, B'_3) → f_2
  86. Dilate(C, If[15 < o ≤ 16, 16 - o, 0], J) → C'_2
  87. Segment(C, C'_2) → h_2
  88. Angle(J, P, B) → ε
  89. Rotate(B, If[16 < o ≤ 17, ((o - 16) * α / 3), α / 3], J) → B'_4
  90. Rotate(B'_3, If[16 < o ≤ 17, ((o - 16) * α / 3), α / 3], J) → Q
  91. Segment(B'_4, Q) → i_2
  92. Dilate(B, If[17 < o ≤ 18, 18 - o, 0], P) → B'_5
  93. Segment(B, B'_5) → l_2
  94. Rotate(B'_4, If[18 < o ≤ 19, ((o - 18) * α / 3), α / 3], C) → R
  95. Rotate(Q, If[18 < o ≤ 19, ((o - 18) * α / 3), α / 3], C) → Q'
  96. Segment(R, Q') → m_2
  97. Dilate(K, If[19 < o ≤ 20, 20 - o, 0], N) → K'_1
  98. Dilate(C, If[19 < o ≤ 20, 20 - o, 0], N) → C'_3
  99. Segment(K, K'_1) → n_2
  100. Segment(C, C'_3) → p_2
  101. Polygon(K, C, N) → poly3, n_3, k_3, c_4
  102. Line(O, l_2) → q_2
  103. Circle(O, 5) → d_2
  104. Intersect(d_2, q_2, 2) → S
  105. Angle(S, O, N) → ζ
  106. Dilate(N, If[20.5 < o ≤ 21.5, 21.5 - o, 0], O) → N'
  107. Segment(N, N') → r_2
  108. Angle(N, O, K) → η
  109. Dilate(B, If[21.5 < o ≤ 22.5, 22.5 - o, 0], O) → B'_6
  110. Vector(B, B'_6) → u_1
  111. Translate(B, u_1) → B'_7
  112. Translate(B'_5, u_1) → T
  113. Segment(B'_7, T) → s_2
  114. Rotate(B'_7, If[22.5 < o ≤ 23.5, ((o - 22.5) * α / 3), α / 3], O) → U
  115. Rotate(T, If[22.5 < o ≤ 23.5, ((o - 22.5) * α / 3), α / 3], O) → T'
  116. Segment(U, T') → t_2

AI 绘图提示语参考

绘制一个三角形ABC，然后围绕点B旋转它108度得到A'，接着画出从B到A'和从B到A的射线，找到它们的交点C。以C为圆心，以BC为半径画圆，再围绕C旋转B得到B'，画出从C到B'的射线，找到它们的交点G，以此类推，构造一系列相关图形，包括圆、线段、角度和缩放变换，最终形成一个动态的几何变换动画。
以上提示语为根据原始绘图命令整理而来，仅供参考。