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正多边形上的舞动点

正多边形上的舞动点
几何画板及绘图内容加载中，请稍等……
2019-02-12 17:26
多边形三角函数微积分几何作图
①本范例演示了在正多边形上运动的点的轨迹与变换现象；②涉及的核心数学概念包括多边形、旋转、对称、三角函数、积分等；③作图要点提炼自GeoGebra命令：Polygon（多边形）、Centroid（重心）、Point（点）、Rotate（旋转）、Angle（角度）、Ray（射线）、Intersect（交点）、Mirror（镜像）；④适合高中及以上数学学习者，用于理解几何变换与动态几何的关联，以及探索积分在几何问题中的应用。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Centroid(poly1) → E
  3. Point(yAxis) → F
  4. Rotate(F, 12°, E) → F'
  5. Angle(F, E, F') → α
  6. Ray(E, F') → i
  7. Intersect(j, i) → G
  8. Mirror(G, yAxis) → G'
  9. Polygon(G', G, 15) → poly2, k, b, c, d, e, f_1, g_1, H, I, J, K, l, L, M, N, O, P, Q, R, S, T, m, n, p, q, r, s, t
  10. Point(poly1, Mod[1.17 - a, 1]) → D
  11. Rotate(F, (360 / 28)°, E) → F'_2
  12. Angle(F, E, F'_2) → β
  13. Ray(E, F'_2) → h_1
  14. Intersect(j, h_1) → V
  15. Mirror(V, yAxis) → V'
  16. Polygon(V', V, 14) → poly3, i_1, s_1, t_1, a_1, b_1, c_1, W, Z, A_1, B_1, C_1, j_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, k_1, l_1, m_1, n_1, p_1, q_1, r_1
  17. Point(poly3, Mod[1.0361 - a / (7 / 3), 1]) → K_1
  18. Point(poly2, Mod[1.0335 - a / 3.5, 1]) → U
  19. Rotate(F, (360 / 26)°, E) → F'_1
  20. Angle(F, E, F'_1) → γ
  21. Ray(E, F'_1) → d_1
  22. Intersect(j, d_1) → L_1
  23. Mirror(L_1, yAxis) → M_1
  24. Polygon(M_1, L_1, 13) → poly4, e_1, n_2, p_2, q_2, r_2, N_1, O_1, P_1, Q_1, R_1, S_1, f_2, T_1, U_1, V_1, W_1, Z_1, g_2, h_2, i_2, j_2, k_2, l_2, m_2
  25. Point(poly4, Mod[1.0385 - a / (7 / 4), 1]) → A_2
  26. Rotate(F, (360 / 24)°, E) → F'_3
  27. Angle(F, E, F'_3) → δ
  28. Ray(E, F'_3) → s_2
  29. Intersect(j, s_2) → B_2
  30. Mirror(B_2, yAxis) → C_2
  31. Polygon(C_2, B_2, 12) → poly5, t_2, i_3, j_3, k_3, D_2, E_2, F_2, G_2, H_2, I_2, J_2, a_2, K_2, L_2, M_2, b_2, c_2, d_2, e_2, f_3, g_3, h_3
  32. Point(poly5, Mod[1.042 - a / (7 / 5), 1]) → N_2
  33. Rotate(F, (360 / 22)°, E) → F'_4
  34. Angle(F, E, F'_4) → ε
  35. Ray(E, F'_4) → l_3
  36. Intersect(l_3, j) → O_2
  37. Mirror(O_2, yAxis) → P_2
  38. Polygon(P_2, O_2, 11) → poly6, m_3, c_3, d_3, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2, n_3, A_3, p_3, q_3, r_3, s_3, t_3, a_3, b_3
  39. Point(poly6, Mod[1.046 - a / (7 / 6), 1]) → B_3
  40. Rotate(F, (360 / 20)°, E) → F'_5
  41. Angle(F, E, F'_5) → ζ
  42. Ray(E, F'_5) → e_3
  43. Intersect(e_3, j) → C_3
  44. Mirror(C_3, yAxis) → D_3
  45. Polygon(D_3, C_3, 10) → poly7, f_4, p_4, E_3, F_3, G_3, H_3, I_3, J_3, K_3, L_3, g_4, h_4, i_4, j_4, k_4, l_4, m_4, n_4
  46. Point(poly7, Mod[1.051 - a, 1]) → M_3
  47. Rotate(F, (360 / 18)°, E) → F'_6
  48. Angle(F, E, F'_6) → η
  49. Ray(E, F'_6) → q_4
  50. Intersect(q_4, j) → N_3
  51. Mirror(N_3, yAxis) → O_3
  52. Polygon(O_3, N_3, 9) → poly8, r_4, P_3, Q_3, R_3, S_3, T_3, U_3, V_3, s_4, t_4, a_4, b_4, c_4, d_4, e_4, f_5
  53. Point(poly8, Mod[1.056 - a / (7 / 8), 1]) → W_3
  54. Rotate(F, (360 / 16)°, E) → F'_7
  55. Angle(F, E, F'_7) → θ
  56. Ray(E, F'_7) → g_5
  57. Intersect(g_5, j) → Z_3
  58. Mirror(Z_3, yAxis) → A_4
  59. Polygon(A_4, Z_3, 8) → poly9, h_5, C_4, D_4, E_4, F_4, G_4, i_5, j_5, k_5, l_5, m_5, n_5, p_5, B_4
  60. Point(poly9, Mod[1.063 - a / (7 / 9), 1]) → H_4
  61. Rotate(F, (360 / 14)°, E) → F'_8
  62. Angle(F, E, F'_8) → ι
  63. Ray(E, F'_8) → q_5
  64. Intersect(q_5, j) → I_4
  65. Mirror(I_4, yAxis) → J_4
  66. Polygon(J_4, I_4, 7) → poly10, r_5, M_4, N_4, O_4, s_5, t_5, a_5, b_5, c_5, d_5, K_4, L_4
  67. Point(poly10, Mod[1.072 - a / (7 / 10), 1]) → P_4
  68. Rotate(F, (360 / 12)°, E) → F'_9
  69. Angle(F, E, F'_9) → κ
  70. Ray(E, F'_9) → e_5
  71. Intersect(e_5, j) → Q_4
  72. Mirror(Q_4, yAxis) → R_4
  73. Polygon(R_4, Q_4, 6) → poly11, f_6, V_4, g_6, h_6, i_6, j_6, k_6, S_4, T_4, U_4
  74. Point(poly11, Mod[1.084 - a / (7 / 11), 1]) → W_4
  75. Rotate(F, (360 / 10)°, E) → F'_{10}
  76. Angle(F, E, F'_{10}) → λ
  77. Ray(E, F'_{10}) → l_6
  78. Intersect(l_6, j) → Z_4
  79. Mirror(Z_4, yAxis) → A_5
  80. Polygon(A_5, Z_4, 5) → poly12, m_6, n_6, p_6, q_6, r_6, B_5, C_5, D_5
  81. Point(poly12, Mod[1.101 - a / (7 / 12), 1]) → E_5
  82. Rotate(F, (360 / 8)°, E) → F'_{11}
  83. Angle(F, E, F'_{11}) → μ
  84. Ray(E, F'_{11}) → s_6
  85. Intersect(s_6, j) → F_5
  86. Mirror(F_5, yAxis) → G_5
  87. Polygon(G_5, F_5, 4) → poly13, t_6, a_6, b_6, c_6, H_5, I_5
  88. Point(poly13, Mod[1.127 - a / (7 / 13), 1]) → J_5