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Dancing Points on Regular Polygons

Dancing Points on Regular Polygons
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2019-02-12 17:26
PolygonTrigonometryCalculusGeometric Construction
This example demonstrates the trajectories and transformations of points moving on regular polygons, involving core math concepts like polygons, rotation, symmetry, trigonometry, and definite integrals. Construction highlights GeoGebra commands: Polygon, Centroid, Point, Rotate, Angle, Ray, Intersect, Mirror. Suitable for high school and above, exploring the connection between geometric transformations and dynamic geometry, and the application of calculus in geometric problems.

GeoGebra Commands & Parameters

  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