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Pursuit Curve Square: Geometric and Function Construction

Pursuit Curve Square: Geometric and Function Construction
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2018-10-14 19:23
FunctionCoordinate GeometryGeometric Construction
This example demonstrates the pursuit curve in a square, involving concepts of geometric trajectories, function graphs, and coordinate transformations. Construction requires using GeoGebra commands like Polygon, Intersect, and Centroid to build dynamic graphics. Suitable for high school students learning geometric construction and function analysis, useful for math teaching and self-exploration.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Point(f) → E
  3. Intersect(h, i) → F
  4. Intersect(f, g) → G
  5. Intersect(f, g) → I
  6. Centroid(poly1) → J
  7. Circle(A, E) → c
  8. Rotate(c, 90°, J) → c'
  9. Intersect(c', g, 1) → H
  10. Polygon(E, H, 4) → poly2, j, k, l, m, K, L
  11. Circle(E, A) → d
  12. Circle(H, B) → e
  13. Intersect(d, j, 1) → M
  14. Intersect(e, k, 1) → N
  15. Polygon(M, N, 4) → poly3, n, p, q, r, O, P
  16. Circle(M, E) → s
  17. Circle(N, H) → t
  18. Intersect(s, n, 1) → Q
  19. Intersect(t, p, 1) → R
  20. Polygon(Q, R, 4) → poly4, a, b, f_1, g_1, S, T
  21. Circle(Q, M) → c_1
  22. Circle(R, N) → d_1
  23. Intersect(c_1, a, 1) → U
  24. Intersect(d_1, b, 1) → V
  25. Polygon(U, V, 4) → poly5, h_1, i_1, j_1, k_1, W, Z
  26. Circle(U, Q) → e_1
  27. Circle(V, R) → p_1
  28. Intersect(e_1, h_1, 1) → A_1
  29. Intersect(p_1, i_1, 1) → B_1
  30. Polygon(A_1, B_1, 4) → poly6, l_1, m_1, n_1, q_1, C_1, D_1
  31. Circle(A_1, U) → r_1
  32. Circle(B_1, V) → s_1
  33. Intersect(r_1, l_1, 1) → E_1
  34. Intersect(s_1, m_1, 1) → F_1
  35. Polygon(E_1, F_1, 4) → poly7, t_1, a_1, b_1, f_2, G_1, H_1
  36. Circle(E_1, A_1) → c_2
  37. Circle(F_1, B_1) → d_2
  38. Intersect(c_2, t_1, 1) → I_1
  39. Intersect(d_2, a_1, 1) → J_1
  40. Polygon(I_1, J_1, 4) → poly8, g_2, h_2, i_2, j_2, K_1, L_1
  41. Circle(I_1, E_1) → e_2
  42. Circle(J_1, F_1) → k_2
  43. Intersect(e_2, g_2, 1) → M_1
  44. Intersect(k_2, h_2, 1) → N_1
  45. Polygon(M_1, N_1, 4) → poly9, l_2, m_2, n_2, p_2, O_1, P_1
  46. Circle(M_1, I_1) → q_2
  47. Circle(N_1, J_1) → r_2
  48. Intersect(q_2, l_2, 1) → Q_1
  49. Intersect(r_2, m_2, 1) → R_1
  50. Polygon(Q_1, R_1, 4) → poly10, s_2, t_2, a_2, b_2, S_1, T_1
  51. Circle(Q_1, M_1) → c_3
  52. Circle(R_1, N_1) → d_3
  53. Intersect(c_3, s_2, 1) → U_1
  54. Intersect(d_3, t_2, 1) → V_1
  55. Polygon(U_1, V_1, 4) → poly11, f_3, g_3, h_3, i_3, W_1, Z_1
  56. Circle(U_1, Q_1) → e_3
  57. Circle(V_1, R_1) → k_3
  58. Intersect(e_3, f_3, 1) → A_2
  59. Intersect(k_3, g_3, 1) → B_2
  60. Polygon(A_2, B_2, 4) → poly12, j_3, l_3, m_3, n_3, C_2, D_2
  61. Circle(A_2, U_1) → p_3
  62. Circle(B_2, V_1) → q_3
  63. Intersect(p_3, j_3, 1) → E_2
  64. Intersect(q_3, l_3, 1) → F_2
  65. Polygon(E_2, F_2, 4) → poly13, r_3, s_3, t_3, a_3, G_2, H_2
  66. Circle(E_2, A_2) → c_4
  67. Circle(F_2, B_2) → d_4
  68. Intersect(c_4, r_3, 1) → I_2
  69. Intersect(d_4, s_3, 1) → J_2
  70. Polygon(I_2, J_2, 4) → poly14, b_3, f_4, g_4, h_4, K_2, L_2
  71. Circle(I_2, E_2) → e_4
  72. Circle(J_2, F_2) → k_4
  73. Intersect(e_4, b_3, 1) → M_2
  74. Intersect(k_4, f_4, 1) → N_2
  75. Polygon(M_2, N_2, 4) → poly15, i_4, j_4, l_4, m_4, O_2, P_2
  76. Circle(M_2, I_2) → p_4
  77. Circle(N_2, J_2) → q_4
  78. Intersect(p_4, i_4, 1) → Q_2
  79. Intersect(q_4, j_4, 1) → R_2
  80. Polygon(Q_2, R_2, 4) → poly16, n_4, r_4, s_4, t_4, S_2, T_2
  81. Circle(Q_2, M_2) → c_5
  82. Circle(R_2, N_2) → d_5
  83. Intersect(c_5, n_4, 1) → U_2
  84. Intersect(d_5, r_4, 1) → V_2
  85. Polygon(U_2, V_2, 4) → poly17, a_4, b_4, f_5, g_5, W_2, Z_2
  86. Circle(U_2, Q_2) → e_5
  87. Circle(V_2, R_2) → h_5
  88. Intersect(e_5, a_4, 1) → A_3
  89. Intersect(h_5, b_4, 1) → B_3
  90. Polygon(A_3, B_3, 4) → poly18, i_5, j_5, k_5, l_5, C_3, D_3
  91. Circle(A_3, U_2) → p_5
  92. Circle(B_3, V_2) → q_5
  93. Intersect(p_5, i_5, 1) → E_3
  94. Intersect(q_5, j_5, 1) → F_3
  95. Polygon(E_3, F_3, 4) → poly19, m_5, n_5, r_5, s_5, G_3, H_3
  96. Circle(E_3, A_3) → t_5
  97. Circle(F_3, B_3) → c_6
  98. Intersect(t_5, m_5, 1) → I_3
  99. Intersect(c_6, n_5, 1) → J_3

AI Drawing Prompt Reference

Draw a square featuring a pursuit curve construction. Begin by creating a square and marking a point on one of its sides. Then, draw a circle centered at that point, intersecting the square's sides to find intersection points, and use these to construct another square. Iterate this process: for each new square, similarly draw circles, locate intersections, and build subsequent squares, forming a sequence that illustrates the trajectory of a moving point. The design incorporates centroid calculations and rotations for added dynamics, showcasing geometric paths and function graphs for math education.
The prompt above was compiled from the original drawing commands and is for reference only.