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Line Symmetry Linkage (Imai's Butterfly)

Line Symmetry Linkage (Imai's Butterfly)
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2016-07-28 18:31
VectorFunctionTrigonometryGeometric Construction
This example demonstrates the geometric construction and motion characteristics of the Line Symmetry Linkage, also known as Imai's Butterfly. It involves core mathematical concepts such as symmetry, geometric transformations, vector operations, function graphs, and three-dimensional vectors. The construction utilizes GeoGebra commands and objects like Circle, Intersect, Point, Segment, Mirror, Angle, Polygon, Dilate, and Midpoint. It is suitable for high school and above learners to understand the relationship between geometric transformations, vectors, and function graphs, as well as to explore the symmetry of complex mechanical structures.

GeoGebra Commands & Parameters

  1. Circle(A, 2) → c
  2. Intersect(c, xAxis, 2) → A'
  3. Circle(A'_1, sqrt(2)) → q_1
  4. Point(c) → B
  5. Circle(A_1, 2) → s
  6. Segment(A_1, B_1) → i
  7. Circle(A_1, sqrt(2)) → q
  8. Intersect(q, i, 1) → F
  9. Circle(F, 2) → p_1
  10. Intersect(p_1, q_1, 2) → E
  11. Mirror(A'_1, E) → D
  12. Circle(D, 2) → r_1
  13. Circle(A_1, (2 * sqrt(2))) → s_1
  14. Intersect(r_1, s_1, 2) → C
  15. Segment(A_1, C) → l_1
  16. Segment(A, B) → f
  17. Mirror(B, xAxis) → B'
  18. Segment(A, B') → g
  19. Angle(A', A, B) → α
  20. Angle(B', A, A') → β
  21. Segment(A, A') → h
  22. Segment(A_1, A'_1) → j
  23. Circle(A'_1, (2 * sqrt(2))) → d
  24. Circle(A_1, (2 * sqrt(2))) → e
  25. Segment(A_1, C) → k
  26. Circle(C, 2) → p
  27. Segment(C, D) → l
  28. Segment(D, A'_1) → m
  29. Segment(A_1, A'_1) → n
  30. Segment(E, F) → r
  31. Polygon(A_1, F, E, A'_1) → poly1, a_1, f_1, e_1, a'_1
  32. Polygon(A_1, A'_1, D, C) → poly2, a_2, a'_2, d_1, c_1
  33. Intersect(s, l_1) → B'_1,
  34. Segment(B'_1, C) → t
  35. Segment(A_1, B'_1) → a
  36. Dilate(G, 1.5, H) → G'
  37. Dilate(G, 1.5, I) → G'_1
  38. Segment(G', H) → b
  39. Segment(I, G'_1) → g_1
  40. Circle(G'_1, G) → h_1
  41. Circle(G', G) → k_1
  42. Intersect(h_1, k_1, 2) → J
  43. Segment(J, G'_1) → i_1
  44. Segment(J, G') → j_1
  45. Segment(G, K) → m_1
  46. Midpoint(L, N) → O
  47. Segment(O, P) → t_1
  48. Segment(P, M) → b_1
  49. Segment(L, M) → f_2
  50. Segment(Q, O) → g_2
  51. Segment(Q, S) → h_2
  52. Segment(R, S) → i_2
  53. Segment(R, P) → j_2
  54. Segment(U, T) → k_2
  55. Segment(O, N) → n_1
  56. Segment(L, O) → l_2
  57. Circle(V, (0.5 * 2)) → c_2
  58. Segment(V, W) → m_2
  59. Point(c_2) → Z
  60. Circle(Z, (0.5 * 3)) → d_2
  61. Segment(V, Z) → n_2
  62. Intersect(d_2, m_2, 1) → C_1
  63. Segment(Z, C_1) → p_2
  64. Circle(C_1, (0.5 * 2)) → e_2
  65. Circle(V, (0.5 * 3)) → q_2
  66. Intersect(e_2, q_2, 2) → D_1
  67. Segment(V, D_1) → r_2
  68. Segment(D_1, C_1) → s_2
  69. Circle(V, (0.5 * 4 / 3)) → t_2
  70. Intersect(t_2, r_2, 1) → E_1
  71. Circle(Z, (0.5 * 4 / 3)) → c_3
  72. Circle(E_1, 1) → d_3
  73. Intersect(c_3, d_3, 2) → F_1
  74. Segment(Z, F_1) → b_2
  75. Segment(F_1, E_1) → f_3
  76. Polygon(F_1, Z, V, E_1) → poly3, f_4, z_1, v, e_3
  77. Angle(F_1, Z, V) → γ
  78. Angle(V, Z, C_1) → δ