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Mechanical Flapping Bird: A 3D Linkage-Driven Animation

Mechanical Flapping Bird: A 3D Linkage-Driven Animation
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2019-12-18 21:28
Solid GeometryTransformationParametric EquationGeometric ConstructionSlider Animation
① This example demonstrates a flapping bird driven by mechanical linkages, using a 3D geometric construction with animated sliders to simulate periodic wing motion. ② Core mathematical concepts include spatial relationships among points, lines, and planes, rotation and reflection transformations, and the use of basic solids like cylinders and cones to build complex mechanical structures; it also touches on linear programming/optimization (to constrain linkage motion) and linear regression (possibly for trajectory fitting). ③ Construction highlights: from the summary, GeoGebra commands such as Circle, Point, Ray, Midpoint, Rotate, Intersect, Segment, Line, Polygon, OrthogonalLine, Cylinder, Cone, Plane, and Mirror are used, creating numerous points (A, B, C, D, E, F, G, H, I, J, K, L, M) and lines (d, e, f, g, h, i, j, k, l, m, t), with rotations and reflections to generate symmetric parts, and polygons/solids to form the bird body and wings. ④ Suitable for high school or university courses in solid geometry, 3D modeling, or mechanical principles, and for interdisciplinary math-engineering projects to understand parametric design and motion simulation.

GeoGebra Commands & Parameters

  1. Circle(A, 2) → c
  2. Point(c) → B
  3. Point(c) → D
  4. Ray(A, D) → f
  5. Midpoint(B, A) → C
  6. Circle(C, A) → d
  7. Rotate(B, (-α), C) → B'
  8. Circle(D, 1) → e
  9. Intersect(e, f, 2) → E
  10. Circle(B', 1.2) → h
  11. Circle(E, 2.2) → g
  12. Intersect(h, g, 2) → F
  13. Segment(F, B') → i
  14. Segment(E, F) → j
  15. Rotate(B', (-(45°)), C) → B''
  16. Circle(B'', 1.3) → p
  17. Intersect(g, p, 1) → H
  18. Ray(H, E) → l
  19. Rotate(F, (-(80°)), E) → F'
  20. Segment(E, F') → k
  21. Intersect(g, l, 2) → G
  22. Circle(G, E) → r
  23. Intersect(r, l, 2) → I
  24. Circle(F', 6) → q
  25. Circle(I, 2.2) → s
  26. Intersect(q, s, 1) → J
  27. Ray(I, J) → m
  28. Circle(J, 15) → t
  29. Intersect(t, m, 1) → K
  30. Segment(I, H) → n
  31. Segment(J, F') → a
  32. Segment(B'', H) → b
  33. Segment(I, K) → f_1
  34. Point(zAxis) → L
  35. Point(zAxis) → M
  36. Line(L, M) → g_1
  37. Line(H, g_1) → h_1
  38. Line(I, g_1) → i_1
  39. Point(i_1) → N
  40. Line(N, n) → j_1
  41. Point(i_1) → O
  42. Line(O, j_1) → k_1
  43. Intersect(j_1, h_1) → P
  44. Intersect(k_1, h_1) → Q
  45. Polygon(P, N, O, Q) → q1, p_1, n_1, o, q_1
  46. Polygon(N, K, O) → t1, o_1, n_2, k_2
  47. Circle(F', 0.2) → c_1
  48. Circle(E, 0.2) → d_1
  49. Circle(H, 0.2) → e_1
  50. OrthogonalLine(F', k) → l_1
  51. OrthogonalLine(E, k) → m_1
  52. Intersect(d_1, m_1, 1) → R
  53. Point(eq1) → S
  54. Line(S, g_1) → r_1
  55. Point(r_1) → T
  56. Point(r_1) → U
  57. Cylinder(T, U, 1) → u, s_1, t_1, v
  58. Point(r_1) → V
  59. Cone(U, V, 1) → w, c_2, a_1
  60. Segment(S, H) → b_1
  61. Point(r_1) → W
  62. Segment(W, P) → f_2
  63. Point(c_2) → Z
  64. Segment(Z, Q) → h_2
  65. Point(r_1) → A_1
  66. Point(r_1) → B_1
  67. Cone(B_1, A_1, 1) → u_1, d_2, v_1
  68. Point(eq1) → C_1
  69. Point(r_1) → D_1
  70. Point(r_1) → E_1
  71. Plane(D_1, C_1, E_1) → p_2
  72. Mirror(q1, p_2) → q1'
  73. Mirror(t1, p_2) → t1'
  74. Mirror(I, p_2) → I'
  75. Mirror(K, p_2) → K'
  76. Segment(I', K') → g_2
  77. Mirror(I, p_2) → I'_1
  78. Mirror(H, p_2) → H'
  79. Segment(I'_1, H') → n'
  80. Mirror(J, p_2) → J'
  81. Mirror(F', p_2) → F''
  82. Segment(J', F'') → a'
  83. Mirror(E, p_2) → E'
  84. Mirror(F', p_2) → F''_1
  85. Segment(E', F''_1) → k'
  86. Mirror(d, p_2) → d'
  87. Mirror(B'', eq1) → B'''
  88. Mirror(H, eq1) → H'_1
  89. Segment(B''', H'_1) → b'
  90. Mirror(F, eq1) → F'_1
  91. Mirror(B', eq1) → B''_1
  92. Segment(F'_1, B''_1) → i'
  93. Mirror(S, eq1) → S'
  94. Mirror(H, eq1) → H'_2
  95. Segment(S', H'_2) → i_2
  96. Mirror(F', eq1) → F''_2
  97. Mirror(d_1, eq1) → e_2
  98. Mirror(c_1, eq1) → q_2
  99. Mirror(E, eq1) → E'_1
  100. Mirror(F, eq1) → F'_2
  101. Segment(E'_1, F'_2) → j'