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Linkage Curves Demo

Linkage Curves Demo
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2020-12-08 20:27
FunctionTrigonometryCoordinate GeometryParametric Equation
This example demonstrates the generation of complex curves through linkage mechanisms. It involves core mathematical concepts such as parametric equations, coordinate geometry, geometric transformations, and trigonometry. Key GeoGebra commands and objects used in the construction include Ray, Midpoint, Line, Intersect, Polygon, Centroid, Mirror, Circle, Point, Rotate, Segment, CurveCartesian, and Semicircle. It is suitable for high school and above students to learn about the creation process of parametric curves and the application of geometric transformations.

GeoGebra Commands & Parameters

  1. Ray(T, C_1) → c_1
  2. Ray(S, B_1) → b_1
  3. Midpoint(J, T) → Z
  4. Midpoint(Z, T) → A_1
  5. Line(H, J) → a_1
  6. Line(U, a_1) → d_1
  7. Intersect(c_1, d_1) → E_1
  8. Intersect(b_1, d_1) → D_1
  9. Midpoint(S, I) → V
  10. Midpoint(V, S) → W
  11. Polygon(D_1, E_1, C_1, A_1, W, B_1) → poly1, e_1, f_2, g_2, h_2, w_1, i_2
  12. Midpoint(f_2) → J_1
  13. Centroid(poly1) → F_1
  14. Mirror(B_1, F_1) → G_1
  15. Midpoint(G_1, J_1) → L_1
  16. Midpoint(G_1, L_1) → M_1
  17. Midpoint(G_1, M_1) → V_1
  18. Line(A, xAxis) → n
  19. Circle(A, R) → c
  20. Intersect(c, n) → E, F
  21. Point(c) → B
  22. Circle(A, R + r1) → d
  23. Rotate(F, (t * w), A) → B'
  24. Ray(A, B') → a
  25. Intersect(d, a) → C,
  26. Circle(C, B') → e
  27. Rotate(B', (t * w1) + 3.141592653589793, C) → B''
  28. Circle(C, r1 + r2) → f
  29. Ray(C, B'') → b
  30. Intersect(f, b) → D,
  31. Circle(D, B'') → g
  32. Rotate(B'', (t * w2) + 3.141592653589793, D) → B'''
  33. Segment(D, B''') → h
  34. Segment(C, B'') → i
  35. Segment(A, B') → j
  36. CurveCartesian(x(A) + ((R + r1) * (cos((w * i)) + cos((w1 * i)))), y(A) + ((R + r1) * (sin((w * i)) + sin((w1 * i)))), i, 0, t) → k
  37. CurveCartesian(x(A) + ((R + r1) * cos((w * i))) - (r1 * cos(((w1 + w) * i) + 3.141592653589793)), y(A) + ((R + r1) * sin((w * i))) - (r1 * sin(((w1 + w) * i) + 3.141592653589793)), i, 0, t) → o
  38. CurveCartesian(x(A) + ((R + r1) * cos((w * i))) - ((r1 + r2) * cos(((w1 + w) * i) + 3.141592653589793)) - (r2 * cos(((w + w1 + w2) * i) + 3.141592653589793)), y(A) + ((R + r1) * sin((w * i))) - ((r1 + r2) * sin(((w1 + w) * i) + 3.141592653589793)) - (r2 * sin(((w + w1 + w2) * i) + 3.141592653589793)), i, 0, t) → l
  39. CurveCartesian(x(A) + ((R + r1) * cos((w * i))) - ((r1 + r2) * cos(((w1 + w) * i) + 3.141592653589793)) - (r2 * cos(((w + w1 + w2) * i) + 3.141592653589793)), y(A) + ((R + r1) * sin((w * i))) - ((r1 + r2) * sin(((w1 + w) * i) + 3.141592653589793)) - (r2 * sin(((w + w1 + w2) * i) + 3.141592653589793)), i, 0, t) → m
  40. Point(j) → G
  41. Segment(E, G) → s
  42. Segment(B', C) → f_1
  43. Segment(B'', D) → g_1
  44. Segment(D, B''') → h_1
  45. Polygon(I, A, J) → t1, p, q, r
  46. Midpoint(q) → K
  47. Midpoint(J, K) → L
  48. Midpoint(p) → M
  49. Midpoint(I, M) → N
  50. Midpoint(q) → O
  51. Midpoint(A, K) → P
  52. Midpoint(A, M) → Q
  53. Segment(I, L) → i_1
  54. Segment(L, M) → j_1
  55. Segment(M, P) → k_1
  56. Segment(P, A) → l_1
  57. Segment(J, N) → m_1
  58. Segment(N, K) → n_1
  59. Segment(K, Q) → p_1
  60. Segment(Q, A) → q_1
  61. Segment(N, L) → r_1
  62. Segment(M, K) → s_1
  63. Segment(P, Q) → t_1
  64. Mirror(A_1, F_1) → H_1
  65. Mirror(W, F_1) → W'
  66. Mirror(C_1, F_1) → I_1
  67. Midpoint(J_1, C_1) → K_1
  68. Midpoint(L_1, J_1) → N_1
  69. Midpoint(J_1, K_1) → O_1
  70. Midpoint(K_1, C_1) → P_1
  71. Segment(Q_1, R_1) → k_2
  72. Midpoint(N_1, J_1) → W_1
  73. Midpoint(J_1, O_1) → Z_1
  74. Midpoint(P_1, C_1) → A_2
  75. Segment(R_1, W_1) → l_2
  76. Segment(Z_1, T_1) → m_2
  77. Segment(T_1, U_1) → n_2
  78. Segment(U_1, A_2) → p_2
  79. Polygon(B_1, W, T, E_1, H_1, I_1) → poly2, q_2, w_2, t_2, r_2, s_2, a_2
  80. Midpoint(E_1, G_1) → B_2
  81. Segment(Q_1, B_2) → j_2
  82. Midpoint(O_1, K_1) → C_2
  83. Midpoint(C_2, O_1) → D_2
  84. Midpoint(M_1, L_1) → E_2
  85. Circle(E_2, V_1) → d_2
  86. Circle(K_1, Distance[E_2, V_1]) → c_2
  87. Circle(E_2, M_1) → e_2
  88. Circle(K_1, Distance[E_2, M_1]) → c_3
  89. Point(a_2) → F_2
  90. Point(a_2) → G_2
  91. Polygon(F_2, G_2, 4) → poly3, b_2, f_3, g_3, h_3, H_2, I_2
  92. Semicircle(I_2, H_2) → d_3
  93. Semicircle(I_2, H_2) → e_3
  94. Point(w_2) → N_2
  95. Intersect(q, m_1) → O_2
  96. Segment(N_2, J_2) → i_3
  97. Segment(J_2, K_2) → j_3
  98. Segment(K_2, L_2) → k_3
  99. Segment(L_2, M_2) → l_3
  100. Segment(M_2, O_2) → m_3
  101. Polygon(Q_2, P_2, S_2, R_2) → q1, n_3, p_3, q_3, r_3

AI Drawing Prompt Reference

Draw a coordinate-plane linkage-style curve composition: start from a reference point A on the plane, and construct several concentric circles centered at A with radii R, R+r1, and R+r1+r2. Define a few rays/lines through A and use their intersections with these circles to obtain key points and rotation centers. Rotate selected points around the appropriate centers with fixed angular parameters (progressing through multiple stages such as B', then B'', then B'''), and connect some of these points with line segments. Finally, create multiple parametric curves using parameter i from 0 to t, each defined relative to A and built from combinations of cosine and sine terms with the given radii and angular frequencies (w, w1, w2), resulting in a set of complex linkage-like trajectories.
The prompt above was compiled from the original drawing commands and is for reference only.