连杆曲线演示

连杆曲线演示
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2020-12-08 20:27
①本范例演示通过连杆机构生成的复杂曲线图形;②涉及的核心数学概念包括参数方程、解析几何、几何变换和三角函数;③作图要点提炼自构造摘要,使用了Ray(射线)、Midpoint(中点)、Line(直线)、Intersect(交点)、Polygon(多边形)、Centroid(重心)、Mirror(镜像)、Circle(圆)、Point(点)、Rotate(旋转)、Segment(线段)、CurveCartesian(笛卡尔曲线)、Semicircle(半圆)等GeoGebra命令和对象;④适合高中及以上学生学习和理解参数曲线的生成过程及几何变换的应用。

GeoGebra 命令及参数

  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 绘图提示语参考

在坐标平面内,先给定一参考点 A 以及以 A 为圆心的若干同心圆,设定半径 R、R+r1、R+r1+r2 等。取与坐标轴成一定关系的若干光线,用两圆与直线的交点生成一组旋转中心与点对。以固定角速度把点围绕旋转中心依次旋转得到多级点(如先从 B' 再到 B'' 再到 B'''),并用若干线段连接这些点。再沿参数 i 从 0 到 t 生成多条曲线:每条曲线分别由与 cos、sin 相关的几何表达式组合而成,曲线以 A 为基准并叠加不同半径项,最终形成类似连杆机构轨迹的复合曲线图。
以上提示语为根据原始绘图命令整理而来,仅供参考。