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星形线（Astroid）的几何构造与周长探究

星形线（Astroid）的几何构造与周长探究
几何画板及绘图内容加载中，请稍等……
2018-06-14 17:54
微积分积分参数方程几何作图轨迹
本范例演示了星形线（Astroid）这一经典内摆线曲线的几何构造过程。星形线是由一个半径为四分之一大圆半径的小圆在大圆内部滚动时，小圆圆周上一点形成的轨迹，其形状类似四角星。核心数学概念包括参数方程、内摆线、切线、中点、垂线以及曲线的周长（弧长）计算。作图要点：利用GeoGebra的线段（Segment）、中点（Midpoint）、直线（Line）、圆（Circle）、交点（Intersect）和点（Point）命令，构造出基础图形，并通过多个辅助点（如A、B、C、D等）和辅助圆（c_1、e_1）逐步生成星形线的关键特征。该范例适合高中或大学低年级学生，用于理解参数曲线的几何生成原理，以及通过几何构造探索曲线周长与参数的关系，也可作为微积分中弧长积分概念的直观演示。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Midpoint(f) → C
  3. Line(C, D) → g
  4. Circle(C, 1) → c
  5. Circle(C, 2) → d
  6. Circle(C, 3) → e
  7. Circle(C, 4) → h
  8. Circle(C, 5) → k
  9. Intersect(k, g, 2) → E
  10. Intersect(c, f, 1) → F
  11. Intersect(h, g, 2) → G
  12. Intersect(d, f, 1) → H
  13. Intersect(e, g, 2) → I
  14. Intersect(e, f, 1) → J
  15. Intersect(d, g, 2) → K
  16. Intersect(h, f, 1) → L
  17. Intersect(c, g, 2) → M
  18. Intersect(k, f, 1) → N
  19. Intersect(k, f, 2) → O
  20. Intersect(h, f, 2) → P
  21. Intersect(e, f, 2) → Q
  22. Intersect(d, f, 2) → R
  23. Intersect(c, f, 2) → S
  24. Intersect(c, g, 1) → T
  25. Intersect(d, g, 1) → U
  26. Intersect(e, g, 1) → V
  27. Intersect(h, g, 1) → W
  28. Intersect(k, g, 1) → Z
  29. Circle(C, 0.5) → c_1
  30. Circle(C, 2.5) → e_1
  31. Circle(C, 3.5) → p_1
  32. Circle(C, 4.5) → q_1
  33. Circle(C, 5.5) → r_1
  34. Circle(C, 6) → s_1
  35. Circle(C, 6.5) → t_1
  36. Intersect(t_1, g, 2) → A_1
  37. Intersect(c_1, f, 1) → B_1
  38. Intersect(t_1, f, 1) → C_1
  39. Intersect(s_1, f, 1) → D_1
  40. Intersect(r_1, f, 1) → E_1
  41. Intersect(q_1, f, 1) → F_1
  42. Intersect(p_1, f, 1) → G_1
  43. Intersect(e_1, f, 1) → H_1
  44. Intersect(c_1, f, 2) → J_1
  45. Circle(C, 1.5) → p
  46. Intersect(p, f, 1) → I_1
  47. Intersect(p, f, 2) → K_1
  48. Intersect(e_1, f, 2) → L_1
  49. Intersect(p_1, f, 2) → M_1
  50. Intersect(q_1, f, 2) → N_1
  51. Intersect(r_1, f, 2) → O_1
  52. Intersect(s_1, f, 2) → P_1
  53. Intersect(s_1, g, 2) → R_1
  54. Intersect(r_1, g, 2) → S_1
  55. Intersect(q_1, g, 2) → T_1
  56. Intersect(p_1, g, 2) → U_1
  57. Intersect(e_1, g, 2) → V_1
  58. Intersect(p, g, 2) → W_1
  59. Intersect(c_1, g, 2) → Z_1
  60. Intersect(c_1, g, 1) → A_2
  61. Intersect(p, g, 1) → B_2
  62. Intersect(e_1, g, 1) → C_2
  63. Intersect(p_1, g, 1) → D_2
  64. Intersect(q_1, g, 1) → E_2
  65. Intersect(r_1, g, 1) → F_2
  66. Intersect(s_1, g, 1) → G_2
  67. Intersect(t_1, g, 1) → H_2
  68. Segment(A_1, B_1) → i
  69. Segment(R_1, F) → j
  70. Segment(S_1, I_1) → l
  71. Segment(E, H) → m
  72. Segment(T_1, H_1) → n
  73. Segment(G, J) → q
  74. Segment(U_1, G_1) → r
  75. Segment(I, L) → s
  76. Segment(V_1, F_1) → t
  77. Segment(K, N) → a
  78. Segment(W_1, E_1) → b
  79. Segment(M, D_1) → f_1
  80. Segment(Z_1, C_1) → g_1
  81. Segment(A_1, J_1) → h_1
  82. Segment(R_1, S) → i_1
  83. Segment(S_1, K_1) → j_1
  84. Segment(E, R) → k_1
  85. Segment(T_1, L_1) → l_1
  86. Segment(G, Q) → m_1
  87. Intersect(t_1, f, 2) → Q_1
  88. Segment(U_1, M_1) → n_1
  89. Segment(I, P) → a_1
  90. Segment(V_1, N_1) → b_1
  91. Segment(K, O) → d_1
  92. Segment(W_1, O_1) → f_2
  93. Segment(M, P_1) → g_2
  94. Segment(Z_1, Q_1) → h_2
  95. Segment(A_2, C_1) → i_2
  96. Segment(T, D_1) → j_2
  97. Segment(B_2, E_1) → k_2
  98. Segment(U, N) → l_2
  99. Segment(C_2, F_1) → m_2
  100. Segment(V, L) → n_2
  101. Segment(D_2, G_1) → p_2
  102. Segment(W, J) → q_2
  103. Segment(E_2, H_1) → r_2
  104. Segment(Z, H) → s_2
  105. Segment(F_2, I_1) → t_2
  106. Segment(G_2, F) → a_2
  107. Segment(H_2, B_1) → b_2
  108. Segment(A_2, Q_1) → c_2
  109. Segment(T, P_1) → d_2
  110. Segment(B_2, O_1) → e_2
  111. Segment(U, O) → f_3
  112. Segment(C_2, N_1) → g_3
  113. Segment(V, P) → h_3
  114. Segment(D_2, M_1) → i_3
  115. Segment(W, Q) → j_3
  116. Segment(E_2, L_1) → k_3
  117. Segment(Z, R) → l_3
  118. Segment(F_2, K_1) → m_3
  119. Segment(G_2, S) → n_3
  120. Segment(H_2, J_1) → p_3
  121. Point(g) → I_2
  122. Segment(D, I_2) → q_3