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平行界面三层介质中的折射现象

平行界面三层介质中的折射现象
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2016-09-15 10:55
角函数变换几何作图滑块动画
该范例演示光在三个平行介质界面间传播时的折射路径，通过几何作图直观展示斯涅尔定律在多层介质中的连续应用。核心数学概念包括折射定律、角度关系、平行线性质以及几何光学中的光路追踪。作图要点：使用GeoGebra的Segment、Point、Intersect、Line、OrthogonalLine、Angle、If、Rotate、Polygon、Vector、Circle、AngularBisector、Mirror、Textfield、Ray、Distance、Midpoint等命令，构建了多层介质界面、法线、入射光线、折射光线以及角度标注，并通过滑块或文本输入动态调整介质折射率与入射角。适合高中物理或大学光学课程中理解多层介质折射规律，也可用于数学几何与物理交叉教学。通过交互式调整参数，学生可观察折射角随介质变化而改变，加深对光路可逆性和界面平行条件下角度传递的理解。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Point(f) → C
  3. Intersect(xAxis, yAxis) → D
  4. Point(yAxis) → E
  5. Point(yAxis) → F
  6. Line(E, F) → g
  7. OrthogonalLine(D, g) → h
  8. Intersect(h, f) → G
  9. Angle(C, D, G) → α
  10. Angle(G, D, C) → β
  11. If(α > 90°, β, α) → γ
  12. Point(h) → K
  13. Rotate(K, (-a), D) → K'_1
  14. Line(D, K'_1) → h_1
  15. Angle(K'_1, D, K) → θ
  16. Polygon(R, S, T, U) → poly2, r, s, t_1, u_1
  17. Intersect(h_1, t_1) → W
  18. Rotate(K, a, D) → K'
  19. Line(D, K') → g_1
  20. Intersect(g_1, t_1) → V
  21. Vector(C, D) → u
  22. Circle(D, 1.2) → c
  23. AngularBisector(C, D, G) → i
  24. Intersect(c, i, 2) → H
  25. Mirror(C, h) → C'
  26. Vector(D, C') → v
  27. AngularBisector(C', D, G) → j
  28. Intersect(c, j, 2) → I
  29. Angle(C', D, G) → δ
  30. Angle(G, D, C') → ε
  31. If(δ > 90°, ε, δ) → ζ
  32. Circle(D, 1.8) → d
  33. Intersect(d, j, 2) → J
  34. Angle(K, D, K') → η
  35. AngularBisector(K'_1, D, K) → k
  36. AngularBisector(K, D, K') → l
  37. Intersect(c, k, 2) → L
  38. Intersect(c, l, 2) → M
  39. Vector(D, K'_1) → w
  40. Vector(D, K') → b
  41. Polygon(N, O, P, Q) → poly1, n, o, p, q
  42. Textfield(n_1) → InputBox1
  43. Textfield(n_2) → InputBox2
  44. Line(C, D) → m
  45. OrthogonalLine(C, m) → f_1
  46. Circle(C, 0.6) → e
  47. Intersect(e, f_1, 2) → Z
  48. Intersect(e, f_1, 1) → A_1
  49. Vector(D, V) → v_1
  50. Vector(D, W) → w_1
  51. Rotate(B_1, (-e_2), W) → C_1
  52. Angle(C_1, W, B_1) → ν
  53. Point(h) → D_1
  54. Point(h) → E_1
  55. Segment(D_1, E_1) → i_1
  56. Vector(W, C_1) → a_1
  57. Rotate(G_1, γ, V) → F_1
  58. Angle(G_1, V, F_1) → ξ
  59. Vector(V, F_1) → b_1
  60. Ray(C, D) → j_1
  61. Segment(H_1, I_1) → k_1
  62. Segment(J_1, K_1) → l_1
  63. AngularBisector(F_1, V, G_1) → m_1
  64. AngularBisector(B_1, W, C_1) → p_1
  65. Circle(W, 1) → c_1
  66. Circle(V, 1) → d_1
  67. Intersect(d_1, m_1, 2) → L_1
  68. Intersect(c_1, p_1, 2) → M_1
  69. Angle(D, W, J_1) → ο
  70. Angle(H_1, V, D) → ρ
  71. AngularBisector(J_1, W, D) → q_1
  72. AngularBisector(D, V, H_1) → r_1
  73. Intersect(d_1, r_1, 2) → N_1
  74. Intersect(c_1, q_1, 2) → O_1
  75. Intersect(t_1, j_1) → P_1
  76. OrthogonalLine(P_1, j_1) → s_1
  77. Line(W, C_1) → e_1
  78. Intersect(s_1, e_1) → Q_1
  79. Distance(P_1, Q_1) → distanceP1Q1
  80. Textfield(t) → InputBox3
  81. Midpoint(P_1, Q_1) → R_1
  82. Vector(P_1, Q_1) → u_2
  83. Vector(Q_1, P_1) → v_2
  84. Line(F_1, V) → f_2
  85. Intersect(f_2, s_1) → S_1
  86. Midpoint(P_1, S_1) → T_1
  87. Vector(S_1, P_1) → w_2
  88. Vector(P_1, S_1) → a_2
  89. Distance(P_1, S_1) → o_1
  90. Distance(P_1, S_1) → b_2
  91. Point(r) → U_1
  92. Point(t_1) → V_1
  93. Vector(U_1, V_1) → c_2
  94. Vector(V_1, U_1) → d_2
  95. Polygon(W_1, Z_1, A_2, B_2) → poly3, w_3, z_1, a_3, b_3
  96. Line(J_1, K_1) → g_2
  97. Mirror(D, g_2) → D'
  98. Vector(W, D') → h_2
  99. Segment(C_2, D_2) → i_2
  100. Rotate(C_2, ζ, D') → F_2
  101. Angle(C_2, D', F_2) → σ
  102. Ray(D', F_2) → k_2
  103. Point(k_2) → G_2
  104. Vector(H_2, I_2) → l_2
  105. AngularBisector(C_2, D', F_2) → m_2
  106. Circle(D', 2) → p_2
  107. Intersect(p_2, m_2, 2) → J_2
  108. Angle(J_1, W, D') → τ
  109. AngularBisector(J_1, W, D') → q_2
  110. Intersect(c_1, q_2, 2) → K_2
  111. Textfield(n_3) → InputBox4
  112. Angle(D_2, D', W) → ϕ
  113. AngularBisector(W, D', D_2) → r_2
  114. Intersect(p_2, r_2, 2) → L_2
  115. Line(B, A) → s_2
  116. Intersect(s_2, k_2) → N_2
  117. Vector(D', N_2) → j_2