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Refraction on Parallel Surfaces

Refraction on Parallel Surfaces
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2016-09-14 23:04
VectorCoordinate GeometryOptimizationGeometric Construction
This example demonstrates the phenomenon of light refraction between parallel surfaces. It involves core mathematical concepts such as the law of refraction, geometric mean, sequences and series, and conic sections. Construction highlights include using commands like Point, Line, Intersect, OrthogonalLine, Segment, Angle, If, Rotate, Polygon, Vector, Circle, AngularBisector, Mirror, Textfield, Ray, Distance, Midpoint from GeoGebra. Suitable for high school and above students studying optics and geometric optimization problems.

GeoGebra Commands & Parameters

  1. Point(yAxis) → F
  2. Point(yAxis) → E
  3. Line(E, F) → g
  4. Intersect(xAxis, yAxis) → D
  5. OrthogonalLine(D, g) → h
  6. Point(h) → K
  7. Segment(A, B) → f
  8. Intersect(h, f) → G
  9. Point(f) → C
  10. Angle(G, D, C) → β
  11. Angle(C, D, G) → α
  12. If(α > 90°, β, α) → γ
  13. Rotate(K, a, D) → K'
  14. Line(D, K') → g_1
  15. Polygon(R, S, T, U) → poly2, r, s, t_1, u_1
  16. Intersect(g_1, t_1) → V
  17. Vector(C, D) → u
  18. Circle(D, 1.2) → c
  19. AngularBisector(C, D, G) → i
  20. Intersect(c, i, 2) → H
  21. Mirror(C, h) → C'
  22. Vector(D, C') → v
  23. AngularBisector(C', D, G) → j
  24. Intersect(c, j, 2) → I
  25. Angle(C', D, G) → δ
  26. Angle(G, D, C') → ε
  27. If(δ > 90°, ε, δ) → ζ
  28. Circle(D, 1.8) → d
  29. Intersect(d, j, 2) → J
  30. Angle(K, D, K') → η
  31. Rotate(K, (-a), D) → K'_1
  32. Angle(K'_1, D, K) → θ
  33. AngularBisector(K'_1, D, K) → k
  34. AngularBisector(K, D, K') → l
  35. Intersect(c, k, 2) → L
  36. Intersect(c, l, 2) → M
  37. Vector(D, K'_1) → w
  38. Vector(D, K') → b
  39. Polygon(N, O, P, Q) → poly1, n, o, p, q
  40. Textfield(n_1) → InputBox1
  41. Textfield(n_2) → InputBox2
  42. Line(C, D) → m
  43. OrthogonalLine(C, m) → f_1
  44. Circle(C, 0.6) → e
  45. Intersect(e, f_1, 2) → Z
  46. Intersect(e, f_1, 1) → A_1
  47. Line(D, K'_1) → h_1
  48. Intersect(h_1, t_1) → W
  49. Vector(D, V) → v_1
  50. Vector(D, W) → w_1
  51. Rotate(B_1, (-γ), 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