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Total Internal Reflection in Prism (Corner Reflector)

Total Internal Reflection in Prism (Corner Reflector)
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2012-08-16 01:11
PolygonAngleTrigonometryTransformationGeometric Construction
① This example demonstrates how light undergoes total internal reflection inside a prism, making the prism act as a corner reflector where the outgoing ray is parallel but opposite to the incoming ray. ② Core mathematical concepts include the law of reflection, conditions for total internal reflection, symmetry in geometric optics, and the use of rotation and angle measurement to verify reflection relationships. ③ Construction details: using GeoGebra commands such as Line, Polygon, Point, OrthogonalLine, Circle, Intersect, Semicircle, Segment, Angle, Rotate, and If, the prism polygon, incident ray, normal lines, reflected rays, and angle markers are constructed, with dynamic light paths achieved through rotation and conditional logic. ④ Suitable for high school physics or mathematics lessons combining geometric optics and transformation geometry, as well as introductory university optics, helping students intuitively understand total internal reflection and corner reflector principles.

GeoGebra Commands & Parameters

  1. Line(A, C) → m_1
  2. Polygon(A, B, C) → poly1, c, a, b
  3. Point(b) → D
  4. OrthogonalLine(D, b) → f
  5. Circle(D, 2) → e
  6. Intersect(e, m_1, 1) → Z
  7. Intersect(e, m_1, 2) → W
  8. Semicircle(W, Z) → p_1
  9. Point(p_1) → G
  10. Circle(D, 0.5) → d
  11. Intersect(d, f, 2) → E
  12. Intersect(d, f, 1) → F
  13. Segment(E, F) → g
  14. Angle(E, D, G) → α
  15. Rotate(F, asin(sin(α) / n), D) → F'
  16. Angle(F, D, F') → β
  17. Line(D, F') → i_3
  18. Intersect(i_3, c) → H
  19. Intersect(i_3, a) → I
  20. Segment(J, K) → j
  21. Angle(D, H, K) → γ
  22. If(γ > asin(1 / n), Rotate[K, γ, H]) → K'
  23. Angle(K, H, K') → δ
  24. Rotate(J, -asin(n * sin(γ)), H) → J'
  25. Angle(J', H, J) → ε
  26. Line(H, K') → k
  27. Intersect(k, a) → L
  28. Segment(D, H) → l
  29. Segment(H, L) → m
  30. Segment(M, N) → p
  31. Angle(H, L, M) → ζ
  32. If(ζ > asin(1 / n), Rotate[M, ζ, L]) → M'
  33. Angle(M, L, M') → η
  34. Rotate(N, -asin(n * sin(ζ)), L) → N'
  35. Angle(N', L, N) → θ
  36. Line(L, M') → q
  37. Intersect(q, b) → O
  38. Segment(L, O) → r_3
  39. Circle(O, 0.5) → s
  40. OrthogonalLine(O, b) → t
  41. Intersect(s, t, 2) → P
  42. Intersect(s, t, 1) → Q
  43. Segment(P, Q) → a_1
  44. Angle(L, O, Q) → ι
  45. Rotate(P, -asin(n * sin(ι)), O) → P'
  46. Angle(P', O, P) → κ
  47. Circle(O, 2) → c_1
  48. Line(O, P') → b_1
  49. Intersect(c_1, b_1, 2) → R
  50. Segment(O, R) → d_1
  51. Circle(H, 1.5) → e_1
  52. Line(H, J') → f_1
  53. Intersect(e_1, f_1, 2) → S
  54. Segment(H, S) → g_1
  55. Circle(L, 1.5) → h_1
  56. Line(L, N') → i_1
  57. Intersect(h_1, i_1, 2) → T
  58. Segment(L, T) → j_1
  59. Segment(D, I) → k_1
  60. Segment(U, V) → l_1
  61. Segment(G, D) → h
  62. Angle(U, I, D) → λ
  63. If(λ > asin(1 / n), Rotate[U, -λ, I]) → U'
  64. Angle(U', I, U) → μ
  65. Rotate(V, asin(n * sin(λ)), I) → V'
  66. Angle(V, I, V') → ν
  67. Circle(I, 1.5) → q_1
  68. Line(I, V') → n_1
  69. Intersect(q_1, n_1, 2) → A_1
  70. Segment(I, A_1) → r_1
  71. Line(I, U') → s_1
  72. Intersect(s_1, c) → B_1
  73. Segment(I, B_1) → t_1
  74. Segment(C_1, D_1) → a_2
  75. Circle(B_1, 1.5) → c_2
  76. Angle(D_1, B_1, I) → ξ
  77. If(ξ > asin(1 / n), Rotate[D_1, -ξ, B_1]) → E_1
  78. Angle(E_1, B_1, D_1) → ο
  79. Line(B_1, E_1) → b_2
  80. Intersect(m_1, b_2) → F_1
  81. Segment(B_1, F_1) → d_2
  82. Rotate(C_1, asin(n * sin(ξ)), B_1) → G_1
  83. Angle(C_1, B_1, G_1) → ρ
  84. Line(B_1, G_1) → e_2
  85. Intersect(c_2, e_2, 2) → H_1
  86. Segment(B_1, H_1) → f_2
  87. Circle(F_1, 0.5) → g_2
  88. Circle(F_1, 2) → h_2
  89. OrthogonalLine(F_1, b) → i_2
  90. Intersect(g_2, i_2, 1) → I_1
  91. Intersect(g_2, i_2, 2) → J_1
  92. Segment(I_1, J_1) → j_2
  93. Angle(I_1, F_1, B_1) → σ
  94. Rotate(J_1, asin(n * sin(σ)), F_1) → K_1
  95. Angle(J_1, F_1, K_1) → τ
  96. Line(F_1, K_1) → k_2
  97. Intersect(h_2, k_2, 2) → L_1
  98. Segment(F_1, L_1) → l_2
  99. Angle(A, B, C) → υ
  100. Angle(B, C, A) → χ
  101. Angle(C, A, B) → ϕ
  102. Line(H, K') → m_2
  103. Line(I, U') → p_2
  104. Line(I, U') → q_2
  105. Intersect(q_2, b) → P_1
  106. Line(P_1, I) → r_2
  107. Intersect(q_1, r_2) → Q_1, R_1
  108. Point(r_2) → S_1
  109. Segment(S_1, Q_1) → s_2
  110. Intersect(b, m_2) → M_1
  111. Line(H, M_1) → n_2
  112. Intersect(e_1, n_2) → N_1, O_1
  113. Segment(H, O_1) → t_2
  114. Angle(E, D, G) → i
  115. Angle(P, O, R) → r

AI Drawing Prompt Reference

Draw a prism with an incident ray, two normal lines, two reflected rays, and angle markings. First, draw a triangular prism, then add the incident ray and normal lines. Use rotation and conditional logic to calculate the reflected rays and angles, ensuring the reflected rays are parallel but in the opposite direction to the incident ray.
The prompt above was compiled from the original drawing commands and is for reference only.