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Total Internal Reflection in a Prism

Total Internal Reflection in a Prism
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2012-08-15 19:46
AngleTrigonometrySolid GeometryGeometric Construction
This example visualizes total internal reflection within a prism, illustrating how light rays reflect entirely inside the prism when the angle of incidence exceeds the critical angle. The construction involves geometric concepts such as the law of refraction, critical angle, sine theorem, and angle relationships, using GeoGebra commands like CircumcircleArc, Point, Segment, Angle, Rotate, Intersect, OrthogonalLine, and conditional statements to define the prism's geometry and trace the light path. It is designed for high‑school and undergraduate students, as well as mathematics and physics educators, to explore the mathematical modeling of optical phenomena and develop geometric reasoning skills.

GeoGebra Commands & Parameters

  1. CircumcircleArc(A, E, B) → d
  2. Point(d) → K
  3. Point(d) → J
  4. CircleSector(D, K, J) → k
  5. Point(k) → F
  6. Segment(B, A) → a
  7. Segment(C, B) → b
  8. Segment(C, A) → c
  9. Segment(E, D) → e
  10. Angle(E, D, F) → α
  11. Segment(D, G) → g
  12. Rotate(G, asin(sin(α) / 1.6), D) → G'
  13. Angle(G, D, G') → β
  14. Line(D, G') → h
  15. Intersect(h, c) → H
  16. OrthogonalLine(H, c) → i_1
  17. Angle(G', H, I) → γ
  18. If(γ > asin(1 / 1.6), Rotate[I, -γ, H]) → I'
  19. Angle(I', H, I) → δ
  20. Segment(F, D) → f
  21. Line(H, I') → j
  22. Intersect(j, b) → L
  23. OrthogonalLine(L, b) → l
  24. Angle(l, j) → ε
  25. Rotate(M, -asin(1.6 * sin(ε)), L) → M'
  26. Angle(M', L, M) → ζ
  27. Rotate(E, -α, D) → E'
  28. Angle(E', D, E) → η
  29. Segment(D, E') → n
  30. Segment(D, H) → p
  31. Segment(H, L) → q
  32. Line(L, M') → m
  33. Segment(I, H) → a_1
  34. Segment(M, L) → b_1
  35. Circle(L, 2.8284271247461903) → r_1
  36. Intersect(r_1, m, 2) → N
  37. Segment(L, N) → s
  38. Segment(L, Q) → c_1
  39. Segment(H, R) → d_1
  40. Rotate(R, asin(1.6 * sin(γ)), H) → R'
  41. Angle(R, H, R') → θ
  42. Circle(H, 2.8284271247461903) → e_1
  43. Line(H, R') → f_1
  44. Intersect(e_1, f_1, 2) → S
  45. Segment(H, S) → g_1
  46. Angle(T, D, F) → i
  47. Angle(U, L, N) → r
  48. Segment(O, D) → t
  49. Segment(T, D) → h_1
  50. Segment(U, L) → j_1
  51. Angle(B, A, C) → κ
  52. Angle(B, C, A) → λ
  53. Angle(A, B, C) → ι
  54. Polygon(A, B, C) → poly1, k_1, l_1, m_1