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Dispersion of White Light by a Prism (Geometric Simulation)

Dispersion of White Light by a Prism (Geometric Simulation)
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2018-07-24 18:28
PolygonAngleTrigonometryTransformationGeometric Construction
① This GeoGebra example simulates the dispersion of white light through a triangular prism, visually demonstrating how different colors separate due to varying refractive indices along geometric paths. ② Core mathematical concepts include basic geometric elements such as points, polygons, angles, straight lines, slopes, perpendiculars, rotations, intersections, and segments, along with trigonometric functions used in angle calculations; it also integrates physical optics principles of refraction and dispersion into a geometric construction. ③ Construction highlights: Based on the summary, the example employs GeoGebra commands like Point, Polygon, OrthogonalLine, Angle, Ray, Slope, Rotate, Intersect, and Segment to build the prism polygon (poly1), incident and refracted rays, normals, and multiple auxiliary points (D, B, A, C, E, F, G, etc.), with angles α, β, γ, δ controlling ray directions and intersections/segments completing the light path. ④ Suitable for middle or high school physics/math classes explaining refraction, dispersion, and geometric construction, as well as introductory university optics or geometric modeling, helping students understand how geometric tools can simulate physical phenomena.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Polygon(A, B, 3) → poly1, c, a, b, C
  3. Point(b) → D_2
  4. OrthogonalLine(D_2, b) → e
  5. Point(e) → F
  6. Point(e) → G
  7. Angle(F, D_2, E) → α
  8. Ray(D_2, E) → d
  9. Slope(d) → m
  10. Rotate(G, asin((n_1 / n_2 * sin(α))), D_2) → G'
  11. Ray(D_2, G') → f
  12. Intersect(a, f) → H
  13. Segment(H, D_2) → g
  14. OrthogonalLine(H, a) → h
  15. Point(h) → I
  16. Angle(D_2, H, I) → γ
  17. Point(h) → J
  18. Rotate(J, (-asin((n_2 / n_1 * sin(γ)))), H) → J'
  19. Angle(J', H, J) → δ
  20. Ray(H, J') → i
  21. Segment(F, G) → j
  22. Segment(J, I) → k
  23. Angle(G, D_2, G') → β
  24. Rotate(G, asin((n_1 / (((1 + (0.002 * D)) * n_2)) * sin(α))), D_2) → P
  25. Rotate(G, asin((n_1 / (((1 + (0.004 * D)) * n_2)) * sin(α))), D_2) → Q
  26. Rotate(G, asin((n_1 / (((1 + (0.006 * D)) * n_2)) * sin(α))), D_2) → R
  27. Rotate(G, asin((n_1 / (((1 + (0.008 * D)) * n_2)) * sin(α))), D_2) → S
  28. Ray(D_2, P) → s
  29. Ray(D_2, Q) → t
  30. Ray(D_2, R) → a_1
  31. Ray(D_2, S) → b_1
  32. Rotate(G, asin((n_1 / (((1 + (0.01 * D)) * n_2)) * sin(α))), D_2) → T
  33. Ray(D_2, T) → c_1
  34. Intersect(a, s) → U
  35. Intersect(a, a_1) → V
  36. Intersect(a, t) → W
  37. Intersect(a, b_1) → Z
  38. Intersect(a, c_1) → A_1
  39. OrthogonalLine(U, a) → d_1
  40. OrthogonalLine(W, a) → e_1
  41. OrthogonalLine(V, a) → f_1
  42. OrthogonalLine(Z, a) → i_1
  43. OrthogonalLine(A_1, a) → l_1
  44. Point(d_1) → B_1
  45. Point(e_1) → C_1
  46. Point(f_1) → D_1
  47. Point(i_1) → E_1
  48. Point(l_1) → F_1
  49. Rotate(B_1, (-asin((((1 + (0.002 * D)) * n_2 / n_1) * sin(γ)))), U) → G_1
  50. Rotate(C_1, (-asin((((1 + (0.004 * D)) * n_2 / n_1) * sin(γ)))), W) → H_1
  51. Rotate(D_1, (-asin((((1 + (0.006 * D)) * n_2 / n_1) * sin(γ)))), V) → I_1
  52. Rotate(E_1, (-asin((((1 + (0.008 * D)) * n_2 / n_1) * sin(γ)))), Z) → J_1
  53. Rotate(F_1, (-asin((((1 + (0.01 * D)) * n_2 / n_1) * sin(γ)))), A_1) → K_1
  54. Ray(U, G_1) → n
  55. Ray(W, H_1) → p
  56. Ray(V, I_1) → q
  57. Ray(Z, J_1) → r
  58. Ray(A_1, K_1) → l
  59. Segment(D_2, U) → g_1
  60. Segment(D_2, V) → h_1
  61. Segment(D_2, Z) → j_1
  62. Segment(D_2, A_1) → k_1
  63. Segment(D_2, W) → m_1
  64. Intersect(b, a) → K
  65. Intersect(b, c) → L
  66. Intersect(a, c) → M
  67. Polygon(K, L, M) → poly2, m_2, k_2, l_2