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Hyperbolic Mirror Reflection

Hyperbolic Mirror Reflection
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2019-05-08 04:28
FunctionCoordinate GeometryTransformation
This example demonstrates the reflection of light on a hyperbolic mirror. Key mathematical concepts include hyperbola, reflection, arithmetic mean, and mirror. Using GeoGebra commands such as Point, Mirror, Intersect, Hyperbola, etc., the reflection effect on a hyperbolic mirror is constructed. Suitable for high school mathematics education, particularly in analytic geometry and optics courses.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Mirror(A, xAxis) → A'
  3. Intersect(xAxis, yAxis) → B
  4. Point(yAxis) → C
  5. Hyperbola(A, A', C) → c
  6. Rotate(yAxis, n, A') → yAxis'
  7. Intersect(c, yAxis', 2) → D
  8. Tangent(D, c) → f,
  9. Mirror(yAxis', f) → yAxis''
  10. Intersect(c, yAxis'', 2) → E
  11. Tangent(E, c) → g,
  12. Mirror(yAxis'', g) → yAxis'''
  13. Intersect(c, yAxis''', 2) → F
  14. Tangent(F, c) → h,
  15. Mirror(yAxis''', h) → i
  16. Intersect(c, i, 2) → G
  17. Tangent(G, c) → j,
  18. Mirror(i, j) → i'
  19. Intersect(c, i', 2) → H
  20. Tangent(H, c) → k,
  21. Mirror(i', k) → i''
  22. Intersect(c, i'', 2) → I
  23. Tangent(I, c) → l,
  24. Mirror(i'', l) → i'''
  25. Intersect(c, i''', 2) → J
  26. Tangent(J, c) → m,
  27. Mirror(i''', m) → p
  28. Intersect(c, p, 2) → K
  29. Tangent(K, c) → q,
  30. Mirror(p, q) → p'
  31. Intersect(c, p', 2) → L
  32. Segment(L, K) → r
  33. Segment(K, J) → s
  34. Segment(J, I) → t
  35. Segment(I, H) → a
  36. Segment(H, G) → b
  37. Segment(G, F) → d
  38. Segment(F, E) → e
  39. Segment(E, D) → f_1
  40. Intersect(c, yAxis', 1) → M
  41. Intersect(c, yAxis', 2) → Q
  42. Point(xAxis) → N
  43. Line(N, yAxis) → g_1
  44. Parabola(A', g_1) → c_1
  45. LineBisector(C, A) → k_1
  46. Intersect(c, k_1, 2) → R
  47. Line(A', R) → l_1
  48. Intersect(c_1, l_1, 2) → S
  49. Line(S, yAxis) → m_1
  50. Intersect(c, k_1, 1) → T
  51. Segment(R, T) → n_1
  52. Intersect(c_1, m_1, 2) → U
  53. Mirror(R, xAxis) → R'
  54. Mirror(T, xAxis) → T'
  55. Segment(R', T') → q_1
  56. Segment(U, S) → p_1
  57. Intersect(c_1, yAxis', 2) → O
  58. Tangent(O, c_1) → h_1,
  59. Mirror(yAxis', h_1) → yAxis''_1
  60. Intersect(yAxis''_1, yAxis) → P
  61. Segment(O, D) → i_1
  62. Ray(O, P) → j_1
  63. Intersect(c, i_1, 1) → V
  64. Segment(V, A') → r_1
  65. Intersect(c, e, 1) → W
  66. Segment(W, A) → s_1
  67. Intersect(c, d, 1) → Z
  68. Segment(Z, A') → t_1
  69. Intersect(c, a, 1) → A_1
  70. Segment(A_1, A') → a_1
  71. Intersect(c, b, 1) → B_1
  72. Segment(B_1, A) → b_1
  73. Intersect(c, t, 1) → C_1
  74. Segment(C_1, A) → d_1
  75. Intersect(c, r, 1) → D_1
  76. Segment(D_1, A) → e_1
  77. Intersect(c, s, 1) → E_1
  78. Segment(E_1, A') → f_2
  79. Intersect(c, r, 2) → F_1
  80. Segment(F_1, A') → g_2
  81. Line(B, xAxis) → h_2

AI Drawing Prompt Reference

Draw a hyperbolic mirror. Start by taking a point A on the y-axis, then take point B on the x-axis as the origin. Next, take another point C on the y-axis. Construct a hyperbola c using points A, A' (the mirror image of A across the x-axis), and C. Rotate A' around the y-axis, and find the intersection of the hyperbola c with the rotated y-axis at point D. Draw the tangent line f at point D on the hyperbola c, then mirror f across the rotated y-axis, and continue this process to create a series of tangents and mirrors, finally connecting the series of intersection points to form a closed shape.
The prompt above was compiled from the original drawing commands and is for reference only.