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3D Rainbow Formation Simulation

3D Rainbow Formation Simulation
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2015-03-05 01:39
VectorSolid GeometryGeometric Construction
This example demonstrates the processes involved in the formation of a rainbow through a 3D simulation. It involves core mathematical concepts such as reflection, area, quadratic functions, differential equations, and 2D vectors. The construction utilizes GeoGebra commands and objects like Circle, Point, Angle, Rotate, Ray, Intersect, Sphere, Line, Plane, Segment, Vector, and Mirror. It is suitable for high school and above students studying optics, geometry, and mathematical modeling.

GeoGebra Commands & Parameters

  1. Circle(Q, R) → c
  2. Point(c) → U
  3. Point(xAxis) → C
  4. Angle(C, Viewer, S) → α
  5. Rotate(U, (-α), Q) → T
  6. Rotate(I_1, (-α), T) → J_1
  7. Ray(Q, T) → r
  8. Ray(T, J_1) → d
  9. Angle(r, d) → δ
  10. Rotate(Q, asin(sin(δ) / 1.33), T) → Q'
  11. Ray(T, Q') → s
  12. Intersect(c, s) → V, W
  13. Angle(Q, T, Q') → ε
  14. Rotate(Q', (2 * ε), W) → Q''
  15. Ray(W, Q'') → a_1
  16. Intersect(c, a_1) → Z, A_1
  17. Ray(Q, A_1) → c_1
  18. Sphere(Viewer, 5.55) → l
  19. Line(S, Viewer) → a
  20. Intersect(a, l) → K, L
  21. Circle(L, B2, a) → C1
  22. Point(xAxis) → F
  23. Point(yAxis) → G
  24. Plane(F, E, G) → b
  25. Point(C1) → A
  26. Line(A, a) → f
  27. Point(f) → B
  28. Ray(A, B) → e
  29. Segment(A, Viewer) → g
  30. Point(e) → D
  31. Point(e) → I
  32. Angle(e, g) → β
  33. Circle(L, B3, a) → C2
  34. Circle(L, B4, a) → C3
  35. Circle(L, B5, a) → C4
  36. Circle(L, B6, a) → C5
  37. Circle(L, B7, a) → C6
  38. Circle(L, B8, a) → C7
  39. Circle(L, B9, a) → C8
  40. Circle(L, B10, a) → C9
  41. Circle(L, B11, a) → C10
  42. Circle(L, B12, a) → C11
  43. Circle(L, B13, a) → C12
  44. Circle(L, B14, a) → C13
  45. Circle(L, B15, a) → C14
  46. Circle(L, B16, a) → C15
  47. Circle(L, B17, a) → C16
  48. Circle(L, B18, a) → C17
  49. Circle(L, B19, a) → C18
  50. Circle(L, B20, a) → C19
  51. Circle(L, B21, a) → C20
  52. Circle(L, B22, a) → C21
  53. Circle(L, B23, a) → C22
  54. Circle(L, B24, a) → C23
  55. Circle(L, B25, a) → C24
  56. Circle(L, B26, a) → C25
  57. Circle(L, B27, a) → C26
  58. Circle(L, B28, a) → C27
  59. Circle(L, B29, a) → C28
  60. Circle(L, B30, a) → C29
  61. Circle(L, B31, a) → C30
  62. Circle(L, B32, a) → C31
  63. Circle(L, B33, a) → C32
  64. Circle(L, B34, a) → C33
  65. Circle(L, B35, a) → C34
  66. Circle(L, B36, a) → C35
  67. Circle(L, B37, a) → C36
  68. Circle(L, B38, a) → C37
  69. Circle(L, B39, a) → C38
  70. Circle(L, B40, a) → C39
  71. Circle(L, B41, a) → C40
  72. Circle(L, B42, a) → C41
  73. Circle(L, B43, a) → C42
  74. Circle(L, B44, a) → C43
  75. Circle(L, B45, a) → C44
  76. Circle(L, B46, a) → C45
  77. Circle(L, B47, a) → C46
  78. Circle(L, B48, a) → C47
  79. Circle(L, B49, a) → C48
  80. Circle(L, B50, a) → C49
  81. Circle(L, B51, a) → C50
  82. Circle(L, B52, a) → C51
  83. Circle(L, B53, a) → C52
  84. Circle(L, B54, a) → C53
  85. Circle(L, B55, a) → C54
  86. Circle(L, B56, a) → C55
  87. Circle(L, B57, a) → C56
  88. Circle(L, B58, a) → C57
  89. Circle(L, B59, a) → C58
  90. Circle(L, B60, a) → C59
  91. Circle(L, B61, a) → C60
  92. Circle(L, B62, a) → C61
  93. Circle(L, B63, a) → C62
  94. Circle(L, B64, a) → C63
  95. Circle(L, B65, a) → C64
  96. Circle(L, B66, a) → C65
  97. Circle(L, B67, a) → C66
  98. Circle(L, B68, a) → C67
  99. Circle(L, B69, a) → C68
  100. Circle(L, B70, a) → C69
  101. Circle(L, B71, a) → C70
  102. Circle(L, B72, a) → C71
  103. Circle(L, B73, a) → C72
  104. Circle(L, B74, a) → C73
  105. Circle(L, B75, a) → C74
  106. Circle(L, B76, a) → C75
  107. Circle(L, B77, a) → C76
  108. Circle(L, B78, a) → C77
  109. Angle(Viewer, A, M) → γ
  110. Point(C77) → O
  111. Segment(O, Viewer) → m
  112. Line(O, e) → p
  113. Point(p) → P
  114. Ray(O, P) → q
  115. Vector(O, N) → n
  116. Vector(A, M) → k
  117. Ray(Q, W) → t
  118. Point(c_1) → B_1
  119. Rotate(B_1, (-δ), A_1) → C_1
  120. Angle(C_1, A_1, B_1) → ζ
  121. Segment(W, A_1) → d_1
  122. Segment(T, Z) → e_1
  123. Rotate(Q, asin(sin(δ) / 1.345), T) → Q'_1
  124. Ray(T, Q'_1) → f_1
  125. Intersect(c, f_1) → D_1, E_1
  126. Segment(T, E_1) → g_1
  127. Ray(Q, E_1) → h_1
  128. Mirror(T, h_1) → T'
  129. Mirror(E_1, h_1) → F_1
  130. Segment(T', F_1) → i_1
  131. Angle(Q, T, E_1) → η
  132. Ray(Q, T') → j_1
  133. Point(j_1) → G_1
  134. Rotate(G_1, (-δ), T') → H_1
  135. Vector(T', H_1) → k_1
  136. Vector(A_1, C_1) → b_1
  137. Point(q) → K_1
  138. Vector(L_1, M_1) → u
  139. Vector((0.3, 0, -0.4), Viewer) → v

AI Drawing Prompt Reference

Create a 3D simulation showing the formation of a rainbow through light reflection in water droplets. Start with a circle and a point on it, then construct reflected rays and vectors using angles and rotations. Determine the rainbow's position by intersecting a sphere with the circle, and simulate the color bands with multiple concentric circles. The final image should depict the dynamic process of light reflection and refraction within water droplets.
The prompt above was compiled from the original drawing commands and is for reference only.