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3D Three-Fold Umbrella

3D Three-Fold Umbrella
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2019-12-16 08:22
TrigonometryCoordinate GeometrySolid GeometryGeometric Construction
This example demonstrates a 3D model of a three-fold umbrella. It involves core mathematical concepts such as solid geometry, linear functions, rotation, midpoint, circle, intersection, polygon, and cylinder. The construction utilizes GeoGebra commands and objects like points, lines, rotation, midpoint, circle, intersection, segments, and polygons. Suitable for high school students learning solid geometry and analytical geometry, or for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Line(B, A) → f
  4. Point(f) → C
  5. Rotate(C, α, A) → C'
  6. Rotate(A, (-((2.3 * α))), C') → A'
  7. Rotate(C', (1.9 * α), A') → C''
  8. Midpoint(A, C') → D
  9. Circle(D, 1.1) → c
  10. Intersect(c, f, 1) → E
  11. Segment(D, E) → g
  12. Circle(B, p) → d
  13. Intersect(d, f, 2) → F
  14. Circle(F, 0.5) → e
  15. Intersect(e, f) → G, H
  16. Segment(A, C') → h
  17. Segment(C', A') → i
  18. Segment(A', C'') → j
  19. Rotate(D, 60°, f) → D'
  20. Rotate(E, 60°, f) → E'
  21. Segment(D', E') → g'
  22. Rotate(D, 120°, f) → D'_1
  23. Rotate(E, 120°, f) → E'_1
  24. Segment(D'_1, E'_1) → g'_1
  25. Rotate(D, 180°, f) → D'_2
  26. Rotate(E, 180°, f) → E'_2
  27. Segment(D'_2, E'_2) → g'_2
  28. Rotate(D, 240°, f) → D'_3
  29. Rotate(E, 240°, f) → E'_3
  30. Segment(D'_3, E'_3) → g'_3
  31. Rotate(D, 300°, f) → D'_4
  32. Rotate(E, 300°, f) → E'_4
  33. Segment(D'_4, E'_4) → g'_4
  34. Rotate(A, 60°, f) → A'_1
  35. Rotate(C', 60°, f) → C''_1
  36. Segment(A'_1, C''_1) → h'
  37. Rotate(A, 120°, f) → A'_2
  38. Rotate(C', 120°, f) → C''_2
  39. Segment(A'_2, C''_2) → h'_1
  40. Rotate(A, 180°, f) → A'_3
  41. Rotate(C', 180°, f) → C''_3
  42. Segment(A'_3, C''_3) → h'_2
  43. Rotate(A, 240°, f) → A'_4
  44. Rotate(C', 240°, f) → C''_4
  45. Segment(A'_4, C''_4) → h'_3
  46. Rotate(A, 300°, f) → A'_5
  47. Rotate(C', 300°, f) → C''_5
  48. Segment(A'_5, C''_5) → h'_4
  49. Rotate(C', 60°, f) → C''_6
  50. Rotate(A', 60°, f) → A''
  51. Segment(C''_6, A'') → i'
  52. Rotate(C', 120°, f) → C''_7
  53. Rotate(A', 120°, f) → A''_1
  54. Segment(C''_7, A''_1) → i'_1
  55. Rotate(C', 180°, f) → C''_8
  56. Rotate(A', 180°, f) → A''_2
  57. Segment(C''_8, A''_2) → i'_2
  58. Rotate(C', 240°, f) → C''_9
  59. Rotate(A', 240°, f) → A''_3
  60. Segment(C''_9, A''_3) → i'_3
  61. Rotate(C', 300°, f) → C''_{10}
  62. Rotate(A', 300°, f) → A''_4
  63. Segment(C''_{10}, A''_4) → i'_4
  64. Rotate(A', 60°, f) → A''_5
  65. Rotate(C'', 60°, f) → C'''
  66. Segment(A''_5, C''') → j'
  67. Rotate(A', 120°, f) → A''_6
  68. Rotate(C'', 120°, f) → C'''_1
  69. Segment(A''_6, C'''_1) → j'_1
  70. Rotate(A', 180°, f) → A''_7
  71. Rotate(C'', 180°, f) → C'''_2
  72. Segment(A''_7, C'''_2) → j'_2
  73. Rotate(A', 240°, f) → A''_8
  74. Rotate(C'', 240°, f) → C'''_3
  75. Segment(A''_8, C'''_3) → j'_3
  76. Rotate(A', 300°, f) → A''_9
  77. Rotate(C'', 300°, f) → C'''_4
  78. Segment(A''_9, C'''_4) → j'_4
  79. Polygon(C''_{10}, C', A'_5) → t1, a'_5, c''_{10}, c'
  80. Polygon(C', C''_6, A) → t2, a, k, c''_6
  81. Polygon(C''_6, C''_7, A) → t3, a_1, l, c''_7
  82. Polygon(C''_8, C''_7, A) → t4, a_2, c''_8, m
  83. Polygon(C''_9, C''_8, A) → t5, a_3, c''_9, n
  84. Polygon(C''_{10}, C''_9, A) → t6, a_4, q, r
  85. Polygon(A''_9, A', C', C''_{10}) → q1, a''_9, a', s, t
  86. Polygon(A', A''_5, C''_6, C') → q2, b, a''_5, f_1, g_1
  87. Polygon(A''_5, A''_6, C''_7, C''_6) → q3, h_1, a''_6, i_1, j_1
  88. Polygon(C''_7, C''_8, A''_7, A''_6) → q4, k_1, l_1, a''_7, m_1
  89. Polygon(C''_9, C''_8, A''_7, A''_8) → q5, n_1, p_1, q_1, a''_8
  90. Polygon(C''_{10}, C''_9, A''_8, A''_9) → q6, r_1, s_1, t_1, b_1
  91. Polygon(C'', A', A''_5, C''') → q7, c'', c_1, d_1, c'''
  92. Polygon(A''_5, C''', C'''_1, A''_6) → q8, e_1, f_2, c'''_1, g_2
  93. Polygon(A''_6, C'''_1, C'''_2, A''_7) → q9, h_2, i_2, c'''_2, j_2
  94. Polygon(C'''_4, A''_9, A', C'') → q10, c'''_4, k_2, l_2, m_2
  95. Polygon(A''_9, A''_8, C'''_3, C'''_4) → q11, n_2, p_2, c'''_3, q_2
  96. Polygon(A''_8, A''_7, C'''_2, C'''_3) → q12, r_2, s_2, t_2, b_2
  97. Cylinder(F, H, 0.2) → o, c_2, d_2, u
  98. Cylinder(H, C, 0.1) → v, e_2, c_3, w
  99. Cylinder(C, A, 0.07) → o_1, d_3, e_3, u_1