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Modifiable Solid: Regular Hexagon Cross Sections Parallel to yAxis

Modifiable Solid: Regular Hexagon Cross Sections Parallel to yAxis
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2019-04-04 00:11
PolygonFunctionCoordinate GeometrySolid GeometryCalculus
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources Cross Sectional Specials (Calculus)
This example demonstrates a modifiable solid where cross sections parallel to the yAxis are regular hexagons. Key math concepts include coordinate geometry and calculus-based cross-section analysis. Construction highlights the use of commands and objects like Segment, Midpoint, Intersect, Line, Sphere, Textfield, Rotate, Polygon, CurveCartesian, and Surface in GeoGebra. Suitable for high school and above students learning the relationship between solid geometry and function graphs.

GeoGebra Commands & Parameters

  1. Segment(H, I) → s_1
  2. Midpoint(s_1) → A_1
  3. Segment(J, K) → q_1
  4. Midpoint(q_1) → Z
  5. Midpoint(s_1) → E
  6. Midpoint(q_1) → F
  7. Intersect(upper, h, (0, 8)) → B
  8. Intersect(lower, h, (0, -4)) → C
  9. Segment(C, B) → q
  10. Intersect(upper, q, (0, 8)) → G
  11. Midpoint(C, G) → T
  12. Line(B, D) → i
  13. Sphere(B, d) → e
  14. Segment(C, B) → p
  15. Textfield(upper) → InputBox1
  16. Textfield(lower) → InputBox2
  17. Textfield(a) → InputBox3
  18. Textfield(b) → InputBox4
  19. Segment(C, G) → k_1
  20. Line(C, G) → j
  21. Sphere(G, T) → k
  22. Line(T, T + (1, 0, 0)) → m
  23. Rotate(G, pi / 3, F_1, m) → G'
  24. Rotate(G', pi / 3, F_1, Line[F_1, F_1 + (1, 0, 0)]) → G''
  25. Line(F_1, F_1 + (1, 0, 0)) → n
  26. Rotate(G'', pi / 3, F_1, n) → G'''
  27. Rotate(G''', pi / 3, F_1, n) → G_1
  28. Polygon(C, G, G', G'', G''', G_1) → poly1, c_1, g_2, g', g'', g''', g_3
  29. Segment(G', G'') → l
  30. Segment(G'', G''') → r
  31. Segment(G''', G_1) → s
  32. Segment(G_1, C) → t
  33. Segment(G', G) → i_1
  34. Line(Z, Z + (1, 0, 0)) → f
  35. Line(D_1, D_1 + (1, 0, 0)) → g
  36. Rotate(J, pi / 3, D_1, g) → J'
  37. Rotate(J', pi / 3, D_1, g) → J''
  38. Rotate(J'', pi / 3, D_1, g) → J'''
  39. Rotate(J''', pi / 3, D_1, g) → E_1
  40. Segment(J, J') → f_1
  41. Segment(J', J'') → g_1
  42. Segment(J'', J''') → j_1
  43. Segment(J''', E_1) → n_1
  44. Segment(E_1, K) → p_1
  45. Line(H_1, H_1 - (1, 0, 0)) → l_1
  46. Rotate(I, pi / 3, H_1, l_1) → I'
  47. Rotate(I', pi / 3, H_1, l_1) → I''
  48. Rotate(I'', pi / 3, H_1, l_1) → I'''
  49. Rotate(I''', pi / 3, H_1, l_1) → I_1
  50. Segment(I, I') → m_1
  51. Segment(I', I'') → r_1
  52. Segment(I'', I''') → a_1
  53. Segment(I''', I_1) → b_1
  54. Segment(I_1, H) → d_1
  55. Polygon(H, I, I', I'', I''', I_1) → poly2, h_2, i_2, i', i'', i''', i_3
  56. Polygon(K, J, J', J'', J''', E_1) → poly3, k_2, j_2, j', j'', j''', e_1
  57. CurveCartesian(t, (0.5 * (lower(t) + upper(t))) - (upper(t) - lower(t)), (((-sqrt(3))) / 2 * (lower(t) - upper(t))), t, a, b) → v
  58. CurveCartesian(t, lower(t), ((-sqrt(3)) * (lower(t) - upper(t))), t, a, b) → c
  59. CurveCartesian(t, upper(t), ((-sqrt(3)) * (lower(t) - upper(t))), t, a, b) → w
  60. CurveCartesian(t, (0.5 * (lower(t) + upper(t))) + upper(t) - lower(t), (((-sqrt(3))) / 2 * (lower(t) - upper(t))), t, a, b) → o_1
  61. Surface(u, lower(u) + ((-v) * (upper(u) - lower(u)) / 2), ((v * sqrt(3)) * (upper(u) - lower(u)) / 2), u, a, b, v, 0, 1) → o
  62. Surface(u, lower(u) - (0.5 * (upper(u) - lower(u))) + ((v * 0.5) * (upper(u) - lower(u))), (sqrt(3) / 2 * (upper(u) - lower(u))) + ((v * sqrt(3) / 2) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u
  63. Surface(u, lower(u) + (v * (upper(u) - lower(u))), (sqrt(3) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u_1
  64. Surface(u, upper(u) + (0.5 * (upper(u) - lower(u))) + (((-v) * 0.5) * (upper(u) - lower(u))), (sqrt(3) / 2 * (upper(u) - lower(u))) + ((v * sqrt(3) / 2) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → v_1
  65. Surface(u, upper(u) + (v * (upper(u) - lower(u)) / 2), ((v * sqrt(3)) * (upper(u) - lower(u)) / 2), u, a, b, v, 0, 1) → w_1