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Modifiable Solid: Regular Heptagon Cross Sections Parallel to yAxis

Modifiable Solid: Regular Heptagon Cross Sections Parallel to yAxis
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2019-04-09 22:20
PolygonSolid 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 heptagons. Core math concepts include integral calculus and polygon construction in geometry. Construction highlights involve GeoGebra commands like Segment, Midpoint, Line, Rotate, Intersect, Polygon, etc., to create cross-sections. Suitable for high school and above students to explore the integration of solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Segment(H, I) → s_1
  2. Midpoint(s_1) → A_1
  3. Line(H_1, H_1 - (1, 0, 0)) → l_1
  4. Rotate(I, (2 * pi / 7), H_1, l_1) → I'
  5. Rotate(I', (2 * pi / 7), H_1, l_1) → I''
  6. Rotate(I'', (2 * pi / 7), H_1, l_1) → I'''
  7. Rotate(I''', (2 * pi / 7), H_1, l_1) → I_1
  8. Rotate(I_1, (2 * pi / 7), H_1, l_1) → L_1
  9. Segment(J, K) → q_1
  10. Midpoint(q_1) → Z
  11. Line(D_1, D_1 + (1, 0, 0)) → g
  12. Rotate(J, (2 * pi / 7), D_1, g) → J'
  13. Rotate(J', (2 * pi / 7), D_1, g) → J''
  14. Rotate(J'', (2 * pi / 7), D_1, g) → J'''
  15. Rotate(J''', (2 * pi / 7), D_1, g) → E_1
  16. Rotate(E_1, (2 * pi / 7), D_1, g) → K_1
  17. Intersect(lower, h, (-10, -10.839071529076453)) → C
  18. Intersect(upper, h, (-10, 0)) → B
  19. Segment(C, B) → q
  20. Intersect(upper, q, (-10, 0)) → G
  21. Midpoint(C, G) → T
  22. Line(T, T + (1, 0, 0)) → m
  23. Line(F_1, F_1 + (1, 0, 0)) → n
  24. Rotate(G, (2 * pi / 7), F_1, n) → G'
  25. Rotate(G', (2 * pi / 7), F_1, n) → G''
  26. Rotate(G'', (2 * pi / 7), F_1, n) → G'''
  27. Rotate(G''', (2 * pi / 7), F_1, n) → G_1
  28. Rotate(G_1, (2 * pi / 7), F_1, n) → J_1
  29. Midpoint(s_1) → E
  30. Midpoint(q_1) → F
  31. Line(B, D) → i
  32. Sphere(B, d) → e
  33. Segment(C, B) → p
  34. Textfield(upper) → InputBox1
  35. Textfield(lower) → InputBox2
  36. Textfield(a) → InputBox3
  37. Textfield(b) → InputBox4
  38. Segment(C, G) → k_1
  39. Line(C, G) → j
  40. Polygon(C, G, G', G'', G''', G_1, J_1) → poly1, m_1, r_1, a_1, b_1, c_1, d_1, f_2
  41. Sphere(G, T) → k
  42. Segment(G', G'') → l
  43. Segment(G'', G''') → r
  44. Segment(G''', G_1) → s
  45. Segment(G', G) → i_1
  46. Line(Z, Z + (1, 0, 0)) → f
  47. Segment(J, J') → f_1
  48. Polygon(H, I, I', I'', I''', I_1, L_1) → poly2, h_2, i_2, s_2, t_2, a_2, b_2, c_2
  49. Polygon(K, J, J', J'', J''', E_1, K_1) → poly3, g_1, j_1, n_1, p_1, e_1, j_2, k_2
  50. Segment(G_1, J_1) → g_2
  51. Segment(J_1, C) → l_2
  52. Segment(J', J'') → m_2
  53. Segment(J'', J''') → n_2
  54. Segment(J''', E_1) → p_2
  55. Segment(E_1, K_1) → q_2
  56. Segment(K_1, K) → r_2
  57. Segment(I', I) → d_2
  58. Segment(I', I'') → e_2
  59. Segment(I'', I''') → f_3
  60. Segment(I''', I_1) → g_3
  61. Segment(I_1, L_1) → h_3
  62. Segment(L_1, H) → i_3
  63. CurveCartesian(t, lower(t) - (cos((2 * pi / 7)) * (upper(t) - lower(t))), (sin((2 * pi / 7)) * (upper(t) - lower(t))), t, a, b) → o_2
  64. CurveCartesian(t, upper(t) + (cos((2 * pi / 7)) * (upper(t) - lower(t))), (sin((2 * pi / 7)) * (upper(t) - lower(t))), t, a, b) → c
  65. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((0.5 * tan((5 * pi / 14))) * (upper(t) - lower(t))) + (0.5 * (upper(t) - lower(t)) / cos((5 * pi / 14))), t, a, b) → v
  66. CurveCartesian(t, ((-(upper(t) - lower(t))) * sin((5 * pi / 14))) + (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * tan((5 * pi / 14))) + (0.5 * (upper(t) - lower(t)) / cos((5 * pi / 14))) - ((upper(t) - lower(t)) * cos((5 * pi / 14))), t, a, b) → t
  67. CurveCartesian(t, ((upper(t) - lower(t)) * sin((5 * pi / 14))) + (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * tan((5 * pi / 14))) + (0.5 * (upper(t) - lower(t)) / cos((5 * pi / 14))) - ((upper(t) - lower(t)) * cos((5 * pi / 14))), t, a, b) → w
  68. Surface(u, lower(u) - ((v * cos((2 * pi / 7))) * (upper(u) - lower(u))), ((v * sin((2 * pi / 7))) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → o
  69. Surface(u, upper(u) + (v * (cos((2 * pi / 7)) * (upper(u) - lower(u)))), ((v * sin((2 * pi / 7))) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u
  70. Surface(u, lower(u) - ((upper(u) - lower(u)) * cos((2 * pi / 7))) + ((v * (upper(u) - lower(u))) * sin(pi / 14)), ((upper(u) - lower(u)) * sin((2 * pi / 7))) + ((v * (upper(u) - lower(u))) * cos(pi / 14)), u, a, b, v, 0, 1) → o_1
  71. Surface(u, upper(u) + ((upper(u) - lower(u)) * cos((2 * pi / 7))) - ((v * (upper(u) - lower(u))) * sin(pi / 14)), ((upper(u) - lower(u)) * sin((2 * pi / 7))) + ((v * (upper(u) - lower(u))) * cos(pi / 14)), u, a, b, v, 0, 1) → u_1
  72. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 7)) - sin(pi / 14))) + (v * ((0.5 * (upper(u) + lower(u))) - (lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 7)) - sin(pi / 14)))))), ((upper(u) - lower(u)) * (sin((2 * pi / 7)) + cos(pi / 14))) + ((v * (upper(u) - lower(u))) * cos((5 * pi / 14))), u, a, b, v, 0, 1) → u_2
  73. Surface(u, (0.5 * (upper(u) + lower(u))) + ((v * (upper(u) - lower(u))) * sin((5 * pi / 14))), ((0.5 * (upper(u) - lower(u))) * tan((5 * pi / 14))) + ((upper(u) - lower(u)) * sin((5 * pi / 14)) / tan((2 * pi / 7))) + (((1 - v) * (upper(u) - lower(u))) * cos((5 * pi / 14))), u, a, b, v, 0, 1) → v_2

AI Drawing Prompt Reference

Draw a modifiable solid with regular heptagon cross sections parallel to the yAxis. First, create two parallel heptagon cross sections to the yAxis and use rotation to generate a series of heptagons, forming one solid. Then, create another heptagon cross section parallel to the yAxis and use rotation to generate another series of heptagons, forming another solid. Finally, combine the two solids to form a modifiable solid.
The prompt above was compiled from the original drawing commands and is for reference only.