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Modifiable Solid: Regular Decagon Cross Sections Parallel to yAxis

Modifiable Solid: Regular Decagon Cross Sections Parallel to yAxis
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2019-05-14 08:38
Coordinate GeometrySolid GeometryCalculusGeometric Construction
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 decagons. Core math concepts include solid geometry, coordinate geometry, and calculus concepts like integration and functions. Construction highlights GeoGebra commands such as Intersect, Segment, Midpoint, Line, Sphere, Rotate, Polygon, CurveCartesian, and Surface. Suitable for high school students and above, particularly those learning calculus and geometric transformations.

GeoGebra Commands & Parameters

  1. Intersect(lower, h, (-10, -11.367936667331891)) → C
  2. Intersect(upper, h, (-10, 1)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 1)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Rotate(J, (2 * pi / 5), Z, c) → J'
  22. Rotate(J', (2 * pi / 5), Z, c) → J''
  23. Rotate(J'', (2 * pi / 5), Z, c) → J'''
  24. Rotate(H, (2 * pi / 5), A_1, r_1) → H'
  25. Rotate(H', (2 * pi / 5), A_1, r_1) → H''
  26. Rotate(H'', (2 * pi / 5), A_1, r_1) → H'''
  27. Rotate(G, (2 * pi / 5), B_1, l_1) → G'
  28. Rotate(G', (2 * pi / 5), B_1, l_1) → G''
  29. Rotate(G'', (2 * pi / 5), B_1, l_1) → G'''
  30. Segment(C, G) → f
  31. Line(C_1, C_1 + (1, 0, 0)) → g
  32. Rotate(G, (2 * pi / 9), C_1, g) → G'_1
  33. Rotate(G'_1, (2 * pi / 9), C_1, g) → D_1
  34. Rotate(D_1, (2 * pi / 9), C_1, g) → E_1
  35. Rotate(E_1, (2 * pi / 9), C_1, g) → F_1
  36. Rotate(F_1, (2 * pi / 9), C_1, g) → G_1
  37. Rotate(G_1, (2 * pi / 9), C_1, g) → H_1
  38. Rotate(H_1, (2 * pi / 9), C_1, g) → I_1
  39. Rotate(J, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → J'_1
  40. Rotate(J'_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → K_1
  41. Rotate(K_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → L_1
  42. Rotate(L_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → M_1
  43. Rotate(M_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → N_1
  44. Rotate(N_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → O_1
  45. Rotate(O_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → P_1
  46. Rotate(H, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → H'_1
  47. Rotate(H'_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → R_1
  48. Rotate(R_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → S_1
  49. Rotate(S_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → T_1
  50. Rotate(T_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → U_1
  51. Rotate(U_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → V_1
  52. Rotate(V_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → W_1
  53. Line(Z_1, Z_1 + (1, 0, 0)) → j
  54. Rotate(J, pi / 5, Z_1, j) → J'_2
  55. Rotate(J'_2, pi / 5, Z_1, j) → A_2
  56. Rotate(A_2, pi / 5, Z_1, j) → B_2
  57. Rotate(B_2, pi / 5, Z_1, j) → C_2
  58. Rotate(C_2, pi / 5, Z_1, j) → D_2
  59. Rotate(D_2, pi / 5, Z_1, j) → E_2
  60. Rotate(E_2, pi / 5, Z_1, j) → F_2
  61. Rotate(F_2, pi / 5, Z_1, j) → G_2
  62. Polygon(K, J, J'_2, A_2, B_2, C_2, D_2, E_2, F_2, G_2) → poly2, k, g_3, j_1, j'_2, a_2, b_2, c_2, d_2, e_2, f_2
  63. Rotate(G, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → G'_2
  64. Rotate(G'_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → I_2
  65. Rotate(I_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → J_2
  66. Rotate(J_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → K_2
  67. Rotate(K_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → L_2
  68. Rotate(L_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → M_2
  69. Rotate(M_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → N_2
  70. Rotate(N_2, pi / 5, H_2, Line[H_2, H_2 + (1, 0, 0)]) → O_2
  71. Polygon(C, O_2, N_2, M_2, L_2, K_2, J_2, I_2, G'_2, G) → poly3, c_4, g_4, o_2, n_2, m_2, l_2, k_2, j_2, i_2, g'_2
  72. Segment(C, O_2) → l
  73. Segment(O_2, N_2) → m
  74. Segment(N_2, M_2) → n
  75. Segment(M_2, L_2) → r
  76. Segment(L_2, K_2) → s
  77. Segment(K_2, J_2) → t
  78. Segment(J_2, I_2) → m_1
  79. Segment(I_2, G'_2) → n_1
  80. Segment(G'_2, G) → p_1
  81. Segment(J, J'_2) → a_1
  82. Segment(J'_2, A_2) → b_1
  83. Segment(A_2, B_2) → p_2
  84. Segment(B_2, C_2) → q_2
  85. Segment(C_2, D_2) → r_2
  86. Segment(D_2, E_2) → s_2
  87. Segment(E_2, F_2) → t_2
  88. Segment(F_2, G_2) → f_3
  89. Segment(G_2, K) → h_3
  90. Rotate(H, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → H'_2
  91. Rotate(H'_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → Q_2
  92. Rotate(Q_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → R_2
  93. Rotate(R_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → S_2
  94. Rotate(S_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → T_2
  95. Rotate(T_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → U_2
  96. Rotate(U_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → V_2
  97. Rotate(V_2, pi / 5, P_2, Line[P_2, P_2 + (1, 0, 0)]) → W_2
  98. Polygon(I, H, H'_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2) → poly4, i_3, w_1, h_4, h'_2, q_3, r_3, s_3, t_3, u_2, v_2
  99. Segment(H, H'_2) → k_3
  100. Segment(H'_2, Q_2) → m_3
  101. Segment(Q_2, R_2) → n_3
  102. Segment(R_2, S_2) → p_3
  103. Segment(S_2, T_2) → f_4
  104. Segment(T_2, U_2) → i_4
  105. Segment(U_2, V_2) → j_4
  106. Segment(V_2, W_2) → k_4
  107. Segment(W_2, I) → l_4
  108. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * sin(pi / 5)), t, a, b) → o
  109. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * sin(pi / 5)), t, a, b) → u
  110. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos(pi / 5) + cos((2 * pi / 5)))), ((upper(t) - lower(t)) * (sin(pi / 5) + sin((2 * pi / 5)))), t, a, b) → v
  111. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos(pi / 5) + cos((2 * pi / 5)))), ((upper(t) - lower(t)) * (sin(pi / 5) + sin((2 * pi / 5)))), t, a, b) → w
  112. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * (sin(pi / 5) + (2 * sin((2 * pi / 5))))), t, a, b) → o_1
  113. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos(pi / 5)), ((upper(t) - lower(t)) * (sin(pi / 5) + (2 * sin((2 * pi / 5))))), t, a, b) → u_1
  114. CurveCartesian(t, lower(t), ((upper(t) - lower(t)) * ((2 * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), t, a, b) → v_1
  115. CurveCartesian(t, upper(t), ((upper(t) - lower(t)) * ((2 * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), t, a, b) → o_3
  116. Surface(u, lower(u) - (((upper(u) - lower(u)) * cos(pi / 5)) * v), (((upper(u) - lower(u)) * sin(pi / 5)) * v), u, a, b, v, 0, 1) → u_3
  117. Surface(u, upper(u) + (((upper(u) - lower(u)) * cos(pi / 5)) * v), (((upper(u) - lower(u)) * sin(pi / 5)) * v), u, a, b, v, 0, 1) → v_3
  118. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos(pi / 5) + (v * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + (v * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → w_3
  119. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos(pi / 5) + (v * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + (v * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → a_4
  120. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos(pi / 5) + ((1 - v) * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + ((1 + v) * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → b_4
  121. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos(pi / 5) + ((1 - v) * cos((2 * pi / 5))))), ((upper(u) - lower(u)) * (sin(pi / 5) + ((1 + v) * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → d_4
  122. Surface(u, lower(u) - ((upper(u) - lower(u)) * ((1 - v) * cos(pi / 5))), ((upper(u) - lower(u)) * (((1 + v) * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → e_4
  123. Surface(u, upper(u) + ((upper(u) - lower(u)) * ((1 - v) * cos(pi / 5))), ((upper(u) - lower(u)) * (((1 + v) * sin(pi / 5)) + (2 * sin((2 * pi / 5))))), u, a, b, v, 0, 1) → m_4
  124. Surface(u, lower(u) + (v * (upper(u) - lower(u))), ((upper(u) - lower(u)) * tan((2 * pi / 5))), u, a, b, v, 0, 1) → n_4

AI Drawing Prompt Reference

Create a modifiable solid where the cross-sections parallel to the y-axis are regular decagons. Start by drawing two regular decagons at the lower and upper positions, then use rotation and translation to generate the solid. Adjust the size and position of the decagons using sliders to observe changes in the solid.
The prompt above was compiled from the original drawing commands and is for reference only.