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Geometric Transformations of a Dome

Geometric Transformations of a Dome
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2020-05-22 20:04
TrigonometrySolid GeometryTransformationGeometric Construction
This example demonstrates the geometric transformations and centroid calculation of a dome structure. It involves core concepts like centroid, geometric transformations (rotation, reflection), and polygon construction. GeoGebra commands and objects used include segment, intersect, plane, circle arc, point, line, rotate, mirror, midpoint, polygon, sequence, and distance. Suitable for high school and above students learning solid geometry, transformations, and centroid computation.

GeoGebra Commands & Parameters

  1. Segment(C, D) → h
  2. Segment(B, C) → g
  3. Intersect(g, yAxis) → G
  4. Plane(B, D, H_1) → diag
  5. CircleArc(G, C, I) → c
  6. Point(c) → K
  7. Line(J_1, h) → s
  8. Rotate(J_1, 90°, zAxis) → P
  9. Point(c) → J
  10. Line(I_1, h) → t
  11. Rotate(I_1, 90°, zAxis) → S_1
  12. Segment(D, A) → i
  13. Line(S_1, i) → g_1
  14. Intersect(g_1, t) → T
  15. Mirror(T, vgl1) → T'
  16. Line(P, i) → f_1
  17. Intersect(f_1, s) → U
  18. Mirror(U, vgl1) → U'
  19. Segment(T', U') → i_2
  20. Segment(A, B) → f
  21. Intersect(f, xAxis) → F
  22. Plane(D, F, F + (0, 0, 1)) → snij
  23. Intersect(i_2, snij) → D_2
  24. Segment(N_3, D_2) → h_6
  25. Mirror(D_2, zAxis) → K_2
  26. Midpoint(D_2, K_2) → L_2
  27. CircleArc(L_2, K_2, M_2) → d_3
  28. Point(d_3) → Q_2
  29. Mirror(V_2, diag) → B_3
  30. Segment(N_3, M_3) → c_5
  31. Point(d_3) → P_2
  32. Mirror(U_2, diag) → A_3
  33. Segment(M_3, L_3) → d_5
  34. Point(d_3) → O_2
  35. Mirror(T_2, diag) → Z_2
  36. Segment(L_3, K_3) → e_5
  37. Point(d_3) → N_2
  38. Mirror(S_2, diag) → W_2
  39. Segment(J_3, E_3) → g_6
  40. Segment(K_3, J_3) → f_6
  41. Intersect(i, yAxis) → E
  42. Point(xAxis) → H
  43. CircleArc(G, I, B) → d
  44. Point(c) → L
  45. Point(c) → M
  46. Point(c) → N
  47. Line(J, g) → j
  48. Line(K, g) → k
  49. Line(L, g) → l
  50. Line(M, g) → m
  51. Line(N, g) → n
  52. Mirror(I_1, vgl1) → N_1
  53. Mirror(J_1, vgl1) → O_1
  54. Mirror(K_1, vgl1) → P_1
  55. Mirror(L_1, vgl1) → Q_1
  56. Mirror(M_1, vgl1) → R_1
  57. Rotate(L_1, 90°, zAxis) → Q
  58. Rotate(M_1, 90°, zAxis) → R
  59. Rotate(I_1, (-(90°)), zAxis) → S
  60. Line(M_1, h) → p
  61. Line(L_1, h) → q
  62. Line(K_1, h) → r
  63. Line(H_1, h) → a
  64. Line(R, i) → b
  65. Line(Q, i) → e
  66. Intersect(r, e) → V
  67. Intersect(p, b) → W
  68. Rotate(K_1, 90°, zAxis) → Z
  69. Line(Z, i) → h_1
  70. Intersect(r, h_1) → A_1
  71. Intersect(q, e) → T_1
  72. Mirror(A_1, vgl1) → U_1
  73. Mirror(T_1, vgl1) → V_1
  74. Mirror(W, vgl1) → W'
  75. Polygon(D, A, W', W) → vierhoek1, d_1, a_1, w', w
  76. Polygon(W, W', V_1, T_1) → vierhoek2, w_1, i_1, v_1, t_1
  77. Polygon(T_1, V_1, U_1, A_1) → vierhoek3, j_1, k_1, u_1, l_1
  78. Polygon(A_1, U_1, U', U) → vierhoek4, m_1, n_1, u', u
  79. Polygon(U, U', T', T) → vierhoek5, u_2, p_1, t', t_2
  80. Polygon(T, T', H_1) → driehoek1, q_1, t_3, r_1
  81. Segment(H_1, T) → s_1
  82. Segment(T, U) → b_1
  83. Segment(U, A_1) → c_1
  84. Segment(A_1, T_1) → e_1
  85. Segment(T_1, W) → f_2
  86. Segment(W, D) → g_2
  87. Segment(H_1, T') → h_2
  88. Segment(U', U_1) → j_2
  89. Segment(U_1, V_1) → k_2
  90. Segment(V_1, W') → l_2
  91. Segment(W', A) → m_2
  92. Sequence(Rotate[vierhoek1, (nn * 90°), zAxis], nn, 1, 4) → l1
  93. Sequence(Rotate[vierhoek2, (nn * 90°), zAxis], nn, 1, 4) → l2
  94. Sequence(Rotate[vierhoek3, (nn * 90°), zAxis], nn, 1, 4) → l3
  95. Sequence(Rotate[vierhoek4, (nn * 90°), zAxis], nn, 1, 4) → l4
  96. Sequence(Rotate[vierhoek5, (nn * 90°), zAxis], nn, 1, 4) → l5
  97. Sequence(Rotate[driehoek1, (nn * 90°), zAxis], nn, 1, 4) → l6
  98. Sequence(Segment[Rotate[H_1, (nn * 90°), zAxis], Rotate[T', (nn * 90°), zAxis]], nn, 1, 2) → l7
  99. Sequence(Segment[Rotate[T', (nn * 90°), zAxis], Rotate[U', (nn * 90°), zAxis]], nn, 1, 2) → l8
  100. Sequence(Segment[Rotate[U', (nn * 90°), zAxis], Rotate[U_1, (nn * 90°), zAxis]], nn, 1, 2) → l9
  101. Sequence(Segment[Rotate[U_1, (nn * 90°), zAxis], Rotate[V_1, (nn * 90°), zAxis]], nn, 1, 2) → l10
  102. Sequence(Segment[Rotate[V_1, (nn * 90°), zAxis], Rotate[W', (nn * 90°), zAxis]], nn, 1, 2) → l11
  103. Sequence(Segment[Rotate[W', (nn * 90°), zAxis], Rotate[A, (nn * 90°), zAxis]], nn, 1, 2) → l12
  104. Midpoint(H, E) → W_1
  105. Line(W_1, zAxis) → n_2
  106. Distance(W_1, E) → horstraal
  107. Rotate(vierhoek1, 90°, zAxis) → vierhoek1'
  108. Rotate(vierhoek2, 90°, zAxis) → vierhoek2'
  109. Rotate(vierhoek3, 90°, zAxis) → vierhoek3'
  110. Rotate(vierhoek4, 90°, zAxis) → vierhoek4'
  111. Rotate(vierhoek5, 90°, zAxis) → vierhoek5'
  112. Intersect(u_2, snij) → Z_1
  113. Intersect(m_1, snij) → A_2
  114. Intersect(j_1, snij) → B_2
  115. Intersect(w_1, snij) → C_2
  116. Intersect(snij, vierhoek4') → E_2, F_2
  117. Intersect(snij, vierhoek2') → G_2, H_2
  118. Segment(D, C_2) → p_2
  119. Segment(C_2, B_2) → q_2
  120. Segment(B_2, A_2) → r_2
  121. Segment(A_2, Z_1) → s_2
  122. Segment(Z_1, D_2) → a_2
  123. Segment(D_2, F_2) → b_2
  124. Segment(F_2, E_2) → c_2
  125. Segment(E_2, H_2) → d_2
  126. Segment(H_2, G_2) → e_2
  127. Segment(G_2, F) → f_3
  128. Mirror(l13, diag) → l13'
  129. Sequence(Rotate[l13, (nn * 90°), zAxis], nn, 1, 3) → l14
  130. Sequence(Rotate[l13', (nn * 90°), zAxis], nn, 1, 3) → l15
  131. Segment(B, E) → g_3
  132. Segment(B, H) → h_3
  133. Segment(C, F) → i_3
  134. Segment(C, E) → j_3
  135. Segment(D, F) → k_3
  136. Segment(D, G) → l_3
  137. Segment(A, G) → m_3
  138. Segment(A, H) → n_3
  139. Rotate(D_2, (-(90°)), zAxis) → I_2
  140. Point(l15) → J_2
  141. Polygon(Z_1, J_2, D_2) → driehoek2, p_3, z_1, q_3
  142. Polygon(J_2, U', D_2) → driehoek3, r_3, s_3, a_3
  143. Plane(D_2, U', D_2 + (0, 0, 1)) → spieg
  144. Mirror(driehoek2, spieg) → driehoek2'
  145. Mirror(driehoek3, spieg) → driehoek3'
  146. Sequence(Rotate[driehoek2, (nn * 90°), zAxis], nn, 1, 3) → l16
  147. Sequence(Rotate[driehoek3, (nn * 90°), zAxis], nn, 1, 3) → l17
  148. Sequence(Rotate[driehoek3', (nn * 90°), zAxis], nn, 1, 3) → l18
  149. Sequence(Rotate[driehoek2', (nn * 90°), zAxis], nn, 1, 3) → l19
  150. Segment(T, H_1) → b_3
  151. Segment(L_2, U) → c_3
  152. Point(s_2) → D_3
  153. Polygon(D_3, D_2, C_3) → driehoek4, e_3, f_4, g_4
  154. Sequence(Rotate[driehoek4, (nn * 90°), zAxis], nn, 1, 3) → l20
  155. Mirror(driehoek4, spieg) → driehoek4'
  156. Sequence(Rotate[driehoek4', (nn * 90°), zAxis], nn, 1, 3) → l21
  157. Polygon(C_3, D_2, N_3, O_3) → vierhoek6, h_4, i_4, j_4, o_3
  158. Rotate(J_3, (-(45°)), zAxis) → P_3
  159. Rotate(K_3, (-(45°)), zAxis) → Q_3
  160. Rotate(L_3, (-(45°)), zAxis) → R_3
  161. Rotate(M_3, (-(45°)), zAxis) → S_3
  162. Rotate(N_3, (-(45°)), zAxis) → T_3
  163. Polygon(P_3, J_3, E_3) → driehoek5, k_4, l_4, m_4
  164. Polygon(P_3, Q_3, K_3, J_3) → vierhoek7, n_4, p_4, q_4, r_4
  165. Polygon(Q_3, R_3, L_3, K_3) → vierhoek8, s_4, t_4, a_4, b_4
  166. Polygon(S_3, M_3, L_3, R_3) → vierhoek9, c_4, d_4, e_4, f_5
  167. Point(o_3) → U_3
  168. Polygon(S_3, U_3, N_3, M_3) → vierhoek10, g_5, u_3, h_5, i_5
  169. Polygon(S_3, T_3, M_3) → driehoek6, j_5, k_5, l_5
  170. Polygon(T_3, N_3, M_3) → driehoek7, m_5, n_5, p_5
  171. Polygon(T_3, C_3, N_3) → driehoek8, q_5, r_5, s_5
  172. Polygon(C_3, N_3, D_2) → driehoek9, t_5, a_5, b_5
  173. Sequence(Rotate[l22, (nn * 90°), zAxis], nn, 1, 3) → l23
  174. Mirror(l22, spieg) → l22'
  175. Sequence(Rotate[l22', (nn * 90°), zAxis], nn, 1, 3) → l24
  176. Sequence(Rotate[l25, (nn * 90°), zAxis], nn, 1, 3) → l26
  177. Segment(D_3, C_3) → i_6
  178. Segment(D_3, T_3) → j_6
  179. Segment(T_3, S_3) → k_6
  180. Segment(S_3, R_3) → l_6
  181. Segment(R_3, Q_3) → m_6
  182. Segment(Q_3, P_3) → n_6
  183. Segment(P_3, E_3) → p_6
  184. Sequence(Rotate[l27, (nn * 90°), zAxis], nn, 1, 3) → l28
  185. Segment(W', A) → q_6
  186. Segment(C_3, D_2) → r_6
  187. Segment(C_3, I_2) → s_6
  188. Sequence(Segment[Rotate[C_3, (nn * 90°), zAxis], Rotate[D_2, (nn * 90°), zAxis]], nn, 1, 3) → l29
  189. Sequence(Segment[Rotate[C_3, (nn * 90°), zAxis], Rotate[I_2, (nn * 90°), zAxis]], nn, 1, 3) → l30

AI Drawing Prompt Reference

Draw a geometric transformation and centroid calculation example of a dome structure. Start by constructing a polygon, then use rotation and reflection to create symmetrical shapes, and calculate the centroid. Show the process of geometric transformations like rotation and reflection, and the construction of polygons along with the centroid's position.
The prompt above was compiled from the original drawing commands and is for reference only.