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Arcs and Domes Construction

Arcs and Domes Construction
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2020-05-23 15:45
CirclePolygonTrigonometryGeometric Construction
This example demonstrates the geometric construction of arcs and domes. It involves core mathematical concepts such as circles, triangles, polygons, and geometric transformations. Key GeoGebra commands and objects used in the construction include Intersect, Circle, Rotate, Mirror, Segment, Line, LineBisector, Sequence, Polygon, Point, Midpoint, Plane, and OrthogonalLine. It is suitable for high school students and teachers learning geometric constructions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. ConstructionStep() → stap
  3. Circle(A, 4) → c
  4. Intersect(c, xAxis, 2) → B
  5. Rotate(B, 90°) → C
  6. Mirror(C, A) → D
  7. Segment(A, C) → f
  8. Circle(A, 1) → d
  9. Intersect(d, f, 1) → E
  10. Circle(A, 2) → e
  11. Intersect(e, f, 1) → F
  12. Circle(A, 3) → g
  13. Intersect(g, f, 1) → G
  14. Circle(A, 3.5) → h
  15. Intersect(h, f, 1) → H
  16. Circle(A, 3.85) → k
  17. Intersect(k, f, 1) → I
  18. Mirror(K, zAxis) → K'
  19. Mirror(L, zAxis) → L'
  20. Mirror(M, zAxis) → M'
  21. Mirror(N, zAxis) → N'
  22. Mirror(O, zAxis) → O'
  23. Segment(J, K) → i
  24. Segment(K, L) → j
  25. Segment(L, M) → l
  26. Segment(M, N) → m
  27. Segment(N, O) → n
  28. Segment(O, C) → p
  29. Segment(J, K') → q
  30. Segment(K', L') → r
  31. Segment(L', M') → s
  32. Segment(M', N') → t
  33. Segment(N', O') → a
  34. Segment(O', D) → b
  35. Line(C, zAxis) → as
  36. Rotate(O, 45°, as) → O'_1
  37. Rotate(N, 45°, as) → N'_1
  38. Rotate(M, 45°, as) → M'_1
  39. Rotate(L, 45°, as) → L'_1
  40. Rotate(K, 45°, as) → K'_1
  41. Rotate(J, 45°, as) → J'
  42. Rotate(K', 45°, as) → K''
  43. Rotate(L', 45°, as) → L''
  44. Rotate(M', 45°, as) → M''
  45. Rotate(N', 45°, as) → N''
  46. Rotate(O', 45°, as) → O''
  47. Rotate(D, 45°, as) → D'
  48. Segment(C, D) → m_{10}
  49. Segment(O'', D') → f_1
  50. Segment(O'', N'') → g_1
  51. Segment(N'', M'') → h_1
  52. Segment(M'', L'') → i_1
  53. Segment(L'', K'') → j_1
  54. Segment(K'', J') → k_1
  55. Segment(J', K'_1) → l_1
  56. Segment(K'_1, L'_1) → m_1
  57. Segment(L'_1, M'_1) → n_1
  58. Segment(M'_1, N'_1) → p_1
  59. Segment(N'_1, O'_1) → q_1
  60. Segment(C, O'_1) → r_1
  61. LineBisector(C, D', xOyPlane) → s_1
  62. LineBisector(C, D) → t_1
  63. Intersect(s_1, t_1) → P
  64. Line(P, zAxis) → rotas
  65. Sequence(Rotate[l1, (nn * 90°), rotas], nn, 1, 7) → l2
  66. Sequence(Rotate[l3, (nn * 90°), rotas], nn, 1, 7) → l4
  67. Rotate(J, (360°) / 8, rotas) → J'_1
  68. Polygon(J, J'_1, 8, xOyPlane) → veelhoek1, a_1, R, S, T, U, V, b_1, c_1, d_1, e_1, f_2, g_2, h_2, Q
  69. Point(rotas) → W
  70. Rotate(C, (360°) / 8, rotas) → C'
  71. Midpoint(C, C') → Z
  72. Plane(Z, P, P + (0, 0, 1)) → spieg
  73. Mirror(L, spieg) → L'_2
  74. Mirror(M, spieg) → M'_2
  75. Mirror(N, spieg) → N'_2
  76. Mirror(O, spieg) → O'_2
  77. Intersect(j, spieg) → C_1
  78. Midpoint(L'_1, L) → A_1
  79. Midpoint(M'_1, M) → B_1
  80. Midpoint(N'_1, N) → D_1
  81. Plane(A_1, B_1, D_1) → p_2
  82. Midpoint(K'_1, K) → E_1
  83. Rotate(spieg, (360°) / 16, rotas) → spieg'
  84. Intersect(spieg', i) → F_1
  85. Mirror(K, spieg) → K'_2
  86. Mirror(F_1, spieg) → G_1
  87. Segment(P, C') → s_2
  88. Segment(P, C) → t_2
  89. OrthogonalLine(C, t_2, xOyPlane) → a_2
  90. OrthogonalLine(C', s_2, xOyPlane) → b_2
  91. Intersect(a_2, b_2) → H_1
  92. Line(O, a_2) → c_2
  93. Line(O'_2, b_2) → f_3
  94. Line(N, a_2) → g_3
  95. Line(N'_2, b_2) → h_3
  96. Line(M, a_2) → i_3
  97. Line(M'_2, b_2) → l_3
  98. Circle(rotas, V) → c_9
  99. Circle(rotas, G_1) → d_9
  100. Circle(rotas, K'_2) → e_9
  101. Circle(rotas, C_1) → p_9
  102. Circle(rotas, L) → q_9
  103. Circle(rotas, M) → r_9
  104. Circle(rotas, N) → s_9
  105. Circle(rotas, O) → t_9
  106. Circle(rotas, C) → c_{10}
  107. Rotate(H_1, (360°) / 8, rotas) → P_2
  108. Polygon(H_1, P_2, 8, xOyPlane) → veelhoek4, n_9, R_2, S_2, T_2, U_2, V_2, a_9, b_9, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, Q_2
  109. Sequence(Segment[Rotate[P, (nn * (360°) / 8), P], Rotate[C, (nn * (360°) / 8), P]], nn, 1, 7) → l15
  110. Segment(C, D) → k_{10}
  111. Rotate(D, 90°, P) → D'_1
  112. Segment(C, D'_1) → l_{10}
  113. Sequence(Segment[Rotate[C, (nn * 45°), P], Rotate[D, (nn * 45°), P]], nn, 1, 7) → l16
  114. Line(L, a_2) → m_3
  115. Line(L'_2, b_2) → n_3
  116. Intersect(c_2, f_3) → I_1
  117. Intersect(g_3, h_3) → J_1
  118. Intersect(i_3, l_3) → K_1
  119. Intersect(m_3, n_3) → L_1
  120. Polygon(C_1, L, L_1) → driehoek1, i_2, j_2, l_2
  121. Polygon(L, M, K_1, L_1) → vierhoek1, l_4, m_2, k_2, n_2
  122. Polygon(M, N, J_1, K_1) → vierhoek2, m_4, n_4, q_2, d_2
  123. Polygon(N, O, I_1, J_1) → vierhoek3, n_5, o, e_2, j_3
  124. Polygon(O, C, H_1, I_1) → vierhoek4, o_1, c_3, k_3, p_3
  125. Polygon(H_1, C', O'_2, I_1) → vierhoek5, q_3, c', o'_2, r_3
  126. Polygon(I_1, O'_2, N'_2, J_1) → vierhoek6, s_3, t_3, n'_2, a_3
  127. Polygon(J_1, N'_2, M'_2, K_1) → vierhoek7, b_3, d_3, m'_2, e_3
  128. Polygon(K_1, M'_2, L'_2, L_1) → vierhoek8, f_4, g_4, l'_2, h_4
  129. Polygon(L_1, L'_2, C_1) → driehoek2, i_4, j_4, k_4
  130. Segment(W, F_1) → q_4
  131. Segment(W, V) → r_4
  132. Segment(V, C_1) → s_4
  133. Rotate(b_2, (360°) / 8, zAxis) → t_4
  134. Mirror(L'_2, spieg') → M_1
  135. Mirror(M'_2, spieg') → N_1
  136. Mirror(N'_2, spieg') → O_1
  137. Mirror(O'_2, spieg') → P_1
  138. Mirror(C_1, spieg') → Q_1
  139. Line(C', rotas) → a_4
  140. Rotate(b_2, (360°) / 32, a_4) → b_4
  141. Polygon(F_1, C_1, Q_1) → driehoek3, c_4, d_4, e_4
  142. Polygon(C_1, L_1, M_1, Q_1) → vierhoek9, f_5, g_5, h_5, i_5
  143. Polygon(L'_2, M'_2, N_1, M_1) → vierhoek10, j_5, k_5, l_5, m_5
  144. Polygon(M'_2, N'_2, O_1, N_1) → vierhoek11, p_5, q_5, r_5, s_5
  145. Line(K'_2, a_2) → t_5
  146. Line(K, b_2) → a_5
  147. Intersect(t_5, a_5) → S_1
  148. Polygon(K'_2, C_1, S_1) → driehoek4, b_5, k'_2, c_5
  149. Polygon(G_1, K'_2, S_1, V) → vierhoek12, d_5, e_5, f_6, v
  150. Polygon(V, S_1, K, F_1) → vierhoek13, v_1, g_6, k_6, h_6
  151. Polygon(S_1, C_1, K) → driehoek5, k_7, i_6, j_6
  152. Segment(C', C) → l_6
  153. Line(H_1, l_6) → m_6
  154. Rotate(M, (360°) / 8, rotas) → M'_3
  155. Rotate(K_1, (360°) / 8, rotas) → R_1
  156. Segment(M'_3, R_1) → n_6
  157. Rotate(m_6, (360°) / 8, rotas) → p_6
  158. Line(C_1, m_6) → q_6
  159. Line(Q_1, p_6) → r_6
  160. Intersect(r_6, q_6) → T_1
  161. Line(L'_2, m_6) → s_6
  162. Line(M'_2, m_6) → t_6
  163. Line(N'_2, m_6) → a_6
  164. Line(O'_2, m_6) → b_6
  165. Plane(W, F_1, C') → p_7
  166. Intersect(s_6, p_7) → U_1
  167. Intersect(t_6, p_7) → V_1
  168. Intersect(a_6, p_7) → W_1
  169. Intersect(b_6, p_7) → Z_1
  170. Segment(F_1, T_1) → c_6
  171. Segment(T_1, U_1) → d_6
  172. Segment(U_1, V_1) → e_6
  173. Segment(V_1, W_1) → f_7
  174. Segment(O'_2, Z_1) → g_7
  175. Segment(Z_1, W_1) → h_7
  176. Segment(Z_1, C') → i_7
  177. Polygon(F_1, C_1, T_1) → driehoek6, j_7, l_7, m_7
  178. Polygon(C_1, L'_2, U_1, T_1) → vierhoek14, n_7, q_7, u_1, r_7
  179. Polygon(L'_2, M'_2, V_1, U_1) → vierhoek15, s_7, t_7, a_7, b_7
  180. Polygon(M'_2, N'_2, W_1, V_1) → vierhoek16, c_7, d_7, w_1, e_7
  181. Polygon(N'_2, O'_2, Z_1, W_1) → vierhoek17, f_8, g_8, z_1, h_8
  182. Polygon(Z_1, O'_2, C') → driehoek7, i_8, j_8, k_8
  183. Segment(B_2, C_2) → p_4
  184. Sequence(Rotate[l5, (nn * (360°) / 8), rotas], nn, 1, 7) → l6
  185. Mirror(l7, spieg') → l7'
  186. Sequence(Rotate[l7, (nn * (360°) / 8), rotas], nn, 1, 7) → l8
  187. Sequence(Rotate[l7', (nn * (360°) / 8), rotas], nn, 1, 7) → l9
  188. Sequence(Rotate[l10, (nn * (360°) / 8), rotas], nn, 1, 7) → l11
  189. Polygon(V, J, 8, xOyPlane) → veelhoek2, l_8, E_2, F_2, G_2, H_2, I_2, m_8, n_8, p_8, q_8, r_8, s_8, t_8, D_2
  190. Polygon(C_2, B_2, 8, xOyPlane) → veelhoek3, a_8, K_2, L_2, M_2, N_2, O_2, b_8, c_8, d_8, e_8, f_9, g_9, h_9, J_2
  191. Polygon(G_1, V, C_2) → driehoek8, i_9, j_9, v_2
  192. Polygon(V, F_1, B_2) → driehoek9, k_9, v_3, l_9
  193. Sequence(Rotate[driehoek8, (nn * (360°) / 8), rotas], nn, 1, 7) → l12
  194. Sequence(Rotate[driehoek9, (nn * (360°) / 8), rotas], nn, 1, 7) → l13
  195. Polygon(W, O_2, C_2) → driehoek10, m_9, w, o_2
  196. Sequence(Rotate[driehoek10, (nn * (360°) / 8), rotas], nn, 1, 7) → l14

AI Drawing Prompt Reference

Draw a geometric structure composed of multiple arcs and domes centered at the intersection of the x-axis and y-axis. Start with a circle of radius 4, then create several triangles and quadrilaterals through rotation and reflection, and finally connect the vertices of these polygons with line segments to form a complex geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.