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iOS Logo Geometric Construction Demo

iOS Logo Geometric Construction Demo
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2020-05-27 23:15
CircleVectorCoordinate GeometryTransformationGeometric Construction
This example demonstrates how to construct the iOS Logo geometrically using GeoGebra, involving concepts like circles, reflections, rotations, similar triangles, and coordinate geometry. Key GeoGebra commands and objects used in the construction include LineBisector, Vector, OrthogonalVector, Line, OrthogonalLine, Intersect, Mirror, CircumcircleArc, Polygon, Rotate, and CircleArc. Suitable for high school students and math enthusiasts learning geometric transformations and coordinate geometry applications.

GeoGebra Commands & Parameters

  1. LineBisector(A, B) → m
  2. Vector(A, B) → u
  3. OrthogonalVector(u) → v
  4. LineBisector(A_1, A) → l
  5. LineBisector(A_1, A) → n
  6. LineBisector(A_1, A) → h
  7. Line(A, B) → s
  8. OrthogonalLine(D, s) → i
  9. Intersect(i, l) → P
  10. Line(Q, l) → t
  11. Mirror(t, l) → t'
  12. OrthogonalLine(A, s) → b
  13. OrthogonalLine(B, u, xOyPlane) → g
  14. Mirror(i, m) → i'
  15. OrthogonalLine(A_1, b) → r
  16. Intersect(g, r) → C
  17. Intersect(b, l) → O
  18. Intersect(b, t) → H
  19. Mirror(H, l) → H'
  20. Intersect(m, l) → L
  21. Mirror(A_1, l) → D_2
  22. Mirror(A_1, l) → F_2
  23. Mirror(C, l) → C'_1
  24. OrthogonalLine(F, b) → f
  25. Mirror(f, l) → f'
  26. Mirror(f, l) → f'_1
  27. Intersect(m, f) → M
  28. Intersect(m, t) → G
  29. OrthogonalLine(E, l) → j
  30. Mirror(F, l) → F'
  31. OrthogonalLine(I, b) → k
  32. Intersect(t, m) → K
  33. OrthogonalLine(N, b) → p
  34. Mirror(j, m) → j'
  35. Intersect(m, f') → S
  36. Intersect(k, j) → T
  37. Intersect(j', k) → U
  38. LineBisector(S, K) → q
  39. CircumcircleArc(U, S, T) → d
  40. Mirror(U, q) → U'
  41. Mirror(S, q) → S'
  42. Mirror(T, q) → T'
  43. CircumcircleArc(U', S', T') → d'
  44. Polygon(U, T, T', U') → c1, u_1, t_1, a, u'
  45. Mirror(U, l) → U'_1
  46. Mirror(S, l) → S'_1
  47. Mirror(T, l) → T'_1
  48. CircumcircleArc(U'_1, S'_1, T'_1) → d'_1
  49. Mirror(U, l) → U'_2
  50. Mirror(T, l) → T'_2
  51. Mirror(T', l) → T''
  52. Mirror(U', l) → U''
  53. Polygon(U'_2, T'_2, T'', U'') → c1', u'_2, t'_2, t'', u''
  54. Mirror(U', l) → U''_1
  55. Mirror(S', l) → S''
  56. Mirror(T', l) → T''_1
  57. CircumcircleArc(U''_1, S'', T''_1) → d''
  58. Rotate(U, 45°, L) → U'_3
  59. Rotate(S, 45°, L) → S'_2
  60. Rotate(T, 45°, L) → T'_3
  61. CircumcircleArc(U'_3, S'_2, T'_3) → d'_2
  62. Rotate(U, 45°, L) → U'_4
  63. Rotate(T, 45°, L) → T'_4
  64. Rotate(T', 45°, L) → T''_2
  65. Rotate(U', 45°, L) → U''_2
  66. Polygon(U'_4, T'_4, T''_2, U''_2) → c1'_1, u'_4, t'_4, t''_2, u''_2
  67. Rotate(U', 45°, L) → U''_3
  68. Rotate(S', 45°, L) → S''_1
  69. Rotate(T', 45°, L) → T''_3
  70. CircumcircleArc(U''_3, S''_1, T''_3) → d''_1
  71. Rotate(U'_3, 45°, L) → R
  72. Rotate(S'_2, 45°, L) → V
  73. Rotate(T'_3, 45°, L) → W
  74. CircumcircleArc(R, V, W) → c
  75. Rotate(U'_4, 45°, L) → Z
  76. Rotate(T'_4, 45°, L) → B_1
  77. Rotate(T''_2, 45°, L) → C_1
  78. Rotate(U''_2, 45°, L) → D_1
  79. Polygon(Z, B_1, C_1, D_1) → c2, z_1, b_1, c_1, d_1
  80. Rotate(U''_3, 45°, L) → E_1
  81. Rotate(S''_1, 45°, L) → F_1
  82. Rotate(T''_3, 45°, L) → G_1
  83. CircumcircleArc(E_1, F_1, G_1) → e
  84. Rotate(R, 45°, L) → R'
  85. Rotate(V, 45°, L) → V'
  86. Rotate(W, 45°, L) → W'
  87. CircumcircleArc(R', V', W') → c'
  88. Rotate(Z, 45°, L) → Z'
  89. Rotate(B_1, 45°, L) → H_1
  90. Rotate(C_1, 45°, L) → I_1
  91. Rotate(D_1, 45°, L) → J_1
  92. Polygon(Z', H_1, I_1, J_1) → c2', z', h_1, i_1, j_1
  93. Rotate(E_1, 45°, L) → K_1
  94. Rotate(F_1, 45°, L) → L_1
  95. Rotate(G_1, 45°, L) → M_1
  96. CircumcircleArc(K_1, L_1, M_1) → e'
  97. Mirror(U'_3, m) → N_1
  98. Mirror(S'_2, m) → O_1
  99. Mirror(T'_3, m) → P_1
  100. CircumcircleArc(N_1, O_1, P_1) → f_1
  101. Mirror(U'_4, m) → Q_1
  102. Mirror(T'_4, m) → R_1
  103. Mirror(T''_2, m) → S_1
  104. Mirror(U''_2, m) → T_1
  105. Polygon(Q_1, R_1, S_1, T_1) → c3, q_1, r_1, s_1, g_1
  106. Mirror(K_1, m) → U_1
  107. Mirror(L_1, m) → V_1
  108. Mirror(M_1, m) → W_1
  109. CircumcircleArc(U_1, V_1, W_1) → e''
  110. Mirror(U''_3, m) → Z_1
  111. Mirror(S''_1, m) → A_2
  112. Mirror(T''_3, m) → B_2
  113. CircumcircleArc(Z_1, A_2, B_2) → k_1
  114. Mirror(Z', m) → Z''
  115. Mirror(H_1, m) → C_2
  116. Mirror(I_1, m) → E_2
  117. Mirror(J_1, m) → G_2
  118. Polygon(Z'', C_2, E_2, G_2) → c2'', z'', c_2, l_1, g_2
  119. Mirror(R', m) → R''
  120. Mirror(V', m) → V''
  121. Mirror(W', m) → W''
  122. CircumcircleArc(R'', V'', W'') → c''
  123. Mirror(R, m) → R'_1
  124. Mirror(V, m) → V'_1
  125. Mirror(W, m) → W'_1
  126. CircumcircleArc(R'_1, V'_1, W'_1) → c'_1
  127. Mirror(Z, m) → Z'_1
  128. Mirror(B_1, m) → H_2
  129. Mirror(C_1, m) → I_2
  130. Mirror(D_1, m) → J_2
  131. Polygon(Z'_1, H_2, I_2, J_2) → c2'_1, z'_1, h_2, i_2, j_2
  132. Mirror(E_1, m) → K_2
  133. Mirror(F_1, m) → L_2
  134. Mirror(G_1, m) → M_2
  135. CircumcircleArc(K_2, L_2, M_2) → e'_1
  136. Polygon(A_1, C, B, A) → c4, a_1, c_3, b_2, a_2
  137. LineBisector(T_1, K) → m_1
  138. Intersect(m_1, j) → N_2
  139. CircleArc(N_2, K, T_1) → p_1
  140. Mirror(p_1, m) → d_2
  141. Mirror(p_1, l) → f_2
  142. Mirror(d_2, l) → k_2
  143. Rotate(p_1, 45°, L) → p_2
  144. Rotate(d_2, 45°, L) → q_2
  145. Rotate(p_2, 45°, L) → r_2
  146. Rotate(q_2, 45°, L) → s_2
  147. Rotate(r_2, 45°, L) → t_2
  148. Rotate(s_2, 45°, L) → d_3
  149. Mirror(p_2, m) → e_3
  150. Mirror(q_2, m) → f_3
  151. Mirror(r_2, m) → g_3
  152. Mirror(s_2, m) → h_3
  153. Rotate(g_3, (-(45°)), L) → k_3
  154. Rotate(h_3, (-(45°)), L) → p_3
  155. Intersect(t_1, u''_2) → O_2
  156. Polygon(O_2, T', S''_1, K, T_1) → polígono1, o_2, n_1, s''_1, k_4, l_2
  157. Rotate(O_2, 45°, L) → P_2
  158. Rotate(T', 45°, L) → T''_4
  159. Rotate(S''_1, 45°, L) → Q_2
  160. Rotate(K, 45°, L) → K'
  161. Rotate(T_1, 45°, L) → R_2
  162. Polygon(P_2, T''_4, Q_2, K', R_2) → polígono1', m_2, t''_4, n_2, k', i_3
  163. Rotate(P_2, 45°, L) → S_2
  164. Rotate(T''_4, 45°, L) → T_2
  165. Rotate(Q_2, 45°, L) → U_2
  166. Rotate(K', 45°, L) → K''
  167. Rotate(R_2, 45°, L) → V_2
  168. Polygon(S_2, T_2, U_2, K'', V_2) → polígono1'', j_3, l_3, u_2, k'', v_2
  169. Rotate(S_2, 45°, L) → W_2
  170. Rotate(T_2, 45°, L) → Z_2
  171. Rotate(U_2, 45°, L) → A_3
  172. Rotate(K'', 45°, L) → K'''
  173. Rotate(V_2, 45°, L) → B_3
  174. Polygon(W_2, Z_2, A_3, K''', B_3) → polígono1''', w_2, z_2, m_3, k''', b_3
  175. Rotate(W_2, 45°, L) → C_3
  176. Rotate(Z_2, 45°, L) → D_3
  177. Rotate(A_3, 45°, L) → E_3
  178. Rotate(K''', 45°, L) → F_3
  179. Rotate(B_3, 45°, L) → G_3
  180. Polygon(C_3, D_3, E_3, F_3, G_3) → polígono2, n_3, q_3, r_3, s_3, t_3
  181. Rotate(C_3, 45°, L) → H_3
  182. Rotate(D_3, 45°, L) → I_3
  183. Rotate(E_3, 45°, L) → J_3
  184. Rotate(F_3, 45°, L) → K_3
  185. Rotate(G_3, 45°, L) → L_3
  186. Polygon(H_3, I_3, J_3, K_3, L_3) → polígono2', f_4, g_4, h_4, i_4, j_4
  187. Rotate(H_3, 45°, L) → M_3
  188. Rotate(I_3, 45°, L) → N_3
  189. Rotate(J_3, 45°, L) → O_3
  190. Rotate(K_3, 45°, L) → P_3
  191. Rotate(L_3, 45°, L) → Q_3
  192. Polygon(M_3, N_3, O_3, P_3, Q_3) → polígono2'', l_4, m_4, o_3, n_4, p_4
  193. Rotate(M_3, 45°, L) → R_3
  194. Rotate(N_3, 45°, L) → S_3
  195. Rotate(O_3, 45°, L) → T_3
  196. Rotate(P_3, 45°, L) → U_3
  197. Rotate(Q_3, 45°, L) → V_3
  198. Polygon(R_3, S_3, T_3, U_3, V_3) → polígono2''', q_4, r_4, s_4, u_3, v_3

AI Drawing Prompt Reference

Draw a geometric construction of the iOS Logo, involving multiple circles, lines, and reflections, to demonstrate how to build the shape using bisectors, vectors, orthogonal lines, and rotations. Start by drawing base lines and bisectors, then construct key points using vectors and orthogonal vectors, generate the various parts of the shape through reflections and rotations, and finally combine them into a polygon similar to the iOS Logo, adding arcs to match the logo's form.
The prompt above was compiled from the original drawing commands and is for reference only.