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单交点簇状圆（Veu110）

单交点簇状圆（Veu110）
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2019-11-16 05:36
圆角变换几何作图轨迹
① 本范例演示了一组相互簇拥的圆，它们彼此之间仅以单个交点相交，形成一种密集而有序的几何图案。② 核心数学概念涉及圆的相交性质、旋转与缩放变换、以及通过中点、距离和角度等工具构造精确的几何关系。③ 作图要点：使用GeoGebra的Circle命令创建基础圆，通过Rotate、Dilate、Midpoint、Intersect等命令生成多个旋转和缩放后的圆，并利用Segment、Angle、Line、Vector等辅助元素控制交点位置，最终形成簇状结构。④ 适合高中或大学阶段的几何教学，可用于探索圆的相交模式、变换组合以及动态几何中的参数化设计。

GeoGebra 命令及参数

  1. Circle(A, 2.613126) → c
  2. Rotate(B, 45°, A) → B'
  3. Rotate(B', 45°, A) → B''
  4. Rotate(B'', 45°, A) → B'''
  5. Rotate(B''', 45°, A) → C
  6. Rotate(C, 45°, A) → C'
  7. Rotate(C', 45°, A) → C''
  8. Rotate(C'', 45°, A) → C'''
  9. Circle(B', 1) → d
  10. Circle(B'', 1) → e
  11. Circle(B''', 1) → f
  12. Circle(C, 1) → g
  13. Circle(C', 1) → h
  14. Circle(C'', 1) → k
  15. Circle(C''', 1) → p
  16. Circle(B, 1) → q
  17. Circle(A, 1.613126) → r
  18. Segment(B, B') → i
  19. Segment(B', B'') → j
  20. Segment(B'', B''') → l
  21. Segment(B''', C) → m
  22. Segment(C, C') → n
  23. Segment(C', C'') → s
  24. Segment(C'', C''') → t
  25. Segment(C''', B) → a
  26. Segment(A, B') → b
  27. Segment(A, B) → f_1
  28. Segment(A, B'') → g_1
  29. Segment(A, B''') → h_1
  30. Segment(A, C) → i_1
  31. Segment(A, C') → j_1
  32. Segment(A, C'') → k_1
  33. Segment(C''', A) → l_1
  34. Angle(B, A, B') → α
  35. Midpoint(i) → D
  36. Segment(A, E) → m_1
  37. Dilate(A, 1.717438935, A) → A'
  38. Dilate(E, 1.717438935, A) → E'
  39. Segment(A', E') → n_1
  40. Circle(E', 1) → c_1
  41. Intersect(c_1, d) → F, G, ,
  42. Rotate(c_1, 45°, A) → d_1
  43. Rotate(d_1, 45°, A) → e_1
  44. Rotate(e_1, 45°, A) → p_1
  45. Rotate(p_1, 45°, A) → q_1
  46. Rotate(q_1, 45°, A) → r_1
  47. Rotate(r_1, 45°, A) → s_1
  48. Rotate(s_1, 45°, A) → t_1
  49. Segment(B, B'') → a_1
  50. Midpoint(a_1) → H
  51. Distance(H, B) → afstandHB
  52. Segment(H, B) → b_1
  53. Rotate(H, 90°, H) → H'
  54. Rotate(B, 90°, H) → B'_1
  55. Segment(H', B'_1) → f_2
  56. Dilate(H', 1.732, H) → H''
  57. Dilate(B'_1, 1.732, H) → I
  58. Segment(H'', I) → g_2
  59. Circle(I, 1) → c_2
  60. Rotate(c_2, 45°, A) → d_2
  61. Rotate(d_2, 45°, A) → e_2
  62. Rotate(e_2, 45°, A) → h_2
  63. Rotate(h_2, 45°, A) → k_2
  64. Rotate(k_2, 45°, A) → p_2
  65. Rotate(p_2, 45°, A) → q_2
  66. Rotate(q_2, 45°, A) → r_2
  67. Intersect(c_2, c_1) → N, O, ,
  68. Circle(A, K) → s_2
  69. Circle(A, 4.132155682847749) → t_2
  70. Circle(A, L) → c_3
  71. Circle(P, M) → d_3
  72. Intersect(d_1, d_3) → S, T, ,
  73. Rotate(d_3, 45°, A) → e_3
  74. Rotate(e_3, 45°, A) → f_3
  75. Rotate(f_3, 45°, A) → g_3
  76. Rotate(g_3, 45°, A) → h_3
  77. Rotate(h_3, 45°, A) → k_3
  78. Rotate(k_3, 45°, A) → p_3
  79. Rotate(p_3, 45°, A) → q_3
  80. Intersect(e_3, d_1) → U, V, ,
  81. Intersect(e_1, c_3, 1) → W
  82. Intersect(e_3, e_1) → Z, A_1, ,
  83. Intersect(f_3, e_1) → B_1, C_1, ,
  84. Intersect(f_3, p_1) → D_1, E_1, ,
  85. Intersect(d_1, d) → F_1, G_1, ,
  86. Intersect(e, d_1) → H_1, I_1, ,
  87. Intersect(e_1, e) → J_1, K_1, ,
  88. Intersect(e_1, f) → L_1, M_1, ,
  89. Intersect(f, p_1) → N_1, O_1, ,
  90. Intersect(p_1, g_3) → P_1, Q_1, ,
  91. Intersect(g_3, q_1) → R_1, S_1, ,
  92. Intersect(g, f) → T_1, , ,
  93. Intersect(g, p_1) → U_1, V_1, ,
  94. Intersect(q_1, h) → W_1, Z_1, ,
  95. Intersect(q_1, h_3) → A_2, B_2, ,
  96. Intersect(h_3, r_1) → C_2, D_2, ,
  97. Intersect(h, r_1) → E_2, F_2, ,
  98. Intersect(k_3, r_1) → G_2, H_2, ,
  99. Intersect(s_1, k_3) → I_2, J_2, ,
  100. Intersect(p_3, s_1) → K_2, L_2, ,
  101. Intersect(p_3, t_1) → M_2, N_2, ,
  102. Intersect(q_3, c_1) → O_2, P_2, ,
  103. Intersect(q_3, t_1) → Q_2, R_2, ,
  104. Intersect(p, t_1) → S_2, T_2, ,
  105. Intersect(s_1, p) → U_2, V_2, ,
  106. Intersect(k, s_1) → W_2, Z_2, ,
  107. Intersect(e_3, s_2) → A_3, B_3, ,
  108. Intersect(f_3, s_2) → C_3, D_3, ,
  109. Intersect(d_3, s_2) → E_3, F_3, ,
  110. Intersect(g_3, s_2) → G_3, H_3, ,
  111. Intersect(h_3, s_2) → I_3, J_3, ,
  112. Intersect(k_3, s_2) → K_3, L_3, ,
  113. Intersect(p_3, s_2) → M_3, N_3, ,
  114. Intersect(q_3, s_2) → O_3, P_3, ,
  115. Line(C', B') → i_2
  116. Line(C''', B''') → j_2
  117. Line(B, C) → l_2
  118. Line(B'', C'') → m_2
  119. Intersect(k, r_1) → Q_3, R_3, ,
  120. Intersect(g, q_1) → S_3, T_3, ,
  121. Intersect(q, c_1) → U_3, V_3, ,
  122. Intersect(t_1, q) → W_3, Z_3, ,
  123. Circle(A_4, 1) → r_3
  124. Circle(P, 2.0144333173267204) → s_3
  125. Intersect(d, r_3) → B_4, C_4, ,
  126. Intersect(r, j_2) → D_4, E_4
  127. Intersect(r, i_2) → F_4, G_4
  128. Intersect(r, l_2) → H_4, I_4
  129. Intersect(r, m_2) → J_4, K_4
  130. Circle(A, 0.5) → t_3
  131. Circle(L_4, 1.8849655) → c_4
  132. Intersect(c_4, q_3) → M_4, H_5, ,
  133. Intersect(c_4, d_3) → N_4, I_5, ,
  134. Intersect(c_4, c_3) → O_4, , ,
  135. Rotate(c_4, 45°, A) → d_4
  136. Rotate(d_4, 45°, A) → e_4
  137. Rotate(e_4, 45°, A) → f_4
  138. Rotate(f_4, 45°, A) → g_4
  139. Rotate(g_4, 45°, A) → h_4
  140. Rotate(h_4, 45°, A) → k_4
  141. Rotate(k_4, 45°, A) → p_4
  142. Intersect(d_4, d_3) → P_4, J_5, ,
  143. Intersect(d_4, e_3) → Q_4, K_5, ,
  144. Intersect(e_4, e_3) → R_4, L_5, ,
  145. Intersect(e_4, f_3) → S_4, M_5, ,
  146. Intersect(e_4, c_3) → T_4, , ,
  147. Intersect(c_3, d_4) → U_4, , ,
  148. Intersect(f_4, f_3) → V_4, N_5, ,
  149. Intersect(f_4, g_3) → W_4, O_5, ,
  150. Intersect(g_4, g_3) → Z_4, P_5, ,
  151. Intersect(g_4, h_3) → A_5, Q_5, ,
  152. Intersect(h_4, h_3) → B_5, R_5, ,
  153. Intersect(k_3, h_4) → C_5, S_5, ,
  154. Intersect(k_4, k_3) → D_5, T_5, ,
  155. Intersect(p_4, p_3) → E_5, U_5, ,
  156. Intersect(k_4, p_3) → F_5, V_5, ,
  157. Intersect(p_4, q_3) → G_5, W_5, ,
  158. Intersect(f_4, c_3) → Z_5, , ,
  159. Intersect(g_4, c_3) → A_6, , ,
  160. Intersect(h_4, c_3) → B_6, , ,
  161. Intersect(k_4, c_3) → C_6, , ,
  162. Intersect(p_4, c_3) → D_6, , ,
  163. Intersect(d, e) → E_6, , ,
  164. Intersect(e, f) → F_6, , ,
  165. Intersect(q, p) → G_6, , ,
  166. Intersect(k, h) → H_6, , ,
  167. Intersect(q, d) → I_6, , ,
  168. Intersect(k, p) → J_6, , ,
  169. Intersect(g, h) → K_6, , ,
  170. Rotate(f, 45°, B''') → f'
  171. Intersect(f, f', 1) → L_6
  172. Rotate(f', 45°, A) → f''
  173. Intersect(f, f', 1) → M_6
  174. Intersect(f, f', 1) → N_6
  175. Intersect(f, f', 1) → O_6
  176. Intersect(f'', f) → P_6, , ,
  177. Circle(S, U, Z) → q_4
  178. Circle(N_1, J_1, F_1) → r_4
  179. Circle(P_6, E_6, I_6) → s_4
  180. Circle(S_4, R_4, Q_4) → t_4
  181. Circle(R, 1.634479683) → d_5
  182. Intersect(c_4, d_5) → Q_6, R_6, ,
  183. Intersect(d_4, d_5) → S_6, T_6, ,
  184. Circle(A, L_4) → c_5
  185. Intersect(d_5, c_5) → U_6, V_6, ,
  186. Rotate(d_5, 45°, A) → e_5
  187. Rotate(e_5, 45°, A) → f_5
  188. Intersect(e_5, d_4) → W_6, Z_6, ,
  189. Intersect(e_4, e_5) → A_7, B_7, ,
  190. Intersect(e_5, c_5) → C_7, D_7, ,
  191. Intersect(f_5, c_5) → G_7, H_7, ,
  192. Intersect(f_5, e_4) → E_7, F_7, ,
  193. Rotate(f_5, 45°, A) → g_5
  194. Intersect(f_5, f_4) → I_7, J_7, ,
  195. Intersect(f_4, g_5) → K_7, L_7, ,
  196. Rotate(g_5, 45°, A) → h_5
  197. Rotate(h_5, 45°, A) → k_5
  198. Intersect(g_5, g_4) → M_7, N_7, ,
  199. Intersect(g_4, h_5) → O_7, P_7, ,
  200. Intersect(h_5, h_4) → Q_7, R_7, ,
  201. Intersect(h_4, k_5) → S_7, T_7, ,
  202. Intersect(k_5, k_4) → U_7, V_7, ,
  203. Rotate(k_5, 45°, A) → p_5
  204. Rotate(p_5, 45°, A) → q_5
  205. Intersect(k_5, p_5) → W_7, , ,
  206. Intersect(k_4, p_5) → Z_7, A_8, ,
  207. Intersect(p_4, p_5) → B_8, C_8, ,
  208. Intersect(p_4, q_5) → D_8, E_8, ,
  209. Intersect(q_5, c_4) → F_8, G_8, ,
  210. Intersect(c_5, g_5) → H_8, I_8, ,
  211. Intersect(c_5, h_5) → J_8, K_8, ,
  212. Intersect(c_5, p_5) → L_8, M_8, ,
  213. Intersect(c_5, k_5) → N_8, O_8, ,
  214. Intersect(q_5, c_5) → P_8, Q_8, ,
  215. Intersect(r_3, d_3) → R_8, S_8, ,
  216. Circle(F_6, E_6, I_6) → r_5
  217. Circle(A, R) → s_5
  218. Line(C_7, O_8) → n_2
  219. Line(H_8, Q_8) → a_2
  220. Circle(A, 10.48172643) → t_5
  221. Rotate(R, 45°, A) → R'
  222. Rotate(R', 45°, A) → R''
  223. Rotate(R'', 45°, A) → R'''
  224. Rotate(R''', 45°, A) → T_8
  225. Rotate(T_8, 45°, A) → U_8
  226. Rotate(U_8, 45°, A) → V_8
  227. Rotate(V_8, 45°, A) → W_8
  228. Vector(T_8, K_8) → u
  229. Dilate(u, 1.3, T_8) → u'
  230. Vector(R, U_6) → v
  231. Dilate(v, 1.3, R) → v'
  232. Vector(V_8, L_8) → w
  233. Dilate(w, 1.34, V_8) → w'
  234. Vector(R''', H_8) → u_1
  235. Vector(R'', G_7) → v_1
  236. Dilate(v_1, 1.3, R'') → w_1