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重复线对称的几何构造：用镜像生成链式图形

重复线对称的几何构造：用镜像生成链式图形
几何画板及绘图内容加载中，请稍等……
2021-01-01 01:47
圆多边形向量变换几何作图
该范例演示如何在同一平面内，沿多条直线进行“重复镜像”，从而把初始几何对象不断映射生成一串具有对称规律的点、线与多边形。核心数学概念包括：轴对称（Reflection/Mirror）、对称变换下的不变量（距离与角保持）、相交关系（Intersection）以及由圆与多边形构成的构造几何。作图要点是使用 GeoGebra 的 Mirror 进行多次镜像生成：先定义若干基础对象（Point、Line、Circle、Polygon），再通过 Intersect 计算镜像后直线与圆（或线与线）得到的新交点，从而逐步形成对象集合（如多边形 poly1/poly2 及其多次镜像版本 poly1'、poly1'' 等），最终叠加得到重复对称链。适合中学高年级到大学的几何构造课与竞赛训练，也适合课堂演示“变换—构造—验证对称性”的教学环节。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Circle(A, B) → c
  4. Line(A, C) → a
  5. Line(A, D) → b
  6. Polygon(E, F, G) → poly1, g, e, f
  7. Mirror(E, a) → E'
  8. Mirror(F, a) → F'
  9. Mirror(G, a) → G'
  10. Polygon(E', F', G') → poly1', g', e', f'
  11. Mirror(E', b) → E''
  12. Mirror(F', b) → F''
  13. Mirror(G', b) → G''
  14. Polygon(E'', F'', G'') → poly1'', g'', e'', f''
  15. Mirror(E'', a) → E'''
  16. Mirror(F'', a) → F'''
  17. Mirror(G'', a) → G'''
  18. Polygon(E''', F''', G''') → poly1''', g''', e''', f'''
  19. Mirror(E''', b) → H
  20. Mirror(F''', b) → I
  21. Mirror(G''', b) → J
  22. Polygon(H, I, J) → poly2, j, h, i
  23. Mirror(H, a) → H'
  24. Mirror(I, a) → I'
  25. Mirror(J, a) → J'
  26. Polygon(H', I', J') → poly2', j', h', i'
  27. Mirror(H', b) → H''
  28. Mirror(I', b) → I''
  29. Mirror(J', b) → J''
  30. Polygon(H'', I'', J'') → poly2'', j'', h'', i''
  31. Mirror(H'', a) → H'''
  32. Mirror(I'', a) → I'''
  33. Mirror(J'', a) → J'''
  34. Polygon(H''', I''', J''') → poly2''', j''', h''', i'''
  35. Mirror(E, b) → E'_1
  36. Mirror(F, b) → F'_1
  37. Mirror(G, b) → G'_1
  38. Polygon(E'_1, F'_1, G'_1) → poly1'_1, g'_1, e'_1, f'_1
  39. Mirror(H''', b) → K
  40. Mirror(I''', b) → L
  41. Mirror(J''', b) → M
  42. Polygon(K, L, M) → poly3, m, k, l
  43. Mirror(E'_1, a) → N
  44. Mirror(F'_1, a) → O
  45. Mirror(G'_1, a) → P
  46. Polygon(N, O, P) → poly4, p, n, o
  47. Mirror(K, a) → K'
  48. Mirror(L, a) → L'
  49. Mirror(M, a) → M'
  50. Polygon(K', L', M') → poly3', m', k', l'
  51. Mirror(K', b) → K''
  52. Mirror(L', b) → L''
  53. Mirror(M', b) → M''
  54. Polygon(K'', L'', M'') → poly3'', m'', k'', l''
  55. Mirror(N, b) → N'
  56. Mirror(O, b) → O'
  57. Mirror(P, b) → P'
  58. Polygon(N', O', P') → poly4', p', n', o'
  59. Mirror(K'', a) → K'''
  60. Mirror(L'', a) → L'''
  61. Mirror(M'', a) → M'''
  62. Polygon(K''', L''', M''') → poly3''', m''', k''', l'''
  63. Mirror(N', a) → N''
  64. Mirror(O', a) → O''
  65. Mirror(P', a) → P''
  66. Polygon(N'', O'', P'') → poly4'', p'', n'', o''
  67. Mirror(K''', b) → Q
  68. Mirror(L''', b) → R
  69. Mirror(M''', b) → S
  70. Polygon(Q, R, S) → poly5, s, q, r
  71. Mirror(N'', b) → N'''
  72. Mirror(O'', b) → O'''
  73. Mirror(P'', b) → P'''
  74. Polygon(N''', O''', P''') → poly4''', p''', n''', o'''
  75. Mirror(Q, a) → Q'
  76. Mirror(R, a) → R'
  77. Mirror(S, a) → S'
  78. Polygon(Q', R', S') → poly5', s', q', r'
  79. Mirror(N''', a) → T
  80. Mirror(O''', a) → U
  81. Mirror(P''', a) → V
  82. Polygon(T, U, V) → poly6, v, t, u
  83. Mirror(T, b) → T'
  84. Mirror(U, b) → U'
  85. Mirror(V, b) → V'
  86. Polygon(T', U', V') → poly6', v', t', u'
  87. Mirror(Q', b) → Q''
  88. Mirror(R', b) → R''
  89. Mirror(S', b) → S''
  90. Polygon(Q'', R'', S'') → poly5'', s'', q'', r''
  91. Mirror(T', a) → T''
  92. Mirror(U', a) → U''
  93. Mirror(V', a) → V''
  94. Polygon(T'', U'', V'') → poly6'', v'', t'', u''
  95. Mirror(Q'', a) → Q'''
  96. Mirror(R'', a) → R'''
  97. Mirror(S'', a) → S'''
  98. Polygon(Q''', R''', S''') → poly5''', s''', q''', r'''
  99. Mirror(T'', b) → T'''
  100. Mirror(U'', b) → U'''
  101. Mirror(V'', b) → V'''
  102. Polygon(T''', U''', V''') → poly6''', v''', t''', u'''
  103. Mirror(T''', a) → W
  104. Mirror(U''', a) → Z
  105. Mirror(V''', a) → A_1
  106. Polygon(W, Z, A_1) → poly7, a_1, w, z
  107. Mirror(Q''', b) → B_1
  108. Mirror(R''', b) → C_1
  109. Mirror(S''', b) → D_1
  110. Polygon(B_1, C_1, D_1) → poly8, d_1, b_1, c_1
  111. Mirror(B_1, a) → E_1
  112. Mirror(C_1, a) → F_1
  113. Mirror(D_1, a) → G_1
  114. Polygon(E_1, F_1, G_1) → poly8', g_1, e_1, f_1
  115. Mirror(E_1, b) → H_1
  116. Mirror(F_1, b) → I_1
  117. Mirror(G_1, b) → J_1
  118. Polygon(H_1, I_1, J_1) → poly8'', j_1, h_1, i_1
  119. Mirror(H_1, a) → K_1
  120. Mirror(I_1, a) → L_1
  121. Mirror(J_1, a) → M_1
  122. Polygon(K_1, L_1, M_1) → poly8''', m_1, k_1, l_1
  123. Mirror(K_1, b) → N_1
  124. Mirror(L_1, b) → O_1
  125. Mirror(M_1, b) → P_1
  126. Polygon(N_1, O_1, P_1) → poly9, p_1, n_1, o_1
  127. Mirror(N_1, a) → Q_1
  128. Mirror(O_1, a) → R_1
  129. Mirror(P_1, a) → S_1
  130. Polygon(Q_1, R_1, S_1) → poly9', s_1, q_1, r_1
  131. Mirror(Q_1, b) → T_1
  132. Mirror(R_1, b) → U_1
  133. Mirror(S_1, b) → V_1
  134. Polygon(T_1, U_1, V_1) → poly9'', v_1, t_1, u_1
  135. Mirror(W, b) → W'
  136. Mirror(Z, b) → Z'
  137. Mirror(A_1, b) → W_1
  138. Polygon(W', Z', W_1) → poly7', w_1, w', z'
  139. Mirror(T_1, a) → Z_1
  140. Mirror(U_1, a) → A_2
  141. Mirror(V_1, a) → B_2
  142. Polygon(Z_1, A_2, B_2) → poly9''', b_2, z_1, a_2
  143. Mirror(W', a) → W''
  144. Mirror(Z', a) → Z''
  145. Mirror(W_1, a) → C_2
  146. Polygon(W'', Z'', C_2) → poly7'', c_2, w'', z''
  147. Mirror(W'', b) → W'''
  148. Mirror(Z'', b) → Z'''
  149. Mirror(C_2, b) → D_2
  150. Polygon(W''', Z''', D_2) → poly7''', d_2, w''', z'''
  151. Mirror(Z_1, b) → E_2
  152. Mirror(A_2, b) → F_2
  153. Mirror(B_2, b) → G_2
  154. Polygon(E_2, F_2, G_2) → poly10, g_2, e_2, f_2
  155. Mirror(W''', a) → H_2
  156. Mirror(Z''', a) → I_2
  157. Mirror(D_2, a) → J_2
  158. Polygon(H_2, I_2, J_2) → poly11, j_2, h_2, i_2
  159. Mirror(E_2, a) → K_2
  160. Mirror(F_2, a) → L_2
  161. Mirror(G_2, a) → M_2
  162. Polygon(K_2, L_2, M_2) → poly10', m_2, k_2, l_2
  163. Mirror(K_2, b) → N_2
  164. Mirror(L_2, b) → O_2
  165. Mirror(M_2, b) → P_2
  166. Polygon(N_2, O_2, P_2) → poly10'', p_2, n_2, o_2
  167. Mirror(N_2, a) → Q_2
  168. Mirror(O_2, a) → R_2
  169. Mirror(P_2, a) → S_2
  170. Polygon(Q_2, R_2, S_2) → poly10''', s_2, q_2, r_2
  171. Mirror(Q_2, b) → T_2
  172. Mirror(R_2, b) → U_2
  173. Mirror(S_2, b) → V_2
  174. Polygon(T_2, U_2, V_2) → poly12, v_2, t_2, u_2
  175. Mirror(T_2, a) → W_2
  176. Mirror(U_2, a) → Z_2
  177. Mirror(V_2, a) → A_3
  178. Polygon(W_2, Z_2, A_3) → poly12', a_3, w_2, z_2
  179. Mirror(W_2, b) → B_3
  180. Mirror(Z_2, b) → C_3
  181. Mirror(A_3, b) → D_3
  182. Polygon(B_3, C_3, D_3) → poly12'', d_3, b_3, c_3
  183. Mirror(B_3, a) → E_3
  184. Mirror(C_3, a) → F_3
  185. Mirror(D_3, a) → G_3
  186. Polygon(E_3, F_3, G_3) → poly12''', g_3, e_3, f_3
  187. Mirror(E_3, b) → H_3
  188. Mirror(F_3, b) → I_3
  189. Mirror(G_3, b) → J_3
  190. Polygon(H_3, I_3, J_3) → poly13, j_3, h_3, i_3
  191. Mirror(H_3, a) → K_3
  192. Mirror(I_3, a) → L_3
  193. Mirror(J_3, a) → M_3
  194. Polygon(K_3, L_3, M_3) → poly13', m_3, k_3, l_3
  195. Mirror(K_3, b) → N_3
  196. Mirror(L_3, b) → O_3
  197. Mirror(M_3, b) → P_3
  198. Polygon(N_3, O_3, P_3) → poly13'', p_3, n_3, o_3
  199. Mirror(N_3, a) → Q_3
  200. Mirror(O_3, a) → R_3
  201. Mirror(P_3, a) → S_3
  202. Polygon(Q_3, R_3, S_3) → poly13''', s_3, q_3, r_3
  203. Mirror(Q_3, b) → T_3
  204. Mirror(R_3, b) → U_3
  205. Mirror(S_3, b) → V_3
  206. Polygon(T_3, U_3, V_3) → poly14, v_3, t_3, u_3
  207. Mirror(T_3, a) → W_3
  208. Mirror(U_3, a) → Z_3
  209. Mirror(V_3, a) → A_4
  210. Polygon(W_3, Z_3, A_4) → poly14', a_4, w_3, z_3
  211. Mirror(W_3, b) → B_4
  212. Mirror(Z_3, b) → C_4
  213. Mirror(A_4, b) → D_4
  214. Polygon(B_4, C_4, D_4) → poly14'', d_4, b_4, c_4
  215. Mirror(B_4, a) → E_4
  216. Mirror(C_4, a) → F_4
  217. Mirror(D_4, a) → G_4
  218. Polygon(E_4, F_4, G_4) → poly14''', g_4, e_4, f_4
  219. Mirror(E_4, b) → H_4
  220. Mirror(F_4, b) → I_4
  221. Mirror(G_4, b) → J_4
  222. Polygon(H_4, I_4, J_4) → poly15, j_4, h_4, i_4
  223. Mirror(H_4, a) → K_4
  224. Mirror(I_4, a) → L_4
  225. Mirror(J_4, a) → M_4
  226. Polygon(K_4, L_4, M_4) → poly15', m_4, k_4, l_4
  227. Mirror(K_4, b) → N_4
  228. Mirror(L_4, b) → O_4
  229. Mirror(M_4, b) → P_4
  230. Polygon(N_4, O_4, P_4) → poly15'', p_4, n_4, o_4
  231. Mirror(N_4, a) → Q_4
  232. Mirror(O_4, a) → R_4
  233. Mirror(P_4, a) → S_4
  234. Polygon(Q_4, R_4, S_4) → poly15''', s_4, q_4, r_4
  235. Mirror(Q_4, b) → T_4
  236. Mirror(R_4, b) → U_4
  237. Mirror(S_4, b) → V_4
  238. Polygon(T_4, U_4, V_4) → poly16, v_4, t_4, u_4
  239. Mirror(T_4, a) → W_4
  240. Mirror(U_4, a) → Z_4
  241. Mirror(V_4, a) → A_5
  242. Polygon(W_4, Z_4, A_5) → poly16', a_5, w_4, z_4
  243. Mirror(W_4, b) → B_5
  244. Mirror(Z_4, b) → C_5
  245. Mirror(A_5, b) → D_5
  246. Polygon(B_5, C_5, D_5) → poly16'', d_5, b_5, c_5
  247. Mirror(B_5, a) → E_5
  248. Mirror(C_5, a) → F_5
  249. Mirror(D_5, a) → G_5
  250. Polygon(E_5, F_5, G_5) → poly16''', g_5, e_5, f_5
  251. Mirror(E_5, b) → H_5
  252. Mirror(F_5, b) → I_5
  253. Mirror(G_5, b) → J_5
  254. Polygon(H_5, I_5, J_5) → poly17, j_5, h_5, i_5
  255. Mirror(H_5, a) → K_5
  256. Mirror(I_5, a) → L_5
  257. Mirror(J_5, a) → M_5
  258. Polygon(K_5, L_5, M_5) → poly17', m_5, k_5, l_5
  259. Mirror(K_5, b) → N_5
  260. Mirror(L_5, b) → O_5
  261. Mirror(M_5, b) → P_5
  262. Polygon(N_5, O_5, P_5) → poly17'', p_5, n_5, o_5
  263. Mirror(N_5, a) → Q_5
  264. Mirror(O_5, a) → R_5
  265. Mirror(P_5, a) → S_5
  266. Polygon(Q_5, R_5, S_5) → poly17''', s_5, q_5, r_5
  267. Mirror(Q_5, b) → T_5
  268. Mirror(R_5, b) → U_5
  269. Mirror(S_5, b) → V_5
  270. Polygon(T_5, U_5, V_5) → poly18, v_5, t_5, u_5
  271. Mirror(T_5, a) → W_5
  272. Mirror(U_5, a) → Z_5
  273. Mirror(V_5, a) → A_6
  274. Polygon(W_5, Z_5, A_6) → poly18', a_6, w_5, z_5
  275. Mirror(W_5, b) → B_6
  276. Mirror(Z_5, b) → C_6
  277. Mirror(A_6, b) → D_6
  278. Polygon(B_6, C_6, D_6) → poly18'', d_6, b_6, c_6
  279. Mirror(B_6, a) → E_6
  280. Mirror(C_6, a) → F_6
  281. Mirror(D_6, a) → G_6
  282. Polygon(E_6, F_6, G_6) → poly18''', g_6, e_6, f_6

AI 绘图提示语参考

以坐标轴交点为圆心，在纵轴上取一点作圆，并过圆心作两条相交直线作为对称轴。先画一个三角形，依次关于第一条轴、第二条轴交替镜像，每次镜像后连接对应点形成新三角形，如此反复生成一串沿两条轴交替翻转的三角形链。同时，将初始三角形关于第二条轴镜像，再关于第一条轴镜像，形成另一组三角形；继续交替镜像并连接，最终得到由多个三角形围绕两轴对称排列的完整图案，所有三角形保持全等，整体呈现重复的轴对称结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。