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环形时钟式对称性的几何作图演示

环形时钟式对称性的几何作图演示
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2021-01-01 09:06
测试改名0多边形变换几何作图
该范例通过“时钟式”环绕排列的图形，展示对称变换如何把点、线与三角形映射到等距离位置，从而形成重复的多边形与相似结构。核心数学概念包括反射（Mirror）、几何作图中的交点与构型（Intersect/Point/Line/Polygon）以及由对称性带来的相似三角形思想（Isosceles Triangles、Similar Triangles 的几何关系）。作图要点是：先用 Point、Line 与 Polygon 构造基础三角形与多边形 poly1、poly2，并通过 Intersect 生成关键交点；随后利用 Mirror 将相关对象（如 D、E、F 及其对应副本 D'、E'、F'，以及由此得到的 poly1'、poly2' 与关联点/线）进行镜像映射；再通过步骤变量/状态点（如 d,e 与 d',e'、g,h 与 g',h' 以及相应的折返集合）把一组对称块拼接成连续的环形结构。适合中学阶段用于几何变换、对称与相似的综合复习，也适合作为课堂交互演示或竞赛建模中的几何构造教学。

GeoGebra 命令及参数

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

AI 绘图提示语参考

绘制一个以坐标轴交点为起点、横轴上取一点连成两条相交直线的几何构图。先构造一个三角形，然后依次以这两条直线为对称轴，对三角形及其镜像进行反复反射，生成多组对称的三角形副本。每次反射后，将新得到的三个顶点连成新的三角形，并继续对它们进行交替反射，逐步形成围绕中心点呈钟表式对称排列的多个三角形图案。整体呈现由中心向外扩展的对称多边形结构，突出反射变换产生的重复几何形态。
以上提示语为根据原始绘图命令整理而来，仅供参考。