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HEX 形几何逻辑构造与向量变换

HEX 形几何逻辑构造与向量变换
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2021-02-18 10:11
多边形向量变换几何作图
所属专题 GeoGebra and Laser Cutting
该范例通过“六边形/HEX”风格的几何拼接与逐步平移构造，演示如何用向量变换在平面中生成一组联动图形。核心数学概念包括：多边形与折线的几何构造、向量的方向与长度刻画、以及平移（Translate）作为刚体变换将图形规则复制到不同位置。作图要点是从下方命令提炼：使用 Point 定义关键点（如 A、B、C…N 等）、用 Polygon 或 PolyLine 形成六边形或折线骨架（poly1~poly4、poly2、poly3），再用 Segment 连接边界，用 Vector 表达平移方向，最后通过 Translate 将由平面几何对象得到的图形按向量重复生成（f_1、g_1、h_1 等在构造链中对应新位置的对象）。适合中学几何、信息化数学建模与探索活动，也可用于课堂上讲解“向量变换-图形复制”的逻辑思维与作图方法。

GeoGebra 命令及参数

  1. Point(yAxis) → A
  2. Polygon(A, B, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  3. Polygon(D, C, 6) → poly2, l, J, m, n, p, q, r, G, H, I
  4. Polygon(I, H, 6) → poly3, s, N, t, a, b, c, d, K, L, M
  5. Polygon(M, L, 6) → poly4, e, R, f_1, g_1, h_1, i_1, j_1, O, P, Q
  6. Polygon(Q, P, 6) → poly5, k_1, V, l_1, m_1, n_1, p_1, q_1, S, T, U
  7. Polygon(U, T, 6) → poly6, r_1, B_1, s_1, t_1, a_1, b_1, c_1, W, Z, A_1
  8. Polygon(A_1, Z, 6) → poly7, d_1, F_1, e_1, f_2, g_2, h_2, i_2, C_1, D_1, E_1
  9. Polygon(E, D, 6) → poly8, j_2, J_1, k_2, l_2, m_2, n_2, p_2, G_1, H_1, I_1
  10. Polygon(J, I, 6) → poly9, q_2, N_1, r_2, s_2, t_2, a_2, b_2, K_1, L_1, M_1
  11. Polygon(N, M, 6) → poly10, c_2, R_1, d_2, e_2, f_3, g_3, h_3, O_1, P_1, Q_1
  12. Polygon(R, Q, 6) → poly11, i_3, V_1, j_3, k_3, l_3, m_3, n_3, S_1, T_1, U_1
  13. Polygon(V, U, 6) → poly12, p_3, B_2, q_3, r_3, s_3, t_3, a_3, W_1, Z_1, A_2
  14. Polygon(B_1, A_1, 6) → poly13, b_3, F_2, c_3, d_3, e_3, f_4, g_4, C_2, D_2, E_2
  15. Polygon(A_2, Z_1, 6) → poly14, h_4, J_2, i_4, j_4, k_4, l_4, m_4, G_2, H_2, I_2
  16. Polygon(U_1, T_1, 6) → poly15, n_4, N_2, p_4, q_4, r_4, s_4, t_4, K_2, L_2, M_2
  17. Polygon(Q_1, P_1, 6) → poly16, a_4, R_2, b_4, c_4, d_4, e_4, f_5, O_2, P_2, Q_2
  18. Polygon(M_1, L_1, 6) → poly17, g_5, V_2, h_5, i_5, j_5, k_5, l_5, S_2, T_2, U_2
  19. Polygon(I_1, H_1, 6) → poly18, m_5, B_3, n_5, p_5, q_5, r_5, s_5, W_2, Z_2, A_3
  20. Polygon(F, E, 6) → poly19, t_5, F_3, a_5, b_5, c_5, d_5, e_5, C_3, D_3, E_3
  21. Polygon(D_3, J_1, 6) → poly20, f_6, J_3, g_6, h_6, i_6, j_6, k_6, G_3, H_3, I_3
  22. Polygon(E_3, D_3, 6) → poly21, l_6, N_3, m_6, n_6, p_6, q_6, r_6, K_3, L_3, M_3
  23. Polygon(M_3, L_3, 6) → poly22, s_6, R_3, t_6, a_6, b_6, c_6, d_6, O_3, P_3, Q_3
  24. Polygon(L_3, J_3, 6) → poly23, e_6, V_3, f_7, g_7, h_7, i_7, j_7, S_3, T_3, U_3
  25. Polygon(I_3, B_3, 6) → poly24, k_7, B_4, l_7, m_7, n_7, p_7, q_7, W_3, Z_3, A_4
  26. Polygon(A_3, V_2, 6) → poly25, r_7, F_4, s_7, t_7, a_7, b_7, c_7, C_4, D_4, E_4
  27. Polygon(U_2, R_2, 6) → poly26, d_7, J_4, e_7, f_8, g_8, h_8, i_8, G_4, H_4, I_4
  28. Polygon(Q_2, N_2, 6) → poly27, j_8, N_4, k_8, l_8, m_8, n_8, p_8, K_4, L_4, M_4
  29. Polygon(M_2, J_2, 6) → poly28, q_8, R_4, r_8, s_8, t_8, a_8, b_8, O_4, P_4, Q_4
  30. Polygon(M_4, L_4, 6) → poly29, c_8, V_4, d_8, e_8, f_9, g_9, h_9, S_4, T_4, U_4
  31. Polygon(I_4, H_4, 6) → poly30, i_9, B_5, j_9, k_9, l_9, m_9, n_9, W_4, Z_4, A_5
  32. Polygon(E_4, D_4, 6) → poly31, p_9, F_5, q_9, r_9, s_9, t_9, a_9, C_5, D_5, E_5
  33. Polygon(A_4, Z_3, 6) → poly32, b_9, J_5, c_9, d_9, e_9, f_{10}, g_{10}, G_5, H_5, I_5
  34. Polygon(U_3, T_3, 6) → poly33, h_{10}, N_5, i_{10}, j_{10}, k_{10}, l_{10}, m_{10}, K_5, L_5, M_5
  35. Polygon(P_3, O_3, 6) → poly34, n_{10}, R_5, p_{10}, q_{10}, r_{10}, s_{10}, t_{10}, O_5, P_5, Q_5
  36. Polygon(Q_3, P_3, 6) → poly35, a_{10}, V_5, b_{10}, c_{10}, d_{10}, e_{10}, f_{11}, S_5, T_5, U_5
  37. Polygon(T_5, R_5, 6) → poly36, g_{11}, B_6, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, W_5, Z_5, A_6
  38. Polygon(Q_5, N_5, 6) → poly37, m_{11}, F_6, n_{11}, p_{11}, q_{11}, r_{11}, s_{11}, C_6, D_6, E_6
  39. Polygon(M_5, J_5, 6) → poly38, t_{11}, J_6, a_{11}, b_{11}, c_{11}, d_{11}, e_{11}, G_6, H_6, I_6
  40. Polygon(I_5, F_5, 6) → poly39, f_{12}, N_6, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, K_6, L_6, M_6
  41. Polygon(E_5, B_5, 6) → poly40, l_{12}, R_6, m_{12}, n_{12}, p_{12}, q_{12}, r_{12}, O_6, P_6, Q_6
  42. Polygon(A_5, V_4, 6) → poly41, s_{12}, V_6, t_{12}, a_{12}, b_{12}, c_{12}, d_{12}, S_6, T_6, U_6
  43. Polygon(Q_6, P_6, 6) → poly42, e_{12}, B_7, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, W_6, Z_6, A_7
  44. Polygon(M_6, L_6, 6) → poly43, k_{13}, F_7, l_{13}, m_{13}, n_{13}, p_{13}, q_{13}, C_7, D_7, E_7
  45. Polygon(I_6, H_6, 6) → poly44, r_{13}, J_7, s_{13}, t_{13}, a_{13}, b_{13}, c_{13}, G_7, H_7, I_7
  46. Polygon(E_6, D_6, 6) → poly45, d_{13}, N_7, e_{13}, f_{14}, g_{14}, h_{14}, i_{14}, K_7, L_7, M_7
  47. Polygon(A_6, Z_5, 6) → poly46, j_{14}, R_7, k_{14}, l_{14}, m_{14}, n_{14}, p_{14}, O_7, P_7, Q_7
  48. Polygon(U_5, T_5, 6) → poly47, q_{14}, V_7, r_{14}, s_{14}, t_{14}, a_{14}, b_{14}, S_7, T_7, U_7
  49. Polygon(T_7, B_6, 6) → poly48, c_{14}, B_8, d_{14}, e_{14}, f_{15}, g_{15}, h_{15}, W_7, Z_7, A_8
  50. Polygon(U_7, T_7, 6) → poly49, i_{15}, F_8, j_{15}, k_{15}, l_{15}, m_{15}, n_{15}, C_8, D_8, E_8
  51. PolyLine(E_8, F_8, E_3, F_3, F, A, B, C, G, H, K, L, U_7, O, P, S, T, W, Z, C_1, D_1, E_1, F_1, V_7, D_2, E_2, H_2, I_2, P_4, Q_4, T_4, U_4, T_6, U_6, U_5, Z_6, A_7, B_7, E_7, F_7, I_7, J_7, M_7, N_7, Q_7, V_5, R_7, A_8, B_8, D_8, Q_3, R_3, M_3, N_3) → p_{15}
  52. Point(p_{15}) → G_8
  53. Point(p_{15}) → H_8
  54. Segment(G_8, H_8) → q_{15}
  55. PolyLine(U_7, T_7, L_6, Q_6, P_6, U_6, B_6, A_6, Z_5, E_6, D_6, I_6, H_6, M_6) → r_{15}
  56. Vector(U_7, U_5) → u
  57. Translate(U_7, u) → I_8
  58. Translate(T_7, u) → J_8
  59. Translate(B_6, u) → K_8
  60. Translate(A_6, u) → L_8
  61. Translate(Z_5, u) → M_8
  62. Translate(E_6, u) → N_8
  63. Translate(D_6, u) → O_8
  64. Translate(I_6, u) → P_8
  65. Translate(H_6, u) → Q_8
  66. Translate(M_6, u) → R_8
  67. Translate(L_6, u) → S_8
  68. Translate(Q_6, u) → T_8
  69. Translate(P_6, u) → U_8
  70. Translate(U_6, u) → V_8
  71. PolyLine(I_8, J_8, S_8, T_8, U_8, V_8, K_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8) → s_{15}
  72. Translate(I_8, u) → W_8
  73. Translate(J_8, u) → Z_8
  74. Translate(K_8, u) → A_9
  75. Translate(L_8, u) → B_9
  76. Translate(M_8, u) → C_9
  77. Translate(N_8, u) → D_9
  78. Translate(O_8, u) → E_9
  79. Translate(P_8, u) → F_9
  80. Translate(Q_8, u) → G_9
  81. Translate(R_8, u) → H_9
  82. Translate(S_8, u) → I_9
  83. Translate(T_8, u) → J_9
  84. Translate(U_8, u) → K_9
  85. Translate(V_8, u) → L_9
  86. PolyLine(W_8, Z_8, I_9, J_9, K_9, L_9, A_9, B_9, C_9, D_9, E_9, F_9, G_9, H_9) → t_{15}
  87. Translate(W_8, u) → M_9
  88. Translate(Z_8, u) → N_9
  89. Translate(A_9, u) → O_9
  90. Translate(B_9, u) → P_9
  91. Translate(C_9, u) → Q_9
  92. Translate(D_9, u) → R_9
  93. Translate(E_9, u) → S_9
  94. Translate(F_9, u) → T_9
  95. Translate(G_9, u) → U_9
  96. Translate(H_9, u) → V_9
  97. Translate(I_9, u) → W_9
  98. Translate(J_9, u) → Z_9
  99. Translate(K_9, u) → A_{10}
  100. Translate(L_9, u) → B_{10}
  101. PolyLine(M_9, N_9, W_9, Z_9, A_{10}, B_{10}, O_9, P_9, Q_9, R_9, S_9, T_9, U_9, V_9) → a_{15}
  102. Translate(M_9, u) → C_{10}
  103. Translate(N_9, u) → D_{10}
  104. Translate(O_9, u) → E_{10}
  105. Translate(P_9, u) → F_{10}
  106. Translate(Q_9, u) → G_{10}
  107. Translate(R_9, u) → H_{10}
  108. Translate(S_9, u) → I_{10}
  109. Translate(T_9, u) → J_{10}
  110. Translate(U_9, u) → K_{10}
  111. Translate(V_9, u) → L_{10}
  112. Translate(W_9, u) → M_{10}
  113. Translate(Z_9, u) → N_{10}
  114. Translate(A_{10}, u) → O_{10}
  115. Translate(B_{10}, u) → P_{10}
  116. PolyLine(C_{10}, D_{10}, M_{10}, N_{10}, O_{10}, P_{10}, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, J_{10}, K_{10}, L_{10}) → b_{15}
  117. Translate(C_{10}, u) → Q_{10}
  118. Translate(D_{10}, u) → R_{10}
  119. Translate(E_{10}, u) → S_{10}
  120. Translate(F_{10}, u) → T_{10}
  121. Translate(G_{10}, u) → U_{10}
  122. Translate(H_{10}, u) → V_{10}
  123. Translate(I_{10}, u) → W_{10}
  124. Translate(J_{10}, u) → Z_{10}
  125. Translate(K_{10}, u) → A_{11}
  126. Translate(L_{10}, u) → B_{11}
  127. Translate(M_{10}, u) → C_{11}
  128. Translate(N_{10}, u) → D_{11}
  129. Translate(O_{10}, u) → E_{11}
  130. Translate(P_{10}, u) → F_{11}
  131. PolyLine(Q_{10}, R_{10}, C_{11}, D_{11}, E_{11}, F_{11}, S_{10}, T_{10}, U_{10}, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}) → c_{15}
  132. PolyLine(B_8, T_7, E_{10}, R_{10}, S_{10}, C, B_6, J_8, K_8, Z_8, A_9, N_9, O_9, D_{10}) → d_{15}
  133. Vector(N_7, J_7) → v
  134. Translate(B_8, v) → G_{11}
  135. Translate(T_7, v) → H_{11}
  136. Translate(B_6, v) → I_{11}
  137. Translate(J_8, v) → J_{11}
  138. Translate(K_8, v) → K_{11}
  139. Translate(Z_8, v) → L_{11}
  140. Translate(A_9, v) → M_{11}
  141. Translate(N_9, v) → N_{11}
  142. Translate(O_9, v) → O_{11}
  143. Translate(D_{10}, v) → P_{11}
  144. Translate(E_{10}, v) → Q_{11}
  145. Translate(R_{10}, v) → R_{11}
  146. Translate(S_{10}, v) → S_{11}
  147. Translate(C, v) → C'
  148. PolyLine(G_{11}, H_{11}, Q_{11}, R_{11}, S_{11}, C', I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}) → e_{15}
  149. Translate(G_{11}, v) → T_{11}
  150. Translate(H_{11}, v) → U_{11}
  151. Translate(I_{11}, v) → V_{11}
  152. Translate(J_{11}, v) → W_{11}
  153. Translate(K_{11}, v) → Z_{11}
  154. Translate(L_{11}, v) → A_{12}
  155. Translate(M_{11}, v) → B_{12}
  156. Translate(N_{11}, v) → C_{12}
  157. Translate(O_{11}, v) → D_{12}
  158. Translate(P_{11}, v) → E_{12}
  159. Translate(Q_{11}, v) → F_{12}
  160. Translate(R_{11}, v) → G_{12}
  161. Translate(S_{11}, v) → H_{12}
  162. Translate(C', v) → C''
  163. PolyLine(T_{11}, U_{11}, F_{12}, G_{12}, H_{12}, C'', V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, D_{12}, E_{12}) → f_{16}
  164. Translate(T_{11}, v) → I_{12}
  165. Translate(U_{11}, v) → J_{12}
  166. Translate(V_{11}, v) → K_{12}
  167. Translate(W_{11}, v) → L_{12}
  168. Translate(Z_{11}, v) → M_{12}
  169. Translate(A_{12}, v) → N_{12}
  170. Translate(B_{12}, v) → O_{12}
  171. Translate(C_{12}, v) → P_{12}
  172. Translate(D_{12}, v) → Q_{12}
  173. Translate(E_{12}, v) → R_{12}
  174. Translate(F_{12}, v) → S_{12}
  175. Translate(G_{12}, v) → T_{12}
  176. Translate(H_{12}, v) → U_{12}
  177. Translate(C'', v) → C'''
  178. PolyLine(I_{12}, J_{12}, S_{12}, T_{12}, U_{12}, C''', K_{12}, L_{12}, M_{12}, N_{12}, O_{12}, P_{12}, Q_{12}, R_{12}) → g_{16}
  179. Translate(I_{12}, v) → V_{12}
  180. Translate(J_{12}, v) → W_{12}
  181. Translate(K_{12}, v) → Z_{12}
  182. Translate(L_{12}, v) → A_{13}
  183. Translate(M_{12}, v) → B_{13}
  184. Translate(N_{12}, v) → C_{13}
  185. Translate(O_{12}, v) → D_{13}
  186. Translate(P_{12}, v) → E_{13}
  187. Translate(Q_{12}, v) → F_{13}
  188. Translate(R_{12}, v) → G_{13}
  189. Translate(S_{12}, v) → H_{13}
  190. Translate(T_{12}, v) → I_{13}
  191. Translate(U_{12}, v) → J_{13}
  192. Translate(C''', v) → K_{13}
  193. PolyLine(V_{12}, W_{12}, H_{13}, I_{13}, J_{13}, K_{13}, Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}, G_{13}) → h_{16}
  194. Translate(V_{12}, v) → L_{13}
  195. Translate(W_{12}, v) → M_{13}
  196. Translate(Z_{12}, v) → N_{13}
  197. Translate(A_{13}, v) → O_{13}
  198. Translate(B_{13}, v) → P_{13}
  199. Translate(C_{13}, v) → Q_{13}
  200. Translate(D_{13}, v) → R_{13}
  201. Translate(E_{13}, v) → S_{13}
  202. Translate(F_{13}, v) → T_{13}
  203. Translate(G_{13}, v) → U_{13}
  204. Translate(H_{13}, v) → V_{13}
  205. Translate(I_{13}, v) → W_{13}
  206. Translate(J_{13}, v) → Z_{13}
  207. Translate(K_{13}, v) → A_{14}
  208. PolyLine(L_{13}, M_{13}, V_{13}, W_{13}, Z_{13}, A_{14}, N_{13}, O_{13}, P_{13}, Q_{13}, R_{13}, S_{13}, T_{13}, U_{13}) → i_{16}

AI 绘图提示语参考

绘制一个由多个正六边形紧密拼接而成的平面几何图案，整体呈蜂窝状结构。从底部中心的一个正六边形开始，依次向上和向两侧逐层添加相邻的正六边形，每个新六边形与已有六边形共享一条边，形成连续扩展的网格。所有六边形大小一致，方向相同，边与边对齐，整体覆盖一个较大的矩形区域。图形中仅包含六边形的边线，不填充颜色，保持线条清晰，便于观察几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。