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战舰几何：极坐标风格的作图与相交轨迹（V2）

战舰几何：极坐标风格的作图与相交轨迹（V2）
几何画板及绘图内容加载中，请稍等……
2023-12-07 06:23
圆解析几何变换几何作图
这个范例展示了用极坐标/旋转构造思想生成“战舰”风格的几何结构：通过旋转、射线与圆弧的组合，形成若干关键点的相交与连线关系，从而呈现出具有对称与约束的图形布局。核心数学概念包括：极坐标式构形的等角旋转、圆与圆弧的交点计算、线段/射线的几何约束，以及由坐标或二次曲线相关性质引出的逻辑连接（如正交中心、二次函数/方程在背景中的几何解释）。作图要点：在 GeoGebra 中主要使用 Rotate（旋转）来复制构造方向；用 Segment（线段）与 Ray（射线）建立连接与半直线约束；用 Circle（圆）与 Intersect（求交）确定关键点；再用 CircleArc（圆弧）刻画圆周上的弧形边界，最终由多个交点与线段组合形成完整“战舰”式轮廓（如构造摘要所列的多组对象点/符号、圆弧段与交点集合）。适合中学高年级到大学预备阶段的几何建模与计算机数学可视化课程使用，也可用于探究“旋转对称—圆的交点—几何约束”之间的关系。

GeoGebra 命令及参数

  1. Rotate(G, pi / 4) → G'_1
  2. Segment(A, G'_1) → j
  3. Rotate(G, (11 * pi / 6)) → G'_5
  4. Segment(A, G'_5) → p
  5. Circle(A, F) → c_1
  6. Intersect(c_1, j, 1) → R
  7. Segment(A, R) → b
  8. Rotate(G, (2 * pi / 3)) → G'_4
  9. Circle(A, B) → q
  10. Segment(A, G'_4) → n
  11. Intersect(q, n, 1) → T
  12. CircleArc(A, B, T) → h_1
  13. Ray((0, 0), (1, 0)) → f
  14. Rotate(G, pi / 6) → G'
  15. Rotate(G, pi / 3) → G'_2
  16. Rotate(G, pi / 2) → G'_3
  17. Segment(A, G') → i
  18. Segment(A, G'_2) → l
  19. Segment(A, G'_3) → m
  20. Rotate(G, (3 * pi / 4)) → G'_{6}
  21. Rotate(G, (5 * pi / 6)) → G'_{7}
  22. Rotate(G, pi) → G'_{8}
  23. Rotate(G, (7 * pi / 6)) → G'_{9}
  24. Rotate(G, (5 * pi / 4)) → G'_{10}
  25. Rotate(G, (4 * pi / 3)) → G'_{11}
  26. Rotate(G, (3 * pi / 2)) → G'_{12}
  27. Rotate(G, (5 * pi / 3)) → G'_{13}
  28. Rotate(G, (7 * pi / 4)) → G'_{14}
  29. Segment(A, G'_{6}) → a
  30. Segment(A, G'_{7}) → f_1
  31. Segment(A, G'_{8}) → g_1
  32. Segment(A, G'_{9}) → i_1
  33. Segment(A, G'_{10}) → j_1
  34. Segment(A, G'_{11}) → k_1
  35. Segment(A, G'_{12}) → l_1
  36. Segment(A, G'_{13}) → m_1
  37. Segment(A, G'_{14}) → n_1
  38. Circle(A, G) → d_1
  39. Circle(A, E) → t
  40. Circle(A, D) → s
  41. Circle(A, C) → r
  42. Intersect(r, i, 1) → H
  43. Intersect(s, i) → I,
  44. Intersect(t, i, 1) → J
  45. Intersect(c_1, i, 1) → K
  46. Intersect(d_1, i, 1) → M
  47. Intersect(d_1, j, 1) → N
  48. Intersect(c_1, j, 1) → O
  49. Intersect(t, j, 1) → P
  50. Intersect(s, j, 1) → Q
  51. Intersect(r, j, 1) → S
  52. Intersect(r, l, 1) → U
  53. Intersect(s, l, 1) → V
  54. Intersect(t, l, 1) → W
  55. Intersect(c_1, l, 1) → Z
  56. Intersect(d_1, l) → A_1,
  57. Intersect(s, m, 1) → C_1
  58. Intersect(r, m, 1) → D_1
  59. Intersect(t, m, 1) → E_1
  60. Intersect(c_1, m, 1) → F_1
  61. Intersect(d_1, m, 1) → G_1
  62. Intersect(r, n, 1) → I_1
  63. Intersect(s, n, 1) → J_1
  64. Intersect(t, n, 1) → K_1
  65. Intersect(c_1, n, 1) → L_1
  66. Intersect(d_1, n, 1) → M_1
  67. Intersect(r, a, 1) → O_1
  68. Intersect(r, f_1, 1) → R_1
  69. Intersect(s, f_1, 1) → S_1
  70. Intersect(t, f_1, 1) → T_1
  71. Intersect(c_1, f_1, 1) → U_1
  72. Intersect(d_1, f_1, 1) → V_1
  73. Intersect(d_1, a) → W_1,
  74. Intersect(c_1, a, 1) → Z_1
  75. Intersect(t, a, 1) → A_2
  76. Intersect(s, a, 1) → B_2
  77. Intersect(r, g_1) → E_2,
  78. Intersect(s, g_1, 1) → F_2
  79. Intersect(t, g_1) → G_2,
  80. Intersect(c_1, g_1, 1) → H_2
  81. Intersect(d_1, g_1, 1) → I_2
  82. Intersect(d_1, i_1, 1) → J_2
  83. Intersect(c_1, i_1, 1) → K_2
  84. Intersect(t, i_1, 1) → L_2
  85. Intersect(s, i_1, 1) → M_2
  86. Intersect(r, i_1, 1) → N_2
  87. Segment(A, H) → p_1
  88. Segment(A, S) → q_1
  89. Segment(A, D_1) → r_1
  90. Segment(A, I_1) → s_1
  91. Segment(A, O_1) → t_1
  92. Segment(A, R_1) → a_1
  93. Segment(A, E_2) → b_1
  94. Segment(A, N_2) → e_1
  95. Intersect(r, l_1, 1) → B_1
  96. Intersect(s, l_1, 1) → H_1
  97. Intersect(t, l_1) → N_1,
  98. Intersect(c_1, l_1, 1) → P_1
  99. Intersect(d_1, l_1, 1) → Q_1
  100. Intersect(d_1, m_1, 1) → C_2
  101. Intersect(c_1, m_1, 1) → D_2
  102. Intersect(t, m_1, 1) → O_2
  103. Intersect(s, m_1, 1) → P_2
  104. Intersect(r, m_1, 1) → Q_2
  105. Intersect(r, n_1) → R_2,
  106. Intersect(r, k_1, 1) → S_2
  107. Intersect(s, k_1, 1) → T_2
  108. Intersect(t, k_1, 1) → U_2
  109. Intersect(c_1, k_1, 1) → V_2
  110. Intersect(d_1, k_1, 1) → W_2
  111. Intersect(d_1, j_1, 1) → Z_2
  112. Intersect(c_1, j_1, 1) → A_3
  113. Intersect(t, j_1, 1) → B_3
  114. Intersect(s, j_1, 1) → C_3
  115. Intersect(r, j_1, 1) → D_3
  116. Intersect(r, p) → E_3,
  117. Intersect(s, p, 1) → F_3
  118. Intersect(t, p, 1) → G_3
  119. Intersect(c_1, p, 1) → H_3
  120. Intersect(d_1, p, 1) → I_3
  121. Intersect(r, f, 1) → J_3
  122. Intersect(s, f, 1) → K_3
  123. Intersect(t, f, 1) → L_3
  124. Intersect(c_1, f) → M_3,
  125. Intersect(d_1, f, 1) → N_3
  126. Segment(A, J_3) → f_2
  127. Segment(A, H) → g_2
  128. Segment(A, S) → h_2
  129. Segment(A, U) → i_2
  130. Segment(A, D_1) → j_2
  131. Segment(A, I_1) → k_2
  132. Segment(A, O_1) → l_2
  133. Segment(A, R_1) → m_2
  134. Segment(A, E_2) → n_2
  135. Segment(A, N_2) → p_2
  136. Segment(A, D_3) → q_2
  137. Segment(A, S_2) → r_2
  138. Segment(A, B_1) → s_2
  139. Segment(A, Q_2) → t_2
  140. Segment(A, R_2) → a_2
  141. Segment(A, E_3) → b_2
  142. Segment(E_3, I_3) → c_2
  143. Segment(J_3, N_3) → d_2
  144. Segment(H, M) → e_2
  145. Segment(S, N) → f_3
  146. Segment(U, A_1) → g_3
  147. Segment(D_1, G_1) → h_3
  148. Segment(I_1, M_1) → i_3
  149. Segment(O_1, W_1) → j_3
  150. Segment(R_1, V_1) → k_3
  151. Segment(E_2, I_2) → l_3
  152. Segment(N_2, J_2) → m_3
  153. Segment(D_3, Z_2) → n_3
  154. Segment(S_2, W_2) → p_3
  155. Segment(B_1, Q_1) → q_3
  156. Segment(Q_2, C_2) → r_3
  157. Segment(R_2, G'_{14}) → s_3
  158. Intersect(s, s_3, 1) → O_3
  159. Intersect(t, s_3, 1) → P_3
  160. Intersect(c_1, s_3, 1) → Q_3

AI 绘图提示语参考

该图形由一系列通过旋转生成的点构成，这些点与原点相连形成线段。多个圆以原点为圆心，半径不同，圆弧用于勾勒边界。圆与线段、圆弧的交点定义关键点，形成多个对称的交点区域。图形整体呈现旋转对称性，线段、圆和圆弧相互连接，交点作为结构节点，最终构成一个复杂而对称的几何图案。
以上提示语为根据原始绘图命令整理而来，仅供参考。