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圆的面积：用几何变换与分割近似/展示

圆的面积：用几何变换与分割近似/展示
几何画板及绘图内容加载中，请稍等……
2023-03-13 04:12
圆向量积分变换几何作图
所属专题 geometriadinamica.es
该范例演示如何从圆及其构造元素出发，直观展示圆的面积与相关测度关系。核心数学概念包括：圆与半径、面积度量的几何解释，以及利用几何构造与变换（如旋转、镜像）生成对称结构来组织面积分割或近似。作图要点：通过 GeoGebra 的 Midpoint 取中点，Circle 绘制圆，Rotate 与 Mirror 实现旋转与镜像对称，Line 与 Polygon 连接并形成多边形分割区域；借助 Vector 标注与刻画方向或度量，使用 If 条件语句在不同情形下控制构造结果，必要时用对象如 α、A、P_0 等控制角度与点的相对位置。对象层面围绕多个点（如 P_1…P_7、A_1…A_8、E_1…E_8）构建对称分段图形，从而帮助理解圆面积的几何来源。适合中学到大学基础几何与积分思想的教学演示，也可用于课堂探究、作业讲评或自学几何作图与面积度量。

GeoGebra 命令及参数

  1. Midpoint(N_1, O_1) → V_1
  2. Midpoint(L_1, M_1) → S_1
  3. Circle(A, P_0) → c
  4. Rotate(P_0, α, A) → A_1
  5. If(a < 1, a, 1) → d
  6. Rotate(A_1, ((-d) * α / 2), P_0) → P_1
  7. Rotate(A, ((-d) * α / 2), P_0) → E_1
  8. Rotate(P_1, α, E_1) → A_2
  9. Rotate(A_2, ((-d) * α), P_1) → P_2
  10. Rotate(E_1, ((-d) * α), P_1) → E_2
  11. Rotate(P_2, α, E_2) → A_3
  12. Rotate(A_3, ((-d) * α), P_2) → P_3
  13. Rotate(E_2, ((-d) * α), P_2) → E_3
  14. Rotate(P_3, α, E_3) → A_4
  15. Rotate(A_4, ((-d) * α), P_3) → P_4
  16. Rotate(E_3, ((-d) * α), P_3) → E_4
  17. Rotate(E_4, ((-d) * α), P_4) → E_5
  18. Rotate(P_4, α, E_4) → A_5
  19. Rotate(A_5, ((-d) * α), P_4) → P_5
  20. Rotate(E_5, ((-d) * α), P_5) → E_6
  21. Rotate(P_5, α, E_5) → A_6
  22. Rotate(A_6, ((-d) * α), P_5) → P_6
  23. Rotate(E_6, ((-d) * α), P_6) → E_7
  24. Rotate(P_6, α, E_6) → A_7
  25. Rotate(A_7, ((-d) * α), P_6) → P_7
  26. Rotate(E_7, ((-d) * α), P_7) → E_8
  27. Rotate(P_7, α, E_7) → A_8
  28. Rotate(A_8, ((-d) * α), P_7) → P_8
  29. Line(P_0, A) → b
  30. Rotate(E_8, ((-d) * α), P_8) → E_9
  31. Rotate(P_8, α, E_8) → A_9
  32. Rotate(A_9, ((-d) * α), P_8) → P_9
  33. Rotate(E_9, ((-d) * α), P_9) → F_0
  34. Rotate(P_9, α, E_9) → B_0
  35. Rotate(B_0, ((-d) * α), P_9) → Q_0
  36. Rotate(F_0, ((-d) * α), Q_0) → F_1
  37. Rotate(Q_0, α, F_0) → B_1
  38. Rotate(B_1, ((-d) * α), Q_0) → Q_1
  39. Rotate(F_1, ((-d) * α), Q_1) → F_2
  40. Rotate(Q_1, α, F_1) → B_2
  41. Rotate(B_2, ((-d) * α), Q_1) → Q_2
  42. Rotate(F_2, ((-d) * α), Q_2) → F_3
  43. Rotate(Q_2, α, F_2) → B_3
  44. Rotate(B_3, ((-d) * α), Q_2) → Q_3
  45. Rotate(F_3, ((-d) * α), Q_3) → F_4
  46. Rotate(Q_3, α, F_3) → B_4
  47. Rotate(B_4, ((-d) * α), Q_3) → Q_4
  48. Rotate(F_4, ((-d) * α), Q_4) → F_5
  49. Rotate(Q_4, α, F_4) → B_5
  50. Rotate(B_5, ((-d) * α), Q_4) → Q_5
  51. Mirror(E_1, b) → B
  52. Mirror(E_2, b) → C
  53. Mirror(E_3, b) → D
  54. Mirror(E_4, b) → E
  55. Mirror(E_6, b) → F
  56. Mirror(E_5, b) → G
  57. Mirror(E_7, b) → H
  58. Mirror(E_8, b) → I
  59. Mirror(E_9, b) → J
  60. Mirror(F_0, b) → K
  61. Mirror(F_2, b) → L
  62. Mirror(F_3, b) → N
  63. Mirror(F_4, b) → O
  64. Mirror(F_5, b) → P
  65. Mirror(F_1, b) → M
  66. Mirror(P_1, b) → Q
  67. Mirror(P_2, b) → R
  68. Mirror(P_3, b) → S
  69. Mirror(P_4, b) → T
  70. Mirror(P_5, b) → U
  71. Mirror(P_6, b) → V
  72. Mirror(P_7, b) → W
  73. Mirror(P_8, b) → Y
  74. Mirror(P_9, b) → Z
  75. Mirror(Q_1, b) → C_1
  76. Mirror(Q_0, b) → D_1
  77. Mirror(Q_2, b) → G_1
  78. Mirror(Q_3, b) → H_1
  79. Mirror(Q_4, b) → I_1
  80. Mirror(Q_5, b) → J_1
  81. Polygon(F_5, Q_5, P_6, P_5, P_4, P_3, P_2, P_1, P_0, Q, R, S, Q_4, T, U, V, W, Y, Z, D_1, C_1, G_1, H_1, Q_3, I_1, J_1, P, I_1, O, H_1, N, G_1, L, C_1, Q_2, M, D_1, K, Z, J, Y, I, W, H, V, Q_1, F, U, G, T, E, S, D, R, C, Q, Q_0, B, P_0, E_1, P_1, E_2, P_2, E_3, P_3, E_4, P_4, P_9, E_5, P_5, E_6, P_6, E_7, P_7, E_8, P_8, E_9, P_9, P_8, F_0, Q_0, F_1, Q_1, F_2, Q_2, F_3, Q_3, F_4, Q_4, P_7) → R_1, e, n, p, q, r, s, t, a_1, b_1, c_1, d_1, f, e_1, f_1, g_1, h_1, i_1, j_1, k_1, l_1, m_1, n_1, g, p_1, q_1, r_1, s_1, t_1, a_2, b_2, c_2, d_2, e_2, h, f_2, g_2, h_2, i_2, j_2, k_2, l_2, m_2, n_2, p_2, i, q_2, r_2, s_2, t_2, a_3, b_3, c_3, d_3, e_3, f_3, j, g_3, h_3, i_3, j_3, k_3, l_3, m_3, n_3, p_3, q_3, k, r_3, s_3, t_3, a_4, b_4, c_4, d_4, e_4, f_4, g_4, l, h_4, i_4, j_4, k_4, l_4, m_4, n_4, p_4, q_4, r_4, m, s_4
  82. If(a < 1.2, 0, a - 1.2) → dd
  83. Midpoint(P_0, B) → K_1
  84. Rotate(F_5, (dd * 3.141592653589793), K_1) → F_5'
  85. Rotate(Q_5, (dd * 3.141592653589793), K_1) → Q_5'
  86. Rotate(Q_4, (dd * 3.141592653589793), K_1) → Q_4'
  87. Rotate(Q_3, (dd * 3.141592653589793), K_1) → Q_3'
  88. Rotate(Q_2, (dd * 3.141592653589793), K_1) → Q_2'
  89. Rotate(Q_1, (dd * 3.141592653589793), K_1) → Q_1'
  90. Rotate(Q_0, (dd * 3.141592653589793), K_1) → Q_0'
  91. Rotate(P_9, (dd * 3.141592653589793), K_1) → P_9'
  92. Rotate(P_8, (dd * 3.141592653589793), K_1) → P_8'
  93. Rotate(P_7, (dd * 3.141592653589793), K_1) → P_7'
  94. Rotate(P_6, (dd * 3.141592653589793), K_1) → P_6'
  95. Rotate(P_5, (dd * 3.141592653589793), K_1) → P_5'
  96. Rotate(P_4, (dd * 3.141592653589793), K_1) → P_4'
  97. Rotate(P_3, (dd * 3.141592653589793), K_1) → P_3'
  98. Rotate(P_2, (dd * 3.141592653589793), K_1) → P_2'
  99. Rotate(P_1, (dd * 3.141592653589793), K_1) → P_1'
  100. Rotate(P_0, (dd * 3.141592653589793), K_1) → P_0'
  101. Rotate(Q, (dd * 3.141592653589793), K_1) → Q'
  102. Rotate(R, (dd * 3.141592653589793), K_1) → R'
  103. Rotate(S, (dd * 3.141592653589793), K_1) → S'
  104. Rotate(T, (dd * 3.141592653589793), K_1) → T'
  105. Rotate(U, (dd * 3.141592653589793), K_1) → U'
  106. Rotate(V, (dd * 3.141592653589793), K_1) → V'
  107. Rotate(W, (dd * 3.141592653589793), K_1) → W'
  108. Rotate(Y, (dd * 3.141592653589793), K_1) → Y'
  109. Rotate(Z, (dd * 3.141592653589793), K_1) → Z'
  110. Rotate(D_1, (dd * 3.141592653589793), K_1) → D_1'
  111. Rotate(C_1, (dd * 3.141592653589793), K_1) → C_1'
  112. Rotate(G_1, (dd * 3.141592653589793), K_1) → G_1'
  113. Rotate(H_1, (dd * 3.141592653589793), K_1) → H_1'
  114. Rotate(I_1, (dd * 3.141592653589793), K_1) → I_1'
  115. Rotate(J_1, (dd * 3.141592653589793), K_1) → J_1'
  116. Rotate(P, (dd * 3.141592653589793), K_1) → P'
  117. Rotate(I_1, (dd * 3.141592653589793), K_1) → I_2
  118. Rotate(O, (dd * 3.141592653589793), K_1) → O'
  119. Rotate(H_1, (dd * 3.141592653589793), K_1) → H_2
  120. Rotate(N, (dd * 3.141592653589793), K_1) → N'
  121. Rotate(G_1, (dd * 3.141592653589793), K_1) → G_2
  122. Rotate(L, (dd * 3.141592653589793), K_1) → L'
  123. Rotate(C_1, (dd * 3.141592653589793), K_1) → C_2
  124. Rotate(M, (dd * 3.141592653589793), K_1) → M'
  125. Rotate(D_1, (dd * 3.141592653589793), K_1) → D_2
  126. Rotate(K, (dd * 3.141592653589793), K_1) → K'
  127. Rotate(Z, (dd * 3.141592653589793), K_1) → Z'_1
  128. Rotate(J, (dd * 3.141592653589793), K_1) → J'
  129. Rotate(Y, (dd * 3.141592653589793), K_1) → Y'_1
  130. Rotate(I, (dd * 3.141592653589793), K_1) → I'
  131. Rotate(W, (dd * 3.141592653589793), K_1) → W'_1
  132. Rotate(H, (dd * 3.141592653589793), K_1) → H'
  133. Rotate(V, (dd * 3.141592653589793), K_1) → V'_1
  134. Rotate(F, (dd * 3.141592653589793), K_1) → F'
  135. Rotate(U, (dd * 3.141592653589793), K_1) → U'_1
  136. Rotate(G, (dd * 3.141592653589793), K_1) → G'
  137. Rotate(T, (dd * 3.141592653589793), K_1) → T'_1
  138. Rotate(E, (dd * 3.141592653589793), K_1) → E'
  139. Rotate(S, (dd * 3.141592653589793), K_1) → S'_1
  140. Rotate(D, (dd * 3.141592653589793), K_1) → D'
  141. Rotate(R, (dd * 3.141592653589793), K_1) → R'_1
  142. Rotate(C, (dd * 3.141592653589793), K_1) → C'
  143. Rotate(Q, (dd * 3.141592653589793), K_1) → Q'_1
  144. Rotate(B, (dd * 3.141592653589793), K_1) → B'
  145. Rotate(P_0, (dd * 3.141592653589793), K_1) → P_{10}
  146. Rotate(E_1, (dd * 3.141592653589793), K_1) → E_1'
  147. Rotate(P_1, (dd * 3.141592653589793), K_1) → P_{11}
  148. Rotate(E_2, (dd * 3.141592653589793), K_1) → E_2'
  149. Rotate(P_2, (dd * 3.141592653589793), K_1) → P_{12}
  150. Rotate(E_3, (dd * 3.141592653589793), K_1) → E_3'
  151. Rotate(P_3, (dd * 3.141592653589793), K_1) → P_{13}
  152. Rotate(E_4, (dd * 3.141592653589793), K_1) → E_4'
  153. Rotate(P_4, (dd * 3.141592653589793), K_1) → P_{14}
  154. Rotate(E_5, (dd * 3.141592653589793), K_1) → E_5'
  155. Rotate(P_5, (dd * 3.141592653589793), K_1) → P_{15}
  156. Rotate(E_6, (dd * 3.141592653589793), K_1) → E_6'
  157. Rotate(P_6, (dd * 3.141592653589793), K_1) → P_{16}
  158. Rotate(E_7, (dd * 3.141592653589793), K_1) → E_7'
  159. Rotate(P_7, (dd * 3.141592653589793), K_1) → P_{17}
  160. Rotate(E_8, (dd * 3.141592653589793), K_1) → E_8'
  161. Rotate(P_8, (dd * 3.141592653589793), K_1) → P_{18}
  162. Rotate(E_9, (dd * 3.141592653589793), K_1) → E_9'
  163. Rotate(P_9, (dd * 3.141592653589793), K_1) → P_{19}
  164. Rotate(F_0, (dd * 3.141592653589793), K_1) → F_0'
  165. Rotate(Q_0, (dd * 3.141592653589793), K_1) → Q_6
  166. Rotate(F_1, (dd * 3.141592653589793), K_1) → F_1'
  167. Rotate(Q_1, (dd * 3.141592653589793), K_1) → Q_7
  168. Rotate(F_2, (dd * 3.141592653589793), K_1) → F_2'
  169. Rotate(Q_2, (dd * 3.141592653589793), K_1) → Q_8
  170. Rotate(F_3, (dd * 3.141592653589793), K_1) → F_3'
  171. Rotate(Q_3, (dd * 3.141592653589793), K_1) → Q_9
  172. Rotate(F_4, (dd * 3.141592653589793), K_1) → F_4'
  173. Rotate(Q_4, (dd * 3.141592653589793), K_1) → Q_{10}
  174. Polygon(F_5', Q_5', P_6', P_5', P_4', P_3', P_2', P_1', P_0', Q', R', S', Q_4', T', U', V', W', Y', Z', D_1', C_1', G_1', H_1', Q_3', I_1', J_1', P', I_2, O', H_2, N', G_2, L', C_2, Q_2', M', D_2, K', Z'_1, J', Y'_1, I', W'_1, H', V'_1, Q_1', F', U'_1, G', T'_1, E', S'_1, D', R'_1, C', Q'_1, Q_0', B', P_{10}, E_1', P_{11}, E_2', P_{12}, E_3', P_{13}, E_4', P_{14}, P_9', E_5', P_{15}, E_6', P_{16}, E_7', P_{17}, E_8', P_{18}, E_9', P_{19}, P_8', F_0', Q_6, F_1', Q_7, F_2', Q_8, F_3', Q_9, F_4', Q_{10}, P_7') → R_1', f_5', p_7', p_6', p_5', p_4', p_3', p_2', p_1', p_0', q', r', q_5', s', t', u', v', w', y', z', d_1', c_1', g_1', q_4', h_1', i_1', j_1', p', i_5, o', h_5, n', g_5, l', q_3', c_5, m', d_5, k', z'_1, j', y'_1, i', w'_1, h', q_2', v'_1, f', u'_1, g', t'_1, e', s'_1, d', r'_1, c', q_1', q'_1, b', p_{10}, e_1', p_{11}, e_2', p_{12}, e_3', p_{13}, e_4', q_0', p_{14}, e_5', p_{15}, e_6', p_{16}, e_7', p_{17}, e_8', p_{18}, e_9', p_9', p_{19}, f_0', q_6, f_1', q_7, f_2', q_8, f_3', q_9, f_4', p_8', q_{10}
  175. Vector(S_1, T_1) → v
  176. Vector(S_1, U_1) → u
  177. Vector(V_1, W_1) → w
  178. Vector(V_1, Y_1) → z

AI 绘图提示语参考

请绘制一个由圆和一系列对称线段组成的图形。圆的中心为原点，半径为单位长度。通过一系列旋转和反射操作，生成多个对称点，这些点用于连接圆周上的位置。最终图形由圆和连接这些点的线段构成，形成多个对称的扇形或区域。构造顺序为：先画圆，然后通过旋转和反射生成对称点，最后用线段连接这些点，形成整体图案。图形应保持对称性，体现几何分割的直观性。
以上提示语为根据原始绘图命令整理而来，仅供参考。