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球体几何构造演示

球体几何构造演示
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2020-05-23 16:24
圆多边形立体几何几何作图
①本范例演示了球体及其相关几何元素的构造与交互；②涉及的核心数学概念包括圆、多边形、立体几何、几何变换等；③构造过程中用到了Circle（圆）、Polygon（多边形）、Rotate（旋转）、Mirror（镜像）、Segment（线段）等GeoGebra命令；④适合高中立体几何教学和几何作图练习。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 4) → c
  3. Intersect(c, xAxis, 2) → B
  4. Rotate(B, 90°) → C
  5. Mirror(C, A) → D
  6. Segment(A, C) → f
  7. Circle(A, 1) → d
  8. Intersect(d, f, 1) → E
  9. Circle(A, 2) → e
  10. Intersect(e, f, 1) → F
  11. Circle(A, 3) → g
  12. Intersect(g, f, 1) → G
  13. Circle(A, 3.5) → h
  14. Intersect(h, f, 1) → H
  15. Circle(A, 3.85) → k
  16. Intersect(k, f, 1) → I
  17. Mirror(K, zAxis) → K'
  18. Mirror(L, zAxis) → L'
  19. Mirror(M, zAxis) → M'
  20. Mirror(N, zAxis) → N'
  21. Mirror(O, zAxis) → O'
  22. Segment(J, K) → i
  23. Segment(K, L) → j
  24. Segment(L, M) → l
  25. Segment(M, N) → m
  26. Segment(N, O) → n
  27. Segment(O, C) → p
  28. Segment(J, K') → q
  29. Segment(K', L') → r
  30. Segment(L', M') → s
  31. Segment(M', N') → t
  32. Segment(N', O') → a
  33. Segment(O', D) → b
  34. Line(C, zAxis) → as
  35. Rotate(O, 45°, as) → O'_1
  36. Rotate(N, 45°, as) → N'_1
  37. Rotate(M, 45°, as) → M'_1
  38. Rotate(L, 45°, as) → L'_1
  39. Rotate(K, 45°, as) → K'_1
  40. Rotate(J, 45°, as) → J'
  41. Rotate(K', 45°, as) → K''
  42. Rotate(L', 45°, as) → L''
  43. Rotate(M', 45°, as) → M''
  44. Rotate(N', 45°, as) → N''
  45. Rotate(O', 45°, as) → O''
  46. Rotate(D, 45°, as) → D'
  47. Segment(O'', D') → f_1
  48. Segment(O'', N'') → g_1
  49. Segment(N'', M'') → h_1
  50. Segment(M'', L'') → i_1
  51. Segment(L'', K'') → j_1
  52. Segment(K'', J') → k_1
  53. Segment(J', K'_1) → l_1
  54. Segment(K'_1, L'_1) → m_1
  55. Segment(L'_1, M'_1) → n_1
  56. Segment(M'_1, N'_1) → p_1
  57. Segment(N'_1, O'_1) → q_1
  58. Segment(C, O'_1) → r_1
  59. LineBisector(C, D', xOyPlane) → s_1
  60. LineBisector(C, D) → t_1
  61. Intersect(s_1, t_1) → P
  62. Line(P, zAxis) → rotas
  63. Sequence(Rotate[l1, (nn * 90°), rotas], nn, 1, 7) → l2
  64. Sequence(Rotate[l3, (nn * 90°), rotas], nn, 1, 7) → l4
  65. Rotate(J, (360°) / 8, rotas) → J'_1
  66. Polygon(J, J'_1, 8, xOyPlane) → veelhoek1, a_1, R, S, T, U, V, b_1, c_1, d_1, e_1, f_2, g_2, h_2, Q
  67. Point(rotas) → W
  68. Rotate(C, (360°) / 8, rotas) → C'
  69. Midpoint(C, C') → Z
  70. Plane(Z, P, P + (0, 0, 1)) → spieg
  71. Mirror(L, spieg) → L'_2
  72. Mirror(M, spieg) → M'_2
  73. Mirror(N, spieg) → N'_2
  74. Mirror(O, spieg) → O'_2
  75. Intersect(j, spieg) → C_1
  76. Midpoint(L'_1, L) → A_1
  77. Midpoint(M'_1, M) → B_1
  78. Midpoint(N'_1, N) → D_1
  79. Plane(A_1, B_1, D_1) → p_2
  80. Midpoint(K'_1, K) → E_1
  81. Rotate(spieg, (360°) / 16, rotas) → spieg'
  82. Intersect(spieg', i) → F_1
  83. Mirror(K, spieg) → K'_2
  84. Mirror(F_1, spieg) → G_1
  85. Segment(P, C') → s_2
  86. Segment(P, C) → t_2
  87. OrthogonalLine(C, t_2, xOyPlane) → a_2
  88. OrthogonalLine(C', s_2, xOyPlane) → b_2
  89. Intersect(a_2, b_2) → H_1
  90. Line(O, a_2) → c_2
  91. Line(O'_2, b_2) → f_3
  92. Line(N, a_2) → g_3
  93. Line(N'_2, b_2) → h_3
  94. Line(M, a_2) → i_3
  95. Line(M'_2, b_2) → l_3
  96. Line(L, a_2) → m_3
  97. Line(L'_2, b_2) → n_3
  98. Intersect(c_2, f_3) → I_1
  99. Intersect(g_3, h_3) → J_1
  100. Intersect(i_3, l_3) → K_1
  101. Intersect(m_3, n_3) → L_1
  102. Polygon(C_1, L, L_1) → driehoek1, i_2, j_2, l_2
  103. Polygon(L, M, K_1, L_1) → vierhoek1, l_4, m_2, k_2, n_2
  104. Polygon(M, N, J_1, K_1) → vierhoek2, m_4, n_4, q_2, d_2
  105. Polygon(N, O, I_1, J_1) → vierhoek3, n_5, o, e_2, j_3
  106. Polygon(O, C, H_1, I_1) → vierhoek4, o_1, c_3, k_3, p_3
  107. Polygon(H_1, C', O'_2, I_1) → vierhoek5, q_3, c', o'_2, r_3
  108. Polygon(I_1, O'_2, N'_2, J_1) → vierhoek6, s_3, t_3, n'_2, a_3
  109. Polygon(J_1, N'_2, M'_2, K_1) → vierhoek7, b_3, d_3, m'_2, e_3
  110. Polygon(K_1, M'_2, L'_2, L_1) → vierhoek8, f_4, g_4, l'_2, h_4
  111. Polygon(L_1, L'_2, C_1) → driehoek2, i_4, j_4, k_4
  112. Segment(W, F_1) → q_4
  113. Segment(W, V) → r_4
  114. Segment(V, C_1) → s_4
  115. Rotate(b_2, (360°) / 8, zAxis) → t_4
  116. Mirror(L'_2, spieg') → M_1
  117. Mirror(M'_2, spieg') → N_1
  118. Mirror(N'_2, spieg') → O_1
  119. Mirror(O'_2, spieg') → P_1
  120. Mirror(C_1, spieg') → Q_1
  121. Line(C', rotas) → a_4
  122. Rotate(b_2, (360°) / 32, a_4) → b_4
  123. Polygon(F_1, C_1, Q_1) → driehoek3, c_4, d_4, e_4
  124. Polygon(C_1, L_1, M_1, Q_1) → vierhoek9, f_5, g_5, h_5, i_5
  125. Polygon(L'_2, M'_2, N_1, M_1) → vierhoek10, j_5, k_5, l_5, m_5
  126. Polygon(M'_2, N'_2, O_1, N_1) → vierhoek11, p_5, q_5, r_5, s_5
  127. Line(K'_2, a_2) → t_5
  128. Line(K, b_2) → a_5
  129. Intersect(t_5, a_5) → S_1
  130. Polygon(K'_2, C_1, S_1) → driehoek4, b_5, k'_2, c_5
  131. Polygon(G_1, K'_2, S_1, V) → vierhoek12, d_5, e_5, f_6, v
  132. Polygon(V, S_1, K, F_1) → vierhoek13, v_1, g_6, k_6, h_6
  133. Polygon(S_1, C_1, K) → driehoek5, k_7, i_6, j_6
  134. Segment(C', C) → l_6
  135. Line(H_1, l_6) → m_6
  136. Rotate(M, (360°) / 8, rotas) → M'_3
  137. Rotate(K_1, (360°) / 8, rotas) → R_1
  138. Segment(M'_3, R_1) → n_6
  139. Rotate(m_6, (360°) / 8, rotas) → p_6
  140. Line(C_1, m_6) → q_6
  141. Line(Q_1, p_6) → r_6
  142. Intersect(r_6, q_6) → T_1
  143. Line(L'_2, m_6) → s_6
  144. Line(M'_2, m_6) → t_6
  145. Line(N'_2, m_6) → a_6
  146. Line(O'_2, m_6) → b_6
  147. Plane(W, F_1, C') → p_7
  148. Intersect(s_6, p_7) → U_1
  149. Intersect(t_6, p_7) → V_1
  150. Intersect(a_6, p_7) → W_1
  151. Intersect(b_6, p_7) → Z_1
  152. Segment(F_1, T_1) → c_6
  153. Segment(T_1, U_1) → d_6
  154. Segment(U_1, V_1) → e_6
  155. Segment(V_1, W_1) → f_7
  156. Segment(O'_2, Z_1) → g_7
  157. Segment(Z_1, W_1) → h_7
  158. Segment(Z_1, C') → i_7
  159. Polygon(F_1, C_1, T_1) → driehoek6, j_7, l_7, m_7
  160. Polygon(C_1, L'_2, U_1, T_1) → vierhoek14, n_7, q_7, u_1, r_7
  161. Polygon(L'_2, M'_2, V_1, U_1) → vierhoek15, s_7, t_7, a_7, b_7
  162. Polygon(M'_2, N'_2, W_1, V_1) → vierhoek16, c_7, d_7, w_1, e_7
  163. Polygon(N'_2, O'_2, Z_1, W_1) → vierhoek17, f_8, g_8, z_1, h_8
  164. Polygon(Z_1, O'_2, C') → driehoek7, i_8, j_8, k_8
  165. Segment(B_2, C_2) → p_4
  166. Sequence(Rotate[l5, (nn * (360°) / 8), rotas], nn, 1, 7) → l6
  167. Mirror(l7, spieg') → l7'
  168. Sequence(Rotate[l7, (nn * (360°) / 8), rotas], nn, 1, 7) → l8
  169. Sequence(Rotate[l7', (nn * (360°) / 8), rotas], nn, 1, 7) → l9
  170. Sequence(Rotate[l10, (nn * (360°) / 8), rotas], nn, 1, 7) → l11
  171. Polygon(V, J, 8, xOyPlane) → veelhoek2, l_8, E_2, F_2, G_2, H_2, I_2, m_8, n_8, p_8, q_8, r_8, s_8, t_8, D_2
  172. Polygon(C_2, B_2, 8, xOyPlane) → veelhoek3, a_8, K_2, L_2, M_2, N_2, O_2, b_8, c_8, d_8, e_8, f_9, g_9, h_9, J_2
  173. Polygon(G_1, V, C_2) → driehoek8, i_9, j_9, v_2
  174. Polygon(V, F_1, B_2) → driehoek9, k_9, v_3, l_9
  175. Sequence(Rotate[driehoek8, (nn * (360°) / 8), rotas], nn, 1, 7) → l12
  176. Sequence(Rotate[driehoek9, (nn * (360°) / 8), rotas], nn, 1, 7) → l13
  177. Polygon(W, O_2, C_2) → driehoek10, m_9, w, o_2
  178. Sequence(Rotate[driehoek10, (nn * (360°) / 8), rotas], nn, 1, 7) → l14
  179. Rotate(H_1, (360°) / 8, P) → P_2
  180. Polygon(H_1, P_2, 8, xOyPlane) → veelhoek4, n_9, R_2, S_2, T_2, U_2, V_2, p_9, q_9, r_9, s_9, t_9, a_9, b_9, Q_2
  181. Segment(C, D) → c_9
  182. Sequence(Segment[Rotate[C, (nn * (360°) / 8), P], Rotate[D, (nn * (360°) / 8), P]], nn, 1, 7) → l15
  183. Sequence(Segment[Rotate[P, (nn * (360°) / 8), P], Rotate[C, (nn * (360°) / 8), P]], nn, 1, 7) → l16
  184. Segment(H_1, P) → d_9
  185. Sequence(Segment[Rotate[H_1, (nn * (360°) / 8), P], Rotate[P, (nn * (360°) / 8), P]], nn, 1, 7) → l17
  186. PointIn(spieg) → W_2
  187. PointIn(spieg) → Z_2
  188. PointIn(spieg) → A_3
  189. Segment(V, A_3) → e_9
  190. Segment(A_3, Z_2) → f_{10}
  191. Segment(Z_2, W_2) → g_{10}
  192. Segment(W_2, W) → h_{10}
  193. Rotate(l18, (360°) / 8, rotas) → l18'
  194. Plane(F_1, W, W + (0, 0, 1)) → p_{10}
  195. Segment(W, F_1) → i_{10}
  196. Line(A_3, p_4) → j_{10}
  197. Line(Z_2, p_4) → k_{10}
  198. Line(W_2, p_4) → l_{10}
  199. Intersect(j_{10}, p_{10}) → B_3
  200. Intersect(k_{10}, p_{10}) → C_3
  201. Intersect(l_{10}, p_{10}) → D_3
  202. Polygon(W, W_2, D_3) → driehoek11, m_{10}, w_2, n_{10}
  203. Polygon(W_2, Z_2, C_3, D_3) → vierhoek18, q_{10}, z_2, r_{10}, s_{10}
  204. Polygon(Z_2, A_3, B_3, C_3) → vierhoek19, t_{10}, a_{10}, b_{10}, c_{10}
  205. Intersect(p_4, p_{10}) → E_3
  206. Polygon(V, E_3, B_3, A_3) → vierhoek20, v_4, d_{10}, e_{10}, f_{11}
  207. Mirror(l19, p_{10}) → l19'
  208. Sequence(Rotate[l19, (nn * (360°) / 8), rotas], nn, 1, 7) → l20
  209. Sequence(Rotate[l19', (nn * (360°) / 8), rotas], nn, 1, 7) → l21

AI 绘图提示语参考

以原点为球心，绘制一个半径为4的球体，其赤道圆与水平轴交于一点。将该点绕球心旋转90度得到顶点，再关于球心对称得到底点，连接顶点与底点形成竖直直径。在直径上从球心向外依次取半径1、2、3、3.5、3.85的点，并分别以这些点为半径画同心圆。将直径上的这些点绕竖直轴镜像到另一侧，然后连接相邻点形成一系列水平线段，构成球面的经线网格。最后将整个网格绕竖直轴旋转45度，得到另一组对称的经线，形成完整的球面经纬线结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。