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经纬度三维球面演示：参数曲线与坐标网格

经纬度三维球面演示：参数曲线与坐标网格
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2017-08-01 00:31
解析几何立体几何变换参数方程几何作图
所属专题 Algebra 1
该范例在三维空间中直观演示了地球经纬度（Latitude and Longitude）的几何构造，通过球体上的参数曲线和正交平面相交路径，展示经线、纬线以及经纬度坐标点的生成过程。核心数学概念包括三维球面坐标、参数曲线、正交平面、旋转与交点计算，以及实数与有理数在坐标表示中的应用。作图要点：使用GeoGebra的Sphere命令创建球体，OrthogonalPlane构造过球心的正交平面，IntersectPath求平面与球面的交线（即大圆或小圆），Rotate实现经度旋转，Plane与Intersect用于定位特定经纬度点，Textfield用于交互输入经纬度数值。该范例适合高中或大学阶段学习立体几何、球面坐标与参数方程的学生，也可用于地理数学或三维可视化教学，帮助理解经纬度与空间点的一一对应关系。

GeoGebra 命令及参数

  1. Sphere(A, 9) → a
  2. OrthogonalPlane(D, zAxis) → b
  3. OrthogonalPlane(E, zAxis) → c
  4. OrthogonalPlane(F, zAxis) → d
  5. OrthogonalPlane(G, zAxis) → e
  6. OrthogonalPlane(H, zAxis) → f
  7. OrthogonalPlane(I, zAxis) → g
  8. OrthogonalPlane(J, zAxis) → h
  9. OrthogonalPlane(K, zAxis) → i
  10. IntersectPath(b, a) → k
  11. IntersectPath(c, a) → p
  12. IntersectPath(d, a) → q
  13. IntersectPath(e, a) → r
  14. IntersectPath(f, a) → s
  15. IntersectPath(g, a) → t
  16. IntersectPath(h, a) → c_1
  17. IntersectPath(i, a) → d_1
  18. OrthogonalPlane(A, zAxis) → j
  19. IntersectPath(j, a) → e_1
  20. OrthogonalPlane(L, zAxis) → l_5
  21. OrthogonalPlane(S, zAxis) → m_4
  22. OrthogonalPlane(R, zAxis) → n_2
  23. OrthogonalPlane(Q, zAxis) → o
  24. OrthogonalPlane(P, zAxis) → u
  25. OrthogonalPlane(O, zAxis) → v
  26. OrthogonalPlane(N, zAxis) → w
  27. OrthogonalPlane(M, zAxis) → a_1
  28. IntersectPath(l_5, a) → f_1
  29. IntersectPath(w, a) → g_1
  30. IntersectPath(v, a) → h_1
  31. IntersectPath(u, a) → k_1
  32. IntersectPath(o, a) → p_1
  33. IntersectPath(n_2, a) → q_1
  34. IntersectPath(m_4, a) → r_1
  35. IntersectPath(a_1, a) → s_1
  36. Rotate(T, 15°, (0, 0)) → T'
  37. Rotate(T', 15°, (0, 0)) → T''
  38. Rotate(T'', 15°, (0, 0)) → T'''
  39. Rotate(T''', 15°, (0, 0)) → U
  40. Rotate(U, 15°, (0, 0)) → U'
  41. Rotate(U', 15°, (0, 0)) → U''
  42. Rotate(U'', 15°, (0, 0)) → U'''
  43. Rotate(U''', 15°, (0, 0)) → V
  44. Rotate(V, 15°, (0, 0)) → V'
  45. Rotate(V', 15°, (0, 0)) → V''
  46. Rotate(V'', 15°, (0, 0)) → V'''
  47. Plane(A, B, T) → b_1
  48. Plane(A, B, T') → i_1
  49. Plane(A, B, T'') → j_1
  50. Plane(A, B, T''') → l_6
  51. Plane(A, B, U) → m_1
  52. Plane(A, B, U') → n_1
  53. Plane(A, B, U'') → o_1
  54. Plane(A, B, U''') → t_1
  55. Plane(A, B, V) → u_1
  56. Plane(A, B, V') → v_1
  57. Plane(A, B, V'') → w_1
  58. Plane(A, B, V''') → a_2
  59. IntersectPath(i_1, a) → c_2
  60. IntersectPath(j_1, a) → d_2
  61. IntersectPath(l_6, a) → e_2
  62. IntersectPath(m_1, a) → f_2
  63. IntersectPath(n_1, a) → g_2
  64. IntersectPath(o_1, a) → h_2
  65. IntersectPath(t_1, a) → k_2
  66. IntersectPath(u_1, a) → p_2
  67. IntersectPath(v_1, a) → q_2
  68. IntersectPath(w_1, a) → r_2
  69. IntersectPath(a_2, a) → s_2
  70. IntersectPath(b_1, a) → t_2
  71. Intersect(d_1, r_2, 1) → W
  72. Intersect(c_1, r_2, 1) → Z
  73. Intersect(d_1, s_2, 1) → A_1
  74. Intersect(c_1, s_2, 1) → B_1
  75. Intersect(t, s_2, 1) → C_1
  76. Intersect(s, s_2, 1) → D_1
  77. Intersect(s_2, r, 2) → E_1
  78. Intersect(q, s_2, 1) → F_1
  79. Intersect(p, s_2, 1) → G_1
  80. Intersect(k, s_2, 1) → H_1
  81. Intersect(d_1, t_2, 2) → J_1
  82. Intersect(t_2, c_1, 1) → K_1
  83. Intersect(t_2, t, 1) → L_1
  84. Intersect(t_2, s, 1) → M_1
  85. Intersect(t_2, r, 1) → N_1
  86. Intersect(q, t_2, 2) → O_1
  87. Intersect(t_2, p, 1) → Q_1
  88. Intersect(k, t_2, 2) → R_1
  89. Intersect(c_2, d_1, 1) → T_1
  90. Intersect(c_2, c_1, 1) → U_1
  91. Intersect(t, c_2, 2) → V_1
  92. Intersect(c_2, s, 1) → W_1
  93. Intersect(r, c_2, 2) → Z_1
  94. Intersect(c_2, q, 1) → A_2
  95. Intersect(p, c_2, 2) → B_2
  96. Intersect(c_2, k, 1) → C_2
  97. Intersect(d_2, d_1, 1) → E_2
  98. Intersect(d_2, c_1, 1) → F_2
  99. Intersect(d_2, t, 1) → G_2
  100. Intersect(d_2, s, 1) → H_2
  101. Intersect(d_2, r, 1) → I_2
  102. Intersect(q, d_2, 2) → J_2
  103. Intersect(d_2, p, 1) → K_2
  104. Intersect(d_2, k, 1) → L_2
  105. Intersect(e_2, d_1, 1) → N_2
  106. Intersect(c_1, e_2, 2) → O_2
  107. Intersect(t, e_2, 2) → Q_2
  108. Intersect(s, e_2, 2) → R_2
  109. Intersect(e_2, r, 1) → S_2
  110. Intersect(q, e_2, 2) → T_2
  111. Intersect(e_2, p, 1) → U_2
  112. Intersect(e_2, k, 1) → V_2
  113. Intersect(f_2, d_1, 1) → Z_2
  114. Intersect(c_1, f_2, 2) → A_3
  115. Intersect(t, f_2, 2) → B_3
  116. Intersect(s, f_2, 2) → C_3
  117. Intersect(r, f_2, 2) → D_3
  118. Intersect(q, f_2, 2) → E_3
  119. Intersect(p, f_2, 2) → F_3
  120. Intersect(k, f_2, 2) → G_3
  121. Point(e_1) → H_3
  122. Intersect(g_2, d_1, 1) → I_3
  123. Intersect(g_2, c_1, 1) → J_3
  124. Intersect(t, g_2, 2) → K_3
  125. Intersect(s, g_2, 2) → L_3
  126. Intersect(g_2, r, 1) → M_3
  127. Intersect(g_2, q, 1) → N_3
  128. Intersect(g_2, p, 1) → O_3
  129. Intersect(g_2, k, 1) → Q_3
  130. Intersect(f_1, t_2, 2) → S_3
  131. Intersect(f_1, c_2, 2) → T_3
  132. Intersect(f_1, d_2, 2) → U_3
  133. Intersect(e_2, f_1, 1) → V_3
  134. Intersect(f_2, f_1, 1) → W_3
  135. Intersect(f_1, g_2, 2) → Z_3
  136. Intersect(g_2, s_1, 1) → A_4
  137. Intersect(g_1, g_2, 2) → B_4
  138. Intersect(k_1, g_2, 2) → C_4
  139. Intersect(f_2, k_1, 1) → D_4
  140. Intersect(f_2, h_1, 1) → E_4
  141. Intersect(f_2, g_1, 1) → F_4
  142. Intersect(s_1, f_2, 2) → G_4
  143. Intersect(e_2, s_1, 1) → H_4
  144. Intersect(g_1, e_2, 2) → I_4
  145. Intersect(h_1, e_2, 2) → J_4
  146. Intersect(k_1, d_2, 2) → K_4
  147. Intersect(d_2, h_1, 1) → L_4
  148. Intersect(g_1, d_2, 2) → M_4
  149. Intersect(s_1, d_2, 2) → N_4
  150. Intersect(s_1, c_2, 2) → O_4
  151. Intersect(g_1, c_2, 2) → P_4
  152. Intersect(h_1, c_2, 2) → Q_4
  153. Intersect(g_1, t_2, 2) → R_4
  154. Intersect(s_1, t_2, 2) → S_4
  155. Textfield(l_1) → InputBox1
  156. Textfield(l_2) → InputBox2
  157. Intersect(t, k_2, 2) → I_1
  158. Intersect(s, k_2, 2) → S_1
  159. Intersect(r, k_2, 2) → D_2
  160. Intersect(k_2, q, 1) → M_2
  161. Intersect(p, k_2, 2) → W_2
  162. Intersect(k, k_2, 2) → R_3
  163. Intersect(f_1, k_2, 2) → T_4
  164. Intersect(s_1, k_2, 2) → U_4
  165. Intersect(g_1, k_2, 2) → V_4
  166. Intersect(h_1, k_2, 2) → W_4
  167. Intersect(t, p_2, 2) → Z_4
  168. Intersect(p_2, s, 1) → A_5
  169. Intersect(r, p_2, 2) → B_5
  170. Intersect(q, p_2, 2) → C_5
  171. Intersect(p_2, p, 1) → D_5
  172. Intersect(k, p_2, 2) → E_5
  173. Intersect(f_1, p_2, 2) → F_5
  174. Intersect(s_1, p_2, 2) → G_5
  175. Textfield(m_{response}) → InputBox3