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Latitude and Longitude: 3D Sphere with Parametric Curves and Coordinate Grid

Latitude and Longitude: 3D Sphere with Parametric Curves and Coordinate Grid
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2017-08-01 00:31
Coordinate GeometrySolid GeometryTransformationParametric EquationGeometric Construction
In collections Algebra 1
This GeoGebra example demonstrates the geometric construction of latitude and longitude on a 3D sphere, showing how meridians (longitude lines) and parallels (latitude lines) are generated via parametric curves and intersections of orthogonal planes with the sphere. Core mathematical concepts include spherical coordinates, parametric curves, orthogonal planes, rotation, and point intersection, along with the role of real and rational numbers in coordinate representation. The construction uses commands such as Sphere to create the globe, OrthogonalPlane to define planes through the center, IntersectPath to obtain the intersection curves (great or small circles), Rotate to sweep longitude angles, Plane and Intersect to locate specific latitude-longitude points, and Textfield for interactive input of coordinate values. This example is suitable for high school or undergraduate students learning solid geometry, spherical coordinates, and parametric equations, and is also valuable for geography-related math visualization, clarifying the one-to-one mapping between (latitude, longitude) and points on the sphere.

GeoGebra Commands & Parameters

  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