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球极平面投影：从球面到平面的保角映射

球极平面投影：从球面到平面的保角映射
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2017-02-09 17:35
圆解析几何立体几何变换轨迹
该范例演示了球极平面投影（Stereographic projection）这一经典几何变换，将球面上的点（除北极点外）通过从北极点发出的射线映射到赤道平面上，直观展示球面与平面之间的对应关系。核心数学概念包括球面几何、投影变换、保角性（共形映射）以及单位圆与复平面的联系。作图要点：使用GeoGebra的Point、Sphere、Line、Rotate、Intersect等命令构造球体、投影射线与交点，通过PointIn、Circle、Plane、OrthogonalLine、Mirror、Locus、Midpoint等命令生成辅助圆、垂线、镜像点与轨迹，并利用滑块或动态对象（如a、f、g、α等）控制投影参数，观察球面上点与平面点的对应运动。适合高中至大学阶段的几何教学，尤其适用于立体几何、解析几何与复分析入门，帮助学生理解球面与平面之间的拓扑与度量关系。

GeoGebra 命令及参数

  1. Point(zAxis) → A
  2. Sphere(A, 1) → a
  3. Line(A, yAxis) → f
  4. Line(A, xAxis) → g
  5. Rotate(a, α, f) → a'
  6. Rotate(a', β, g) → a''
  7. Rotate(a'', γ, zAxis) → a'''
  8. Intersect(f, a''') → B, C
  9. PointIn(a''') → D1
  10. PointIn(a''') → D2
  11. PointIn(a''') → D3
  12. PointIn(a''') → D4
  13. PointIn(a''') → D5
  14. PointIn(a''') → D6
  15. Line(D1, D2) → h
  16. Circle(A, 1, h) → d3
  17. Plane(A, D1, D4) → b
  18. OrthogonalLine(A, b) → i
  19. Circle(A, 1 / 4, i) → e1
  20. Intersect(h, e1) → D, C1
  21. OrthogonalLine(C1, b) → j
  22. Circle(C1, 1 / 4, j) → e2
  23. Intersect(h, e2) → F, C2
  24. OrthogonalLine(C2, b) → k
  25. Circle(C2, 1 / 4, k) → e3
  26. Intersect(h, e3) → H, C3
  27. Mirror(C1, A) → C1'
  28. Mirror(C2, A) → C2'
  29. Mirror(C3, A) → C3'
  30. Intersect(k, a''') → E, G
  31. Intersect(j, a''') → I, J
  32. OrthogonalLine(C3, b) → l
  33. Intersect(l, a''') → K, L
  34. Circle(C3, Distance[C3, K], h) → d4
  35. Circle(C2, Distance[C2, E], h) → d6
  36. Circle(C1, Distance[C1, I], h) → d8
  37. Mirror(d4, A) → d5
  38. Mirror(d6, A) → d7
  39. Mirror(d8, A) → d9
  40. Rotate(D3, (-((45 * ° / 2))), h) → E1
  41. Rotate(D3, (-(45°)), h) → E2
  42. Rotate(D3, (-(45° + (45 * ° / 2))), h) → E3
  43. Line(D3, A) → m
  44. Circle(D3, D1, D2) → c9
  45. Circle(D1, E1, D2) → c2
  46. Circle(D1, E2, D2) → c3
  47. Circle(D1, E3, D2) → c4
  48. Circle(D5, D1, D2) → c5
  49. Rotate(D5, (-((45 * ° / 2))), h) → E4
  50. Rotate(E4, (-(45°)), h) → E5
  51. Rotate(D5, (-(45°)), h) → E6
  52. Circle(E4, D1, D2) → c6
  53. Circle(E6, D1, D2) → c7
  54. Circle(E5, D1, D2) → c8
  55. Point(d4) → M1
  56. Point(d6) → M2
  57. Point(d8) → M3
  58. Point(d3) → M4
  59. Point(d9) → M5
  60. Point(d7) → M6
  61. Point(d5) → M7
  62. Line(Z, M1) → n
  63. Intersect(n, xOyPlane) → T
  64. Locus(T, M1) → loc1
  65. Line(Z, M2) → i_1
  66. Intersect(i_1, xOyPlane) → U
  67. Locus(U, M2) → loc2
  68. Line(Z, M3) → j_1
  69. Intersect(j_1, xOyPlane) → V
  70. Locus(V, M3) → loc3
  71. Line(Z, M4) → l_1
  72. Intersect(l_1, xOyPlane) → W
  73. Locus(W, M4) → loc4
  74. Line(Z, M5) → m_1
  75. Intersect(m_1, xOyPlane) → A_1
  76. Locus(A_1, M5) → loc5
  77. Line(Z, M6) → n_1
  78. Intersect(n_1, xOyPlane) → B_1
  79. Locus(B_1, M6) → loc6
  80. Line(M7, Z) → p_1
  81. Intersect(p_1, xOyPlane) → C_1
  82. Locus(C_1, M7) → loc7
  83. Point(c2) → F1
  84. Point(c3) → F2
  85. Point(c4) → F3
  86. Point(c5) → F4
  87. Point(c6) → F5
  88. Point(c9) → F6
  89. Point(c7) → F7
  90. Point(c8) → F8
  91. Line(F6, Z) → a4
  92. Line(F1, Z) → a5
  93. Line(Z, F2) → a6
  94. Line(Z, F3) → a7
  95. Line(Z, F4) → a1
  96. Line(Z, F5) → b_1
  97. Line(Z, F7) → a2
  98. Line(Z, F8) → a3
  99. Intersect(a4, xOyPlane) → L_1
  100. Intersect(a5, xOyPlane) → M_1
  101. Intersect(a6, xOyPlane) → N_1
  102. Intersect(a7, xOyPlane) → O_1
  103. Intersect(a1, xOyPlane) → P_1
  104. Intersect(b_1, xOyPlane) → Q_1
  105. Intersect(a2, xOyPlane) → R_1
  106. Intersect(a3, xOyPlane) → S_1
  107. Locus(L_1, F6) → loc8
  108. Locus(M_1, F1) → loc9
  109. Locus(N_1, F2) → loc10
  110. Locus(O_1, F3) → loc11
  111. Locus(P_1, F4) → loc12
  112. Locus(Q_1, F5) → loc13
  113. Locus(R_1, F7) → loc14
  114. Locus(S_1, F8) → loc15
  115. Midpoint(C3, D1) → M
  116. OrthogonalLine(M, b) → t
  117. Intersect(t, a''') → N, O
  118. Circle(M, Distance[M, N], h) → d1
  119. Mirror(d1, A) → d2
  120. Point(d1) → M8
  121. Point(d2) → M9
  122. Line(M8, Z) → f_1
  123. Line(Z, M9) → g_1
  124. Intersect(f_1, xOyPlane) → P
  125. Intersect(g_1, xOyPlane) → Q
  126. Locus(P, M8) → loc16
  127. Locus(Q, M9) → loc17

AI 绘图提示语参考

绘制一个立体几何中的射影变换，将球面上的点通过射线映射到平面。首先画一个单位球体，然后画出与球体相交的两条射线，接着围绕这些射线旋转球体，生成一系列交点和辅助圆。最后通过镜像和旋转操作，生成一系列新的点和圆，展示射影变换的性质。
以上提示语为根据原始绘图命令整理而来，仅供参考。