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球极平面投影演示

球极平面投影演示
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2016-04-13 23:38
圆解析几何立体几何变换几何作图
所属专题 intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
该范例演示了球极平面投影（Stereographic projection）这一经典几何变换，将球面上的点映射到平面上，并展示其保角性质。核心数学概念包括三维球面与二维平面的映射关系、正交直线、圆弧、平移变换以及球体与平面的相交结构。作图要点：使用GeoGebra命令OrthogonalLine构造正交线，Arc绘制圆弧，Point定义关键点，Translate实现平移，Sphere创建球体，If进行条件判断，Segment连接线段；对象包括多个点（A1-E1、A2-E2等）和辅助几何元素，通过平移和投影关系构建出完整的投影示意图。适合高中及以上学段学习立体几何、解析几何或复变函数入门时使用，也可用于大学数学中理解黎曼球面与复平面的对应关系。

GeoGebra 命令及参数

  1. OrthogonalLine((0, 0, 0), Plane[D1, D2, (0, 0, 0)]) → F1
  2. OrthogonalLine((0, 0, 0), Plane[D2, D7, (0, 0, 0)]) → F2
  3. OrthogonalLine((0, 0, 0), Plane[D7, D8, (0, 0, 0)]) → F5
  4. Arc(Circle[(0, 0, 0), 1, F1], D1, D2) → G1
  5. Arc(Circle[(0, 0, 0), 1, F2], D2, D7) → G2
  6. OrthogonalLine((0, 0, 0), Plane[D3, D1, (0, 0, 0)]) → F3
  7. Arc(Circle[(0, 0, 0), 1, F3], D3, D1) → G3
  8. OrthogonalLine((0, 0, 0), Plane[D7, D3, (0, 0, 0)]) → F4
  9. Arc(Circle[(0, 0, 0), 1, F4], D7, D3) → G4
  10. Point((0, 0, 0), Vector[UnitVector[Vector[D1] + Vector[D2]]]) → H1
  11. Point((0, 0, 0), Vector[UnitVector[Vector[D2] + Vector[D7]]]) → H2
  12. Arc(Circle[E1, E2, I1], E2, E1) → J1
  13. Arc(Circle[E2, E7, I2], E7, E2) → J2
  14. Point((0, 0, 0), Vector[UnitVector[Vector[D3] + Vector[D1]]]) → H3
  15. Arc(Circle[E3, E1, I3], E1, E3) → J3
  16. Point((0, 0, 0), Vector[UnitVector[Vector[D7] + Vector[D3]]]) → H4
  17. Arc(Circle[E7, E3, I4], E3, E7) → J4
  18. Arc(Circle[(0, 0, 0), 1, F5], D7, D8) → G5
  19. Point((0, 0, 0), Vector[UnitVector[Vector[D8] + Vector[D7]]]) → H5
  20. Arc(Circle[E8, E7, I5], E7, E8) → J5
  21. OrthogonalLine((0, 0, 0), Plane[D8, D6, (0, 0, 0)]) → F6
  22. Arc(Circle[(0, 0, 0), 1, F6], D8, D6) → G6
  23. Point((0, 0, 0), Vector[UnitVector[Vector[D6] + Vector[D8]]]) → H6
  24. Arc(Circle[E6, E8, I6], E8, E6) → J6
  25. OrthogonalLine((0, 0, 0), Plane[D2, D6, (0, 0, 0)]) → F7
  26. Arc(Circle[(0, 0, 0), 1, F7], D2, D6) → G7
  27. Point((0, 0, 0), Vector[UnitVector[Vector[D2] + Vector[D6]]]) → H7
  28. Arc(Circle[E2, E6, I7], E6, E2) → J7
  29. OrthogonalLine((0, 0, 0), Plane[D3, D5, (0, 0, 0)]) → F8
  30. Arc(Circle[(0, 0, 0), 1, F8], D3, D5) → G8
  31. Point((0, 0, 0), Vector[UnitVector[Vector[D3] + Vector[D5]]]) → H8
  32. Arc(Circle[E3, E5, I8], E5, E3) → J8
  33. OrthogonalLine((0, 0, 0), Plane[D4, D1, (0, 0, 0)]) → F9
  34. Arc(Circle[(0, 0, 0), 1, F9], D4, D1) → G9
  35. Point((0, 0, 0), Vector[UnitVector[Vector[D4] + Vector[D1]]]) → H9
  36. Arc(Circle[E4, E1, I9], E1, E4) → J9
  37. OrthogonalLine((0, 0, 0), Plane[D6, D4, (0, 0, 0)]) → F10
  38. Arc(Circle[(0, 0, 0), 1, F10], D6, D4) → G10
  39. Point((0, 0, 0), Vector[UnitVector[Vector[D6] + Vector[D4]]]) → H10
  40. Arc(Circle[E6, E4, I10], E4, E6) → J10
  41. OrthogonalLine((0, 0, 0), Plane[D4, D5, (0, 0, 0)]) → F11
  42. Arc(Circle[(0, 0, 0), 1, F11], D4, D5) → G11
  43. Point((0, 0, 0), Vector[UnitVector[Vector[D4] + Vector[D5]]]) → H11
  44. Arc(Circle[E4, E5, I11], E5, E4) → J11
  45. OrthogonalLine((0, 0, 0), Plane[D5, D8, (0, 0, 0)]) → F12
  46. Arc(Circle[(0, 0, 0), 1, F12], D5, D8) → G12
  47. Point((0, 0, 0), Vector[UnitVector[Vector[D5] + Vector[D8]]]) → H12
  48. Arc(Circle[E5, E8, I12], E8, E5) → J12
  49. Translate(J1, Vector[(0, 0, -1)]) → K1
  50. Translate(J2, Vector[(0, 0, -1)]) → K2
  51. Translate(J3, Vector[(0, 0, -1)]) → K3
  52. Sphere((0, 0, 0), 1) → e
  53. If(¬Defined[J5], Segment[E7, E8]) → K14
  54. If(¬Defined[J7], Segment[E2, E6]) → K15
  55. If(¬Defined[J8], Segment[E3, E5]) → K16
  56. If(¬Defined[J9], Segment[E4, E1]) → K17
  57. Segment((0, 0, 1), E1) → L1
  58. Segment((0, 0, 1), E2) → L2
  59. Segment((0, 0, 1), E3) → L3
  60. Segment((0, 0, 1), E4) → L4
  61. Segment((0, 0, 1), E5) → L5
  62. Segment((0, 0, 1), E6) → L6
  63. Segment((0, 0, 1), E7) → L7
  64. Segment((0, 0, 1), E8) → L8