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Stereographic Projection: Mapping the Sphere to the Plane

Stereographic Projection: Mapping the Sphere to the Plane
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2017-02-09 17:35
CircleCoordinate GeometrySolid GeometryTransformationLocus
This GeoGebra example demonstrates the stereographic projection, a classical geometric transformation that maps points on a sphere (except the north pole) to the equatorial plane via rays emanating from the north pole. It visually illustrates the correspondence between spherical and planar points, highlighting the conformal (angle-preserving) nature of the projection. Core mathematical concepts include spherical geometry, projection transformations, conformal mapping, and the connection to the unit circle and complex plane. The construction uses GeoGebra commands such as Point, Sphere, Line, Rotate, Intersect, PointIn, Circle, Plane, OrthogonalLine, Mirror, Locus, and Midpoint to build the sphere, projection rays, intersection points, auxiliary circles, perpendiculars, mirrored points, and loci. Dynamic objects (a, f, g, α, etc.) allow interactive exploration of how points move between the sphere and the plane. Suitable for high school to university-level geometry courses, particularly for teaching solid geometry, analytic geometry, and introductory complex analysis, helping students grasp the topological and metric relationships between the sphere and the plane.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create a stereographic projection, mapping points on a unit sphere to a plane via rays. Start by drawing a unit sphere, then draw two intersecting rays with the sphere. Rotate the sphere around these rays to generate intersection points and auxiliary circles. Finally, use mirroring and rotation to create additional points and circles, illustrating the properties of the projection.
The prompt above was compiled from the original drawing commands and is for reference only.