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Circle Section of a 3-Sphere

Circle Section of a 3-Sphere
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2019-08-29 21:37
CircleCoordinate GeometrySolid GeometryParametric EquationGeometric Construction
In collections Spherical sections of a 3-Sphere
① This example demonstrates how to obtain a circular cross-section of a 4-dimensional unit sphere (3-Sphere) by intersecting it with a 3-dimensional hyperplane, and shows the projection of that circle into 3D space. ② Core mathematical concepts include higher-dimensional spheres, hyperplanes, orthogonal projection, generation of circles and ellipses, and interactions among points, lines, and planes in four dimensions. ③ Construction highlights: using GeoGebra commands such as Point, Intersect, Sphere, Line, Vector, Plane, OrthogonalLine, IntersectPath, Center, Radius, Distance, Segment, OrthogonalPlane, Midpoint, Circle, and Ellipse, along with numerous auxiliary objects (e.g., S_4, Σ_4, S_3, Σ_3, D, f, E, g, h, p) and direction vectors (xdir, ydir, wdir, zdir) to precisely control the section position, finally generating the circle c and its radius r. ④ Suitable for advanced undergraduate or graduate students studying higher-dimensional geometry, topology, or mathematical visualization, and also useful as an interactive teaching tool for instructors to illustrate abstract high-dimensional concepts.

GeoGebra Commands & Parameters

  1. Point(zAxis) → S_4
  2. Intersect(zAxis, yAxis) → O
  3. Sphere(S_4, 1) → Σ_4
  4. Point(zAxis) → S_3
  5. Sphere(S_3, 1) → Σ_3
  6. Point(xAxis) → D
  7. Line(O, D) → f
  8. Point(yAxis) → E
  9. Line(O, E) → g
  10. Line(O, S_4) → h
  11. Vector(O, D) → xdir
  12. Vector(O, E) → ydir
  13. Vector(O, S_4) → wdir
  14. Vector(O, S_3) → zdir
  15. PointIn(p) → A
  16. PointIn(p) → B
  17. Line(A, B) → z_4
  18. Line(G, H) → w_3
  19. Plane(C, z_4) → α_4
  20. Plane(F, w_3) → α_3
  21. OrthogonalLine(B, xOyPlane) → i
  22. OrthogonalLine(A, xOyPlane) → j
  23. Intersect(j, α_3) → I
  24. Intersect(i, α_3) → J
  25. Line(I, J) → z_3
  26. OrthogonalLine(G, xOyPlane) → w_1
  27. OrthogonalLine(H, xOyPlane) → l
  28. Intersect(l, α_4) → K
  29. Intersect(w_1, α_4) → L
  30. Line(L, K) → w_4
  31. IntersectPath(α_3, Σ_3) → c
  32. Center(c) → C_3
  33. Radius(c) → r
  34. OrthogonalLine(C_3, α_3) → k
  35. Distance(O, S_4) → wS
  36. Sphere(C_3, wS) → b_1
  37. Intersect(k, b_1) → C_r, N
  38. Sphere(C_r, r) → d
  39. OrthogonalLine(C_3, xOyPlane) → m
  40. Intersect(m, α_4) → P_4
  41. Plane(S_4, xOyPlane) → q_2
  42. Intersect(m, q_2) → C_4
  43. Segment(C_4, C_3) → l_2
  44. Intersect(m, xOyPlane) → C_0
  45. Distance(P_4, C_0) → wC
  46. Sphere(C_3, wC) → a_1
  47. Intersect(k, a_1) → P_r, P
  48. Plane(P_r, z_3) → α_r
  49. IntersectPath(α_r, d) → k_r
  50. OrthogonalPlane(C_3, z_3) → q
  51. Intersect(q, k_r) → M, Q
  52. Midpoint(M, Q) → O_r
  53. OrthogonalLine(O_r, q) → t
  54. Line(M, Q) → s
  55. Intersect(t, k_r) → R, S
  56. Line(O_r, k) → n
  57. Intersect(n, α_3) → O_3
  58. Line(M, k) → g_1
  59. Line(Q, k) → i_1
  60. Intersect(i_1, α_3) → T
  61. Intersect(g_1, α_3) → W
  62. Distance(T, C_3) → b_2
  63. Line(R, k) → a_2
  64. Line(S, k) → e
  65. Intersect(a_2, α_3) → U
  66. Intersect(e, α_3) → V
  67. Line(U, V) → f_1
  68. Line(W, T) → h_1
  69. Distance(V, O_3) → a
  70. Distance(O_3, T) → b
  71. Circle(T, a, i_1) → c_1
  72. Intersect(f_1, c_1) → Z, A_1
  73. Segment(M, W) → m_2
  74. Segment(R, U) → n_2
  75. Segment(O_r, A_1) → r_2
  76. Segment(S, V) → s_2
  77. Segment(Q, T) → t_2
  78. Segment(O_r, O_3) → f_3
  79. Ellipse(Z, A_1, W) → d_1
  80. OrthogonalLine(W, xOyPlane) → j_1
  81. OrthogonalLine(T, xOyPlane) → k_1
  82. OrthogonalLine(V, xOyPlane) → l_1
  83. OrthogonalLine(U, xOyPlane) → m_1
  84. Intersect(j_1, α_4) → B_1
  85. Intersect(l_1, α_4) → C_1
  86. Intersect(m_1, α_4) → D_1
  87. Intersect(k_1, α_4) → E_1
  88. OrthogonalLine(O_3, xOyPlane) → n_1
  89. Intersect(n_1, α_4) → O_4
  90. Segment(T, E_1) → g_3
  91. Segment(V, C_1) → h_3
  92. Segment(U, D_1) → i_3
  93. Segment(W, B_1) → j_3
  94. Segment(O_3, O_4) → k_3
  95. Line(E_1, B_1) → p_1
  96. Line(C_1, D_1) → q_1
  97. OrthogonalPlane(O_4, q_1) → r_1
  98. IntersectPath(r_1, α_4) → s_1
  99. OrthogonalLine(O_4, α_4) → t_1
  100. Circle(t_1, C_1) → e_1
  101. Intersect(s_1, e_1) → F_1, G_1
  102. Line(G_1, E_1) → f_2
  103. Midpoint(E_1, G_1) → H_1
  104. OrthogonalLine(H_1, α_4) → g_2
  105. Circle(g_2, O_4) → c_2
  106. Intersect(f_2, c_2) → I_1, J_1
  107. Line(J_1, O_4) → h_2
  108. Line(I_1, O_4) → i_2
  109. Distance(J_1, E_1) → a_4
  110. Distance(G_1, J_1) → b_4
  111. Circle(O_4, a_4, t_1) → d_2
  112. Circle(O_4, b_4, t_1) → e_2
  113. Intersect(i_2, d_2) → K_1, L_1
  114. Intersect(h_2, e_2) → M_1, N_1
  115. OrthogonalLine(N_1, α_4) → j_2
  116. Circle(N_1, a_4, j_2) → k_2
  117. Intersect(i_2, k_2) → O_1, P_1
  118. Ellipse(P_1, O_1, N_1) → p_2