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Two-point section of a 3-Sphere

Two-point section of a 3-Sphere
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2019-08-30 17:36
VectorSolid GeometryParametric EquationGeometric Construction
In collections Spherical sections of a 3-Sphere
This example demonstrates the cross-section of a 3-Sphere at two points, involving core mathematical concepts like solid geometry, vector operations, parametric equations, and spatial geometric relationships. It utilizes commands and objects such as Point, Line, Sphere, Vector, OrthogonalLine, Plane, and IntersectPath in GeoGebra. Suitable for high school students and enthusiasts of spatial geometry.

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. OrthogonalLine(B_1, xOyPlane) → i
  16. OrthogonalLine(A_1, xOyPlane) → j
  17. Point(j) → A_4
  18. Point(i) → B_4
  19. Point(j) → A_3
  20. Point(i) → B_3
  21. Line(A_4, B_4) → p_4
  22. Line(A_3, B_3) → p_3
  23. Segment(A_3, A_4) → k_1
  24. Segment(B_3, B_4) → l_1
  25. Intersect(p_4, xOyPlane) → Z_4
  26. Intersect(p_3, xOyPlane) → W_3
  27. OrthogonalLine(W_3, xOyPlane) → m
  28. OrthogonalLine(Z_4, xOyPlane) → n
  29. Intersect(n, p_3) → Z_3
  30. Intersect(p_4, m) → W_4
  31. Segment(Z_3, Z_4) → k
  32. Segment(W_4, W_3) → l
  33. Plane(p_3, p_4) → ρ
  34. IntersectPath(ρ, Σ_3) → σ_3
  35. Center(σ_3) → C_3
  36. OrthogonalLine(C_3, ρ) → r_1
  37. Distance(S_4, O) → r
  38. Sphere(C_3, r) → a
  39. Intersect(r_1, a) → C_r, C
  40. Segment(C_3, C_r) → m_1
  41. Radius(σ_3) → q
  42. Sphere(C_r, q) → σ_r
  43. OrthogonalLine(W_3, ρ) → s
  44. Distance(W_3, W_4) → w
  45. Sphere(W_3, w) → d
  46. Intersect(s, d) → A, W_r
  47. Segment(W_r, W_3) → c
  48. Line(W_r, Z_3) → p_r
  49. Intersect(p_r, σ_r) → V_r, U_r
  50. OrthogonalLine(V_r, ρ) → e
  51. OrthogonalLine(U_r, ρ) → f_1
  52. Intersect(f_1, p_3) → U_3
  53. Intersect(e, p_3) → V_3
  54. Segment(U_3, V_3) → j_1
  55. OrthogonalLine(V_3, xOyPlane) → g_1
  56. OrthogonalLine(U_3, xOyPlane) → h_1
  57. Intersect(h_1, p_4) → U_4
  58. Intersect(g_1, p_4) → V_4
  59. Segment(V_3, V_4) → t
  60. Segment(U_4, U_3) → b
  61. Segment(U_4, V_4) → i_1