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Spherical Section of a 3-Sphere

Spherical Section of a 3-Sphere
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2019-08-28 21:18
VectorCoordinate GeometrySolid GeometryGeometric Construction
In collections Spherical sections of a 3-Sphere
This example demonstrates the cross-section of a 3-Sphere, showcasing its geometric properties and transformations. Key mathematical concepts include solid geometry, spherical geometry, vectors, and coordinate geometry. GeoGebra commands and objects used in construction are Point, Intersect, Sphere, Line, Vector, Rotate, Translate, etc. Suitable for high school students and math enthusiasts exploring spatial geometry and parametric equations.

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) → i
  18. PointIn(p) → C
  19. OrthogonalLine(C, p) → j
  20. Point(j) → F
  21. Point(j) → G
  22. Plane(F, i) → q
  23. Plane(G, i) → r
  24. OrthogonalPlane(S_3, i) → p_1
  25. OrthogonalLine(S_3, r) → n
  26. IntersectPath(r, p_1) → k
  27. Line(S_3, k) → l
  28. Intersect(l, Σ_3, 2) → S_0
  29. Sphere(S_0, S_3) → Σ_0
  30. Intersect(l, p) → I
  31. OrthogonalLine(I, p) → m
  32. Intersect(m, q) → J
  33. Segment(I, J) → j_3
  34. Plane(J, p) → q_1
  35. Intersect(h, q_1) → P_4
  36. Plane(S_3, r) → r_1
  37. Sphere(S_3, z(P_4)) → d
  38. Intersect(l, d) → L, P_0
  39. Intersect(n, r) → N
  40. Plane(N, p) → p_2
  41. IntersectPath(p_2, r) → s
  42. Plane(s, n) → r_2
  43. Plane(P_0, s) → q_2
  44. IntersectPath(q_2, Σ_0) → k_0
  45. Intersect(p_1, k_0) → P, Q
  46. Line(P, l) → t
  47. Center(k_0) → C_0
  48. Line(C_0, l) → f_1
  49. Intersect(f_1, n) → C_3
  50. Intersect(t, n) → T
  51. OrthogonalLine(C_0, p_1) → g_1
  52. Intersect(f_1, k_0) → U,
  53. Intersect(g_1, k_0) → V, W
  54. Line(V, l) → h_1
  55. Intersect(h_1, r_2) → Z
  56. Line(P, C_0) → i_1
  57. AngularBisector(i_1, t) → j_1, k_1
  58. Intersect(k_1, f_1) → A_1
  59. OrthogonalLine(A_1, q_2) → l_1
  60. Distance(C_3, Z) → a
  61. Circle(T, a, t) → c_1
  62. Line(C_3, Z) → m_1
  63. Intersect(m_1, c_1) → B_1, C_1
  64. Ellipse(C_1, B_1, T) → k_3
  65. OrthogonalLine(N, p) → b_3
  66. Intersect(b_3, p) → H
  67. IntersectPath(p, r_2) → c_5
  68. Point(c_5) → D_1
  69. OrthogonalLine(D_1, p) → e
  70. Intersect(e, q) → E_1
  71. Line(E_1, c_5) → n_1
  72. Plane(H, n_1) → p_3
  73. Line(C_3, h) → s_1
  74. Line(T, wdir) → t_1
  75. Line(Z, wdir) → a_1
  76. Intersect(a_1, p_3) → F_1
  77. Intersect(s_1, p_3) → C_4
  78. Intersect(t_1, p_3) → H_1
  79. Line(C_4, H_1) → b_1
  80. Line(C_4, F_1) → d_1
  81. OrthogonalLine(H_1, p_3) → e_1
  82. Circle(H_1, a, e_1) → c_2
  83. Intersect(d_1, c_2) → I_1, J_1
  84. Ellipse(J_1, I_1, H_1) → k_4
  85. Plane(C_4, p) → q_3
  86. IntersectPath(q_3, Σ_4) → e_2
  87. Radius(e_2) → o
  88. Sphere(S_3, o) → u
  89. Intersect(f_1, q_3) → K_1
  90. Intersect(f_1, u) → L_1, M_1
  91. Plane(M_1, m_1) → r_3
  92. OrthogonalLine(M_1, r_3) → f_2
  93. Circle(n, Z) → l_3
  94. OrthogonalLine(T, p_1) → h_2
  95. Circle(T, a, h_2) → k_2
  96. Intersect(r_3, k_2) → N_1, O_1
  97. Ellipse(N_1, O_1, T) → m_3
  98. IntersectPath(p_1, q_3) → i_2
  99. OrthogonalLine(M_1, p) → m_2
  100. OrthogonalLine(L_1, p) → n_2
  101. Intersect(n_2, q_3) → P_1
  102. Intersect(m_2, q_3) → Q_1
  103. Segment(P_1, Q_1) → g_2
  104. OrthogonalLine(C_4, p_3) → j_2
  105. Circle(d_1, H_1) → t_2
  106. Intersect(j_2, t_2) → R_1, S_1
  107. Line(R_1, P_1) → l_2
  108. Midpoint(R_1, P_1) → T_1
  109. OrthogonalLine(T_1, p_1) → a_2
  110. Circle(a_2, C_4) → c_3
  111. Intersect(l_2, c_3) → U_1, V_1
  112. Line(C_4, U_1) → b_2
  113. Line(C_4, V_1) → f_3
  114. Distance(V_1, P_1) → e1
  115. Distance(R_1, V_1) → e2
  116. Circle(C_4, e1, p_3) → d_3
  117. Circle(C_4, e2, d_1) → e_3
  118. Intersect(b_2, t_2) → W_1, Z_1
  119. Intersect(f_3, e_3) → A_2, B_2
  120. OrthogonalLine(A_2, p_1) → g_3
  121. Circle(A_2, a, g_3) → h_3
  122. Intersect(b_2, h_3) → C_2, D_2
  123. Distance(F_1, C_4) → c_6
  124. Plane(F_1, f_3) → p_4
  125. OrthogonalLine(A_2, p_4) → i_3
  126. Circle(A_2, a, i_3) → t_3
  127. Intersect(d_1, t_3) → E_2, F_2
  128. Ellipse(E_2, F_2, A_2) → c_4
  129. Circle(C_4, a, f_3) → d_4
  130. Distance(C_3, T) → b
  131. Ellipse(C_2, D_2, A_2) → b_4
  132. Surface(((a * cos(u)) * sin(v)), ((a * sin(u)) * sin(v)), (b * cos(v)), u, 0, (2 * π), v, 0, π) → v
  133. Vector(O, C_3) → w
  134. Translate(v, w) → v'
  135. Angle(r, p) → α
  136. Rotate(v', (-α), m_1) → σ_3
  137. Distance(C_4, A_2) → c
  138. Surface(((a * cos(u)) * sin(v)), ((a * sin(u)) * sin(v)), (c * cos(v)), u, 0, (2 * π), v, 0, π) → o_1
  139. Vector(O, C_4) → u_1
  140. Translate(o_1, u_1) → v_1
  141. Angle(q, p) → β
  142. Rotate(v_1, β, d_1) → σ_4
  143. Intersect(s_1, p) → K
  144. Distance(K, C_3) → zc
  145. Circle(C_4, zc, f_3) → d_2
  146. Intersect(b_2, d_2) → M, R
  147. Intersect(f_3, q) → S
  148. OrthogonalLine(S, p_1) → s_2
  149. Circle(s_2, R) → s_3
  150. Intersect(f_3, s_3) → C_r, G_2
  151. Sphere(C_r, a) → σ_r