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三维球面截面

三维球面截面
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2019-08-28 21:18
向量解析几何立体几何几何作图
所属专题 Spherical sections of a 3-Sphere
①本范例演示三维球面（3-Sphere）的截面图形及其变化；②涉及核心数学概念包括立体几何、球面几何、向量与坐标几何；③构造过程中运用了点、线、面、向量、旋转、平移等GeoGebra命令与对象，如Sphere, Line, Vector, Rotate, Translate等；④适合高中及以上学生及数学爱好者学习空间几何与参数化图形。

GeoGebra 命令及参数

  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