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球体穿过三角形的概率与积分计算

球体穿过三角形的概率与积分计算
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2019-07-05 15:08
向量微积分概率统计最值优化几何作图
①本范例演示球体穿过三角形区域的过程，并计算其概率；②涉及核心数学概念包括概率统计、对数函数、积分计算、最值优化、差分与斜率、平方运算；③作图要点：利用Vector（向量）、Point（点）、Circle（圆）、OrthogonalLine（垂线）、Intersect（交点）、Segment（线段）、Polygon（多边形）、Line（直线）等GeoGebra命令构建图形；④适合高中及以上学生学习微积分、概率统计及几何作图。

GeoGebra 命令及参数

  1. Vector(A, B) → u
  2. Vector(B, C) → v
  3. Vector(C, D) → w
  4. Vector(D, E) → a
  5. Vector(E, F) → b
  6. Vector(F, G) → c
  7. Point(xAxis) → H
  8. Point(xAxis) → I
  9. Point(xAxis) → J
  10. Point(xAxis) → K
  11. Point(xAxis) → L
  12. Point(xAxis) → M
  13. Point(xAxis) → N
  14. Circle(H, r1) → d
  15. Circle(I, r2) → e
  16. Circle(J, r3) → g
  17. Circle(K, r4) → h
  18. Circle(L, r5) → k
  19. Circle(M, r6) → p
  20. Circle(N, r7) → q
  21. OrthogonalLine(H, xAxis) → i
  22. OrthogonalLine(I, xAxis) → j
  23. OrthogonalLine(J, xAxis) → l
  24. OrthogonalLine(K, xAxis) → m
  25. OrthogonalLine(L, xAxis) → n_1
  26. OrthogonalLine(M, xAxis) → r
  27. OrthogonalLine(N, xAxis) → s
  28. Intersect(d, i, 1) → O
  29. Segment(H, O) → t
  30. Intersect(e, j, 1) → P
  31. Segment(I, P) → f_1
  32. Intersect(g, l, 1) → Q
  33. Segment(J, Q) → g_1
  34. Intersect(h, m, 1) → R
  35. Segment(K, R) → h_1
  36. Intersect(k, n_1, 1) → S
  37. Segment(L, S) → i_1
  38. Intersect(p, r, 1) → T
  39. Segment(M, T) → j_1
  40. Intersect(q, s, 1) → U
  41. Segment(N, U) → k_1
  42. Polygon(V, W, Z) → t8, z_1, v_1, w_1
  43. Point(v_1) → A_1
  44. Point(v_1) → B_1
  45. Point(v_1) → C_1
  46. Point(v_1) → D_1
  47. Point(v_1) → E_1
  48. Line(A_1, z_1) → m_1
  49. Line(B_1, z_1) → p_1
  50. Line(C_1, z_1) → q_1
  51. Line(D_1, z_1) → r_1
  52. Line(E_1, z_1) → s_1
  53. Line(E_1, w_1) → t_1
  54. Line(D_1, w_1) → a_1
  55. Line(C_1, w_1) → b_1
  56. Line(B_1, w_1) → c_1
  57. Line(A_1, w_1) → d_1
  58. Intersect(m_1, w_1) → F_1
  59. Segment(A_1, F_1) → e_1
  60. Intersect(p_1, w_1) → G_1
  61. Segment(B_1, G_1) → f_2
  62. Intersect(q_1, w_1) → H_1
  63. Segment(C_1, H_1) → g_2
  64. Intersect(r_1, w_1) → I_1
  65. Segment(D_1, I_1) → h_2
  66. Intersect(s_1, w_1) → J_1
  67. Segment(E_1, J_1) → i_2
  68. Intersect(t_1, z_1) → K_1
  69. Segment(E_1, K_1) → j_2
  70. Intersect(a_1, z_1) → L_1
  71. Segment(D_1, L_1) → k_2
  72. Intersect(b_1, z_1) → M_1
  73. Segment(C_1, M_1) → l_2
  74. Intersect(c_1, z_1) → N_1
  75. Segment(B_1, N_1) → m_2
  76. Intersect(d_1, z_1) → O_1
  77. Segment(A_1, O_1) → n_2
  78. Line(P_1, Q_1) → p_2
  79. Line(R, p_2) → q_2
  80. Point(p_2) → R_1
  81. Point(p_2) → S_1
  82. Point(p_2) → T_1
  83. Point(p_2) → U_1
  84. Point(p_2) → V_1
  85. Point(p_2) → W_1
  86. OrthogonalLine(T_1, p_2) → r_2
  87. OrthogonalLine(U_1, p_2) → s_2
  88. OrthogonalLine(R_1, p_2) → t_2
  89. OrthogonalLine(S_1, p_2) → a_2
  90. OrthogonalLine(V_1, p_2) → b_2
  91. OrthogonalLine(W_1, p_2) → c_2
  92. Point(p_2) → Z_1
  93. Point(p_2) → A_2
  94. OrthogonalLine(Z_1, p_2) → d_2
  95. OrthogonalLine(A_2, p_2) → e_2
  96. Intersect(q_2, t_2) → B_2
  97. Intersect(q_2, a_2) → C_2
  98. Polygon(B_2, C_2, S_1, R_1) → q1, f_3, g_3, h_3, i_3
  99. Line(Q, p_2) → j_3
  100. Intersect(s_2, j_3) → D_2
  101. Intersect(t_2, j_3) → E_2
  102. Polygon(D_2, E_2, R_1, U_1) → q2, k_3, l_3, m_3, u_1
  103. Line(S, p_2) → n_3
  104. Intersect(a_2, n_3) → F_2
  105. Intersect(b_2, n_3) → G_2
  106. Polygon(F_2, G_2, V_1, S_1) → q3, p_3, q_3, r_3, s_3
  107. Line(P, p_2) → t_3
  108. Intersect(t_3, r_2) → H_2
  109. Intersect(t_3, s_2) → I_2
  110. Polygon(H_2, I_2, U_1, T_1) → q4, a_3, b_3, c_3, d_3
  111. Line(T, p_2) → e_3
  112. Intersect(b_2, e_3) → J_2
  113. Intersect(c_2, e_3) → K_2
  114. Polygon(J_2, K_2, W_1, V_1) → q5, f_4, g_4, h_4, i_4
  115. Line(O, p_2) → j_4
  116. Intersect(d_2, j_4) → L_2
  117. Intersect(r_2, j_4) → M_2
  118. Polygon(L_2, M_2, T_1, Z_1) → q6, k_4, l_4, m_4, n_4
  119. Line(U, p_2) → p_4
  120. Intersect(c_2, p_4) → N_2
  121. Intersect(e_2, p_4) → O_2
  122. Polygon(N_2, O_2, A_2, W_1) → q7, q_4, o_2, r_4, s_4