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可修改立体：平行于y轴的等腰梯形截面

可修改立体：平行于y轴的等腰梯形截面
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2019-07-09 00:31
函数立体几何微积分积分滑块动画
所属专题 Cross Sectional Specials (Calculus)
本范例演示一个由上下两条函数曲线围成的立体，其所有平行于y轴的截面均为等腰梯形，且梯形的较长底边等于上函数值减去下函数值。核心数学概念是积分学中的截面法求体积，通过动态调整上下函数、旋转角度和高度等参数，直观展示立体形状随截面变化的过程。作图要点包括：使用Line、Ray、Circle、Segment等构造基础几何元素，通过Rotate、Translate、Mirror实现空间变换，利用CurveCartesian和Surface生成曲面，并借助Textfield、Slider等交互控件实现参数调节。该范例适合高中至大学阶段学习微积分、立体几何和函数图像的学生，帮助理解截面面积与体积积分的关系。

GeoGebra 命令及参数

  1. Line(I, I + (1, 0, 0)) → f_1
  2. Rotate(H, (ANG * pi / 180), I, f_1) → H'
  3. Rotate(I, ((-ANG) * pi / 180), H, Plane[H', I, H]) → I'_1
  4. Ray(H, I'_1) → p_2
  5. Ray(I, H') → q_2
  6. Intersect(q_2, p_2) → W_1
  7. Circle(W_1, (SLratio * Distance[W_1, I]), Plane[W_1, I, H]) → e_2
  8. Intersect(q_2, e_2) → B_2, C_2
  9. Intersect(p_2, e_2) → Z_1, A_2
  10. Line(K, K + (1, 0, 0)) → t
  11. Rotate(J, (ANG * pi / 180), K, t) → J'_1
  12. Ray(K, J'_1) → k_2
  13. Rotate(K, ((-ANG) * pi / 180), J, Plane[J'_1, K, J]) → K'
  14. Ray(J, K') → j_2
  15. Intersect(k_2, j_2) → R_1
  16. Circle(R_1, (SLratio * Distance[R_1, K]), Plane[J'_1, K, J]) → d_2
  17. Intersect(j_2, d_2) → U_1, V_1
  18. Intersect(k_2, d_2) → S_1, T_1
  19. Intersect(lower, h, (0, -9)) → C
  20. Line(C, C + (1, 0, 0)) → c
  21. Intersect(upper, h, (0, 5)) → B
  22. Segment(C, B) → q
  23. Intersect(upper, q, (0, 5)) → G
  24. Rotate(G, (ANG * pi / 180), C, c) → G'_1
  25. Rotate(C, (ANG * pi / 180), G, Plane[G, C, G'_1]) → C'
  26. Ray(G, C') → f_2
  27. Ray(C, G'_1) → g_2
  28. Intersect(g_2, f_2) → M_1
  29. Circle(M_1, (SLratio * Distance[M_1, C]), Plane[C, G'_1, G]) → c_2
  30. Intersect(g_2, c_2) → N_1, O_1
  31. Intersect(f_2, c_2) → P_1, Q_1
  32. Translate(G'_1, Vector[C, G]) → J_1
  33. Segment(H, I) → s_1
  34. Midpoint(s_1) → A_1
  35. Segment(J, K) → q_1
  36. Midpoint(q_1) → Z
  37. Midpoint(s_1) → E
  38. Midpoint(q_1) → F
  39. Midpoint(C, G) → T
  40. Line(B, D) → i
  41. Sphere(B, d) → e
  42. Segment(C, B) → p
  43. Textfield(upper) → InputBox1
  44. Textfield(lower) → InputBox2
  45. Textfield(a) → InputBox3
  46. Textfield(b) → InputBox4
  47. Segment(C, G) → k_1
  48. Line(C, G) → j
  49. Sphere(G, T) → k
  50. Line(T, T + (1, 0, 0)) → m
  51. Rotate(G, pi / 3, F_1, m) → G'
  52. Rotate(G', pi / 3, F_1, Line[F_1, F_1 + (1, 0, 0)]) → G''
  53. Line(F_1, F_1 + (1, 0, 0)) → n
  54. Polygon(C, G, Q_1, O_1) → poly1, g_1, m_1, r_1, a_1
  55. Rotate(G'', pi / 3, F_1, n) → G'''
  56. Rotate(G''', pi / 3, F_1, n) → G_1
  57. Line(Z, Z + (1, 0, 0)) → f
  58. Line(D_1, D_1 + (1, 0, 0)) → g
  59. Rotate(J, pi / 3, D_1, g) → J'
  60. Rotate(J', pi / 3, D_1, g) → J''
  61. Rotate(J'', pi / 3, D_1, g) → J'''
  62. Rotate(J''', pi / 3, D_1, g) → E_1
  63. Line(H_1, H_1 - (1, 0, 0)) → l_1
  64. Rotate(I, pi / 3, H_1, l_1) → I'
  65. Rotate(I', pi / 3, H_1, l_1) → I''
  66. Rotate(I'', pi / 3, H_1, l_1) → I'''
  67. Rotate(I''', pi / 3, H_1, l_1) → I_1
  68. Translate(J'_1, Vector[K, J]) → K_1
  69. Polygon(K, J, V_1, T_1) → q1, j_1, l_2, m_2, n_2
  70. Polygon(I, H, A_2, C_2) → q2, e_1, h_2, i_2, f_3
  71. Translate(H', Vector[I, H]) → H''
  72. Intersect(t, q_1) → L_1
  73. Angle(G, C, G'_1) → α
  74. Segment(O_1, Q_1) → r
  75. Segment(G, Q_1) → l
  76. Angle(Q_1, G, C) → β
  77. Segment(O_1, C) → s
  78. Segment(T_1, L_1) → d_1
  79. Segment(J, V_1) → b_1
  80. Segment(T_1, V_1) → c_1
  81. Segment(C_2, I) → p_1
  82. Segment(H, A_2) → i_1
  83. Segment(A_2, C_2) → n_1
  84. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) - ((SLratio * 0.5) * (upper(t) - lower(t))), ((((1 - SLratio) * 0.5) * (upper(t) - lower(t))) * tan((ANG * pi / 180))), t, a, b) → o
  85. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) + ((SLratio * 0.5) * (upper(t) - lower(t))), ((((1 - SLratio) * 0.5) * (upper(t) - lower(t))) * tan((ANG * pi / 180))), t, a, b) → u
  86. Surface(u, lower(u) + ((((1 - SLratio) * 0.5) * (upper(u) - lower(u))) * v), (((((1 - SLratio) * 0.5) * (upper(u) - lower(u))) * tan((ANG * pi / 180))) * v), u, a, b, v, 0, 1) → o_1
  87. Surface(u, upper(u) - ((((1 - SLratio) * 0.5) * (upper(u) - lower(u))) * v), (((((1 - SLratio) * 0.5) * (upper(u) - lower(u))) * tan((ANG * pi / 180))) * v), u, a, b, v, 0, 1) → u_1
  88. Surface(u, lower(u) + (((1 - SLratio) * 0.5) * (upper(u) - lower(u))) + ((SLratio * (upper(u) - lower(u))) * v), ((((1 - SLratio) * 0.5) * (upper(u) - lower(u))) * tan((ANG * pi / 180))), u, a, b, v, 0, 1) → v_1
  89. Mirror(o_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → w_1
  90. Mirror(o, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → o'
  91. Mirror(u_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → a_2
  92. Mirror(v_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → b_2
  93. Mirror(J, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → J'_2
  94. Mirror(V_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → D_2
  95. Segment(J'_2, D_2) → r_2
  96. Mirror(T_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → E_2
  97. Mirror(L_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → F_2
  98. Segment(E_2, F_2) → s_2
  99. Mirror(T_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → G_2
  100. Mirror(V_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → H_2
  101. Segment(G_2, H_2) → t_2
  102. Mirror(poly1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → poly1'
  103. Mirror(O_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → I_2
  104. Mirror(C, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → C'_1
  105. Segment(I_2, C'_1) → s'
  106. Mirror(O_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → J_2
  107. Mirror(Q_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → K_2
  108. Segment(J_2, K_2) → r'
  109. Mirror(G, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → G'_2
  110. Mirror(Q_1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → L_2
  111. Segment(G'_2, L_2) → l'
  112. Mirror(u, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → u'
  113. Mirror(q1, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → q1'
  114. Mirror(q2, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → q2'
  115. Mirror(C_2, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → M_2
  116. Mirror(I, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → I'_2
  117. Segment(M_2, I'_2) → g_3
  118. Mirror(H, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → H'_1
  119. Mirror(A_2, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → N_2
  120. Segment(H'_1, N_2) → h_3
  121. Mirror(A_2, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → O_2
  122. Mirror(C_2, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → P_2
  123. Segment(O_2, P_2) → i_3
  124. Angle(J_2, C, G) → γ
  125. Angle(L_2, G, C) → δ
  126. Textfield(SLratio) → InputBox5

AI 绘图提示语参考

绘制一个三维立体图形，其垂直于y轴的截面为等腰梯形。该立体由上下两条函数曲线界定，梯形的较长底边等于上下函数之差。通过滑块调整上下函数、旋转角度和高度，观察截面变化。图形包含两个主要截面区域，每个区域由两条射线和圆弧构成，形成对称的梯形轮廓。立体表面由多个平面和曲面拼接而成，并显示关键点、线段和角度标记。整体呈现动态可调的几何体，用于演示积分求体积的截面法。
以上提示语为根据原始绘图命令整理而来，仅供参考。