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可修改固体：垂直于y轴的规则五边形截面

可修改固体：垂直于y轴的规则五边形截面
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2019-05-10 07:07
多边形立体几何微积分
所属专题 GeoGebra 3D & AR: PreCalc & Calculus Resources Cross Sectional Specials (Calculus)
这个范例演示了一个可修改固体，其垂直于y轴的截面是规则五边形。涉及的核心数学概念包括微积分中的截面分析、几何中的多边形性质以及函数的积分应用。作图要点包括使用GeoGebra命令如Intersect（相交）、Segment（线段）、Midpoint（中点）、Line（直线）、Sphere（球体）等来构建截面和固体形状。适合高中及以上学生用于学习立体几何和微积分中的截面问题。

GeoGebra 命令及参数

  1. Intersect(lower, h, (-10, -2)) → C
  2. Intersect(upper, h, (-10, 6.159318853398537)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 6.159318853398537)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Rotate(J, (2 * pi / 5), Z, c) → J'
  22. Rotate(J', (2 * pi / 5), Z, c) → J''
  23. Rotate(J'', (2 * pi / 5), Z, c) → J'''
  24. Polygon(K, J, J', J'', J''') → poly1, k, j, j', j'', j'''
  25. Rotate(H, (2 * pi / 5), A_1, r_1) → H'
  26. Rotate(H', (2 * pi / 5), A_1, r_1) → H''
  27. Rotate(H'', (2 * pi / 5), A_1, r_1) → H'''
  28. Polygon(H, H', H'', H''', I) → poly2, h_2, h', h'', h''', i_1
  29. Rotate(G, (2 * pi / 5), B_1, l_1) → G'
  30. Rotate(G', (2 * pi / 5), B_1, l_1) → G''
  31. Rotate(G'', (2 * pi / 5), B_1, l_1) → G'''
  32. Polygon(C, G, G', G'', G''') → poly3, c_1, g, g', g'', g'''
  33. Segment(K, J''') → f
  34. Segment(J''', J'') → l
  35. Segment(J'', J') → m
  36. Segment(J', J) → n
  37. Segment(H''', I) → r
  38. Segment(H''', H'') → s
  39. Segment(H'', H') → t
  40. Segment(H', H) → f_1
  41. Segment(C, G''') → g_1
  42. Segment(G''', G'') → j_1
  43. Segment(G'', G') → m_1
  44. Segment(G', G) → n_1
  45. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * (tan((3 * pi / 10)) + 1 / cos((3 * pi / 10)))), t, a, b) → u
  46. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) - (sin((3 * pi / 10)) * (upper(t) - lower(t))), ((upper(t) - lower(t)) * sin((2 * pi / 5))), t, a, b) → o
  47. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) + (sin((3 * pi / 10)) * (upper(t) - lower(t))), ((upper(t) - lower(t)) * sin((2 * pi / 5))), t, a, b) → v
  48. Surface(u, lower(u) - ((cos((2 * pi / 5)) * (upper(u) - lower(u))) * v), (((upper(u) - lower(u)) * sin((2 * pi / 5))) * v), u, a, b, v, 0, 1) → w
  49. Surface(u, upper(u) + ((cos((2 * pi / 5)) * (upper(u) - lower(u))) * v), (((upper(u) - lower(u)) * sin((2 * pi / 5))) * v), u, a, b, v, 0, 1) → a_1
  50. Surface(u, (0.5 * (upper(u) + lower(u))) - ((upper(u) - lower(u)) * sin((3 * pi / 10))) + ((sin((3 * pi / 10)) * (upper(u) - lower(u))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 5))) + (((upper(u) - lower(u)) * cos((3 * pi / 10))) * v), u, a, b, v, 0, 1) → b_1
  51. Surface(u, (0.5 * (upper(u) + lower(u))) + ((sin((3 * pi / 10)) * (upper(u) - lower(u))) * v), ((0.5 * (upper(u) - lower(u))) * (tan((3 * pi / 10)) + 1 / cos((3 * pi / 10)))) - (((upper(u) - lower(u)) * cos((3 * pi / 10))) * v), u, a, b, v, 0, 1) → d_1

AI 绘图提示语参考

绘制一个可修改的立体图形，其横截面平行于y轴的截面是正五边形。首先确定上下边界，然后通过计算找到关键点并连接成线段，接着以这些线段为基础构造正五边形。将这些正五边形围绕中心轴旋转，形成连续的立体结构。最后，将所有旋转后的正五边形组合成一个整体，展示出立体图形的完整形态。
以上提示语为根据原始绘图命令整理而来，仅供参考。