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两正交圆柱的交集

两正交圆柱的交集
几何画板及绘图内容加载中，请稍等……
2015-06-01 23:35
立体几何几何作图函数图像
所属专题 intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
①本范例演示了两个正交圆柱的交集图形；②涉及的核心数学概念包括立体几何、交集、圆柱体；③作图要点——从下方『构造摘要』提炼用到了 Ray, Point, CurveCartesian, Cylinder, Surface, If 等GeoGebra命令/对象；④适合高中立体几何学习或数学建模入门。

GeoGebra 命令及参数

  1. Ray((0, b, c), (0, b + 1, c)) → dda'
  2. Point(dda') → A'
  3. Ray((0, (-b), c), (0, (-b) - 1, c)) → dda
  4. Point(dda) → A
  5. Ray(((-a), 0, (-c)), ((-a) - 1, 0, (-c))) → ddb
  6. Point(ddb) → B
  7. Ray((a, 0, (-c)), (a + 1, 0, (-c))) → ddb'
  8. Point(ddb') → B'
  9. CurveCartesian((a * cos(t)), sqrt(b^(2) - ((2 * c) + (a * sin(t)))^(2)), c + (a * sin(t)), t, 0, 6.283185307179586) → d
  10. CurveCartesian((a * cos(t)), (-sqrt(b^(2) - ((2 * c) + (a * sin(t)))^(2))), c + (a * sin(t)), t, 0, 6.28319) → e
  11. Cylinder(B', B, b) → cylb, f, g, j
  12. Cylinder(A, A', a) → cyla, k, h, i
  13. Surface((a * cos(u)), (y(A') * (1 - v)) + (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 0, 6.28319, v, 0, 1) → cpa1
  14. Surface((a * cos(u)), (y(A) * (1 - v)) - (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 0, 6.28319, v, 0, 1) → cpa2
  15. Surface(x(A) + (a * cos(u)), (y(A') * v) + ((1 - v) * y(A)), z(A) + (a * sin(u)), u, π - asin(((-b) - (2 * c)) / a), (2 * π) + asin(((-b) - (2 * c)) / a), v, 0, 1) → cpa4
  16. Surface((x(B') * (1 - v)) + ((v * a) * cos(u)), sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), c + (a * sin(u)), u, -1.5707963267948966, 1.5708, v, 0, 1) → cpb1
  17. Surface((x(B) * (1 - v)) + ((v * a) * cos(u)), sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), c + (a * sin(u)), u, 1.5708, 4.71238898038469, v, 0, 1) → cpb2
  18. Surface((x(B') * (1 - v)) + ((v * a) * cos(u)), (-sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, -1.5707963267948966, 1.5707963267948966, v, 0, 1) → cpb3
  19. Surface((x(B) * (1 - v)) + ((v * a) * cos(u)), (-sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 1.5707963267948966, 4.71238898038469, v, 0, 1) → cpb4
  20. Surface((x(B') * (1 - v)) + (v * x(B)), (b * cos(u)), z(B') + (b * sin(u)), u, asin((a + (2 * c)) / b), π - asin((a + (2 * c)) / b), v, 0, 1) → cpb5
  21. If(-1 ≤ (b - (2 * c)) / a ≤ 1, asin((b - (2 * c)) / a), If[-1 ≤ (b + (2 * c)) / a ≤ 1, π + asin((b + (2 * c)) / a), 0]) → uad
  22. If(-1 ≤ (b - (2 * c)) / a ≤ 1, π - asin((b - (2 * c)) / a), If[-1 ≤ (b + (2 * c)) / a ≤ 1, (2 * π) - asin((b + (2 * c)) / a), 0]) → uaf
  23. Surface(x(A) + (a * cos(u)), (y(A') * v) + ((1 - v) * y(A)), z(A) + (a * sin(u)), u, uad, uaf, v, 0, 1) → cpa3
  24. If(-1 ≤ (a - (2 * c)) / b ≤ 1, π + asin((a - (2 * c)) / b), If[-1 ≤ (a + (2 * c)) / b ≤ 1, asin((a + (2 * c)) / b), 0]) → ubd
  25. If(-1 ≤ (a - (2 * c)) / b ≤ 1, (2 * π) - asin((a - (2 * c)) / b), If[-1 ≤ (a + (2 * c)) / b ≤ 1, π - asin((a + (2 * c)) / b), 0]) → ubf
  26. Surface((x(B') * (1 - v)) + (v * x(B)), (b * cos(u)), z(B') + (b * sin(u)), u, ubd, ubf, v, 0, 1) → cpb
  27. Surface((10 * p) + ((v * a) * cos(u)), (-10 * p) - sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), (10 * p) + c + (a * sin(u)), u, 0, 6.28319, v, -1, 1) → sb'
  28. Surface((10 * p) + (a * cos(u)), (20 * q) - (10 * p) + (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), (10 * p) + c + (a * sin(u)), u, 0, 6.28319, v, -1, 1) → sa
  29. Surface((10 * p) + ((v * a) * cos(u)), (-10 * p) + sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), (10 * p) + c + (a * sin(u)), u, 0, 6.283185307179586, v, -1, 1) → sb

AI 绘图提示语参考

画两个正交的圆柱体，展示它们的交集。首先画出两条垂直的射线，分别从点(0, b, c)和(0, -b, c)出发，长度为1。然后在这两条射线上分别取点A'和A，再画出两条垂直的射线，分别从点(-a, 0, -c)和(a, 0, -c)出发，长度为1，并在射线上取点B和B'。接着，画出两个圆柱体，一个以A和B为底面，半径为a，另一个以A'和B'为底面，半径为b。最后，用曲面展示两个圆柱体的交集部分。
以上提示语为根据原始绘图命令整理而来，仅供参考。