参数环面

参数环面
几何画板及绘图内容加载中,请稍等……
2015-05-13 03:47
这个GeoGebra范例展示了如何通过参数方程描述环面及其他旋转曲面。它涉及参数曲线和曲面的概念,使用了Segment, Point, Polygon, Surface, CurveCartesian等命令。适合高中及以上学生学习微积分和立体几何。

GeoGebra 命令及参数

  1. Segment(A, ((2 * π), y(A))) → a
  2. Point(a) → B
  3. Segment(A, (x(A), (2 * π))) → b
  4. Point(b) → C
  5. Polygon(A, B, D, C) → poly1, a_1, c, f, e
  6. Surface(((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), u, x(A), x(B), t, y(A), y(C)) → d
  7. Segment(A, B) → g
  8. Segment(A, C) → h
  9. CurveCartesian(((5 + (2 * cos(u))) * cos(y(A))), ((5 + (2 * cos(u))) * sin(y(A))), (2 * sin(u)), u, x(A), x(B)) → i
  10. CurveCartesian(((5 + (2 * cos(x(A)))) * cos(t)), ((5 + (2 * cos(x(A)))) * sin(t)), (2 * sin(x(A))), t, y(A), y(C)) → j
  11. Segment(C, D) → k
  12. Segment(D, B) → l
  13. CurveCartesian(((5 + (2 * cos(u))) * cos(y(C))), ((5 + (2 * cos(u))) * sin(y(C))), (2 * sin(u)), u, x(A), x(B)) → m
  14. CurveCartesian(((5 + (2 * cos(x(B)))) * cos(t)), ((5 + (2 * cos(x(B)))) * sin(t)), (2 * sin(x(B))), t, y(A), y(C)) → n
  15. Intersect(p, g) → F
  16. Intersect(p, k) → G
  17. Segment(F, G) → q
  18. Segment(G, F) → r
  19. CurveCartesian(((5 + (2 * cos(x(H)))) * cos(t)), ((5 + (2 * cos(x(H)))) * sin(t)), (2 * sin(x(H))), t, y(A), y(C)) → u
  20. CurveCartesian(((5 + (2 * cos(u))) * cos(y(H))), ((5 + (2 * cos(u))) * sin(y(H))), (2 * sin(u)), u, x(A), x(B)) → v
  21. Intersect(b_1, k) → J
  22. Intersect(b_1, g) → K
  23. Segment(J, K) → c_1
  24. Intersect(d_1, h) → L
  25. Intersect(d_1, l) → M
  26. Segment(L, M) → e_1
  27. CurveCartesian(((5 + (2 * cos(x(F)))) * cos(t)), ((5 + (2 * cos(x(F)))) * sin(t)), (2 * sin(x(F))), t, y(A), y(C)) → o
  28. Sequence(CurveCartesian[((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), u, x(A), x(B)], t, y(A), y(C), 0.2617993877991494) → list1
  29. Sequence(CurveCartesian[((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), t, y(A), y(C)], u, x(A), x(B), 0.2618) → list2
  30. Sequence(Segment[(u, y(A)), (u, y(C))], u, x(A), x(B), 0.2617993877991494) → list3
  31. Sequence(Segment[(x(A), v), (x(B), v)], v, y(A), y(C), 0.2618) → list4

AI 绘图提示语参考

绘制一个环面,使用参数方程 (5 + 2*cos(u)) * cos(t) 和 (5 + 2*cos(u)) * sin(t) 表示环面的横截面,2*sin(u) 表示高度。首先画出环面的横截面,然后围绕中心轴旋转形成环面。接着画出环面的边界线和辅助线,最后用参数方程绘制环面的表面。
以上提示语为根据原始绘图命令整理而来,仅供参考。