积极 AI Logo 积极 AI

节日趣味（二）：圆柱与矩形截面的动态构造

节日趣味（二）：圆柱与矩形截面的动态构造
几何画板及绘图内容加载中，请稍等……
2017-12-26 09:16
立体几何变换参数方程几何作图滑块动画
① 本范例通过GeoGebra展示了一个以圆柱体为基础、结合矩形截面与平移、镜像等变换的趣味立体几何构造，呈现节日主题下的数学之美。② 核心数学概念包括圆柱体的三维表示、平面与曲面的交线（IntersectPath）、镜像对称（Mirror）、平移（Translation）以及减法运算（Subtraction）在几何体中的应用，涉及立体几何与变换几何的交叉。③ 作图要点：使用Surface命令创建圆柱曲面，通过Plane定义截面，利用IntersectPath求出交线，再借助Mirror和Line实现对称与辅助线，对象列表包含h, r, R等参数及多个点、线、面对象，体现了参数化建模与多步骤构造。④ 适合中学高年级或大学低年级学生作为立体几何与动态数学的拓展练习，也可用于节日主题的数学活动，激发学习兴趣。

GeoGebra 命令及参数

  1. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, 0, π / 6) → g
  2. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, π / 6, π / 3) → b
  3. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, π / 3, π / 2) → c
  4. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, π / 2, (2 * π / 3)) → d
  5. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (2 * π / 3), (5 * π / 6)) → e
  6. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (5 * π / 6), π) → f
  7. IntersectPath(y = 0, i) → k
  8. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, 0, h / 6) → j
  9. Plane((0, 0, 0), (3, 0, 0), (10, 0, 5)) → l
  10. Mirror(j, l) → j'
  11. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, h / 6, h / 3) → m
  12. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, h / 3, h / 2) → n
  13. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, h / 2, (2 * h / 3)) → o
  14. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, (2 * h / 3), (5 * h / 6)) → p
  15. Surface((-R) + (r * cos(u)), v, (r * sin(u)), u, 0, u_1, v, (5 * h / 6), h) → q
  16. Mirror(m, l) → m'
  17. Mirror(n, l) → n'
  18. Mirror(o, l) → o'
  19. Mirror(p, l) → p'
  20. Mirror(q, l) → q'
  21. Line((0, 0, 0), (0, 0, 1)) → s
  22. Line((0, 0, 0), (1, 0, 0)) → t
  23. Line((0, 0, 0), (0, 1, 0)) → a
  24. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, π, (7 * π / 6)) → v
  25. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (7 * π / 6), (4 * π / 3)) → w
  26. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (4 * π / 3), (3 * π / 2)) → a_1
  27. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (3 * π / 2), (5 * π / 3)) → b_1
  28. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (5 * π / 3), (11 * π / 6)) → c_1
  29. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, u_1, v, (11 * π / 6), (2 * π)) → d_1