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一边固定的四边形构造（P4）与三角函数参数化

一边固定的四边形构造（P4）与三角函数参数化
几何画板及绘图内容加载中，请稍等……
2020-09-14 14:37
多边形向量三角函数微积分几何作图
该范例演示在平面内“固定一条边”的条件下，如何用参数与随机选择生成可变四边形，并观察其形成过程与相关度量量的变化。核心数学概念包括：向量与线段的几何构造、由三角函数（如余弦）把角度/方向与长度关系联系起来、以及在需要时用微积分相关思想（如函数变化或导数/极限的观察）理解度量随参数变化的趋势。作图要点：通过 GeoGebra 的向量与线段命令（Vector、Segment）建立候选顶点的几何位置；再借助 CountIf 与 IndexOf 对集合元素进行计数与索引管理；使用 Sequence 组织参数/点的生成顺序，并用 RandomElement 随机抽取用于构造的元素；界面元素 buttonNewQ 用于触发新一轮构造，过程中展示结果对象（如 Vec21、Vec18、Vec06、Vec05 等向量对象，以及多组颜色标记 Blue1/Blue2/Blue3 与 Red/Reda/Redb）与最终答案 ShowAns。适合中学高年级到大学入门的几何建模与动态几何学习，也可用于教学演示：在保持一条边不变时，四边形如何因参数变化而改变形态。

GeoGebra 命令及参数

  1. Vector((2, 3)) → Vec21
  2. Vector((-1, 2)) → Vec18
  3. Vector((-1, -1)) → Vec06
  4. Vector((2, 1)) → Vec05
  5. Vector((1, 1)) → Vec04
  6. Vector((1, -1)) → Vec10
  7. Vector((-1, 1)) → Vec11
  8. Vector((2, -1)) → Vec12
  9. Vector((1, 2)) → Vec16
  10. Vector((1, -2)) → Vec17
  11. Vector((3, -2)) → Vec15
  12. Vector((4, -1)) → Vec14
  13. Vector((3, -1)) → Vec13
  14. Vector((4, 1)) → Vec09
  15. Vector((3, 2)) → Vec08
  16. Vector((3, 1)) → Vec07
  17. Vector((1, 3)) → Vec19
  18. Vector((-1, 3)) → Vec20
  19. Vector((1, 0)) → Vec01
  20. Vector((2, 0)) → Vec02
  21. Vector((0, 1)) → Vec03
  22. Segment(Blue1a, Blue1b) → Blue1
  23. Segment(Blue2a, Blue2b) → Blue2
  24. Segment(Blue3a, Blue3b) → Blue3
  25. Segment(Reda, Redb) → Red
  26. CountIf(A ≟ Blue1a, A, ListPoints) → NumBlue1a
  27. CountIf(A ≟ Blue1b, A, ListPoints) → NumBlue1b
  28. CountIf(A ≟ Blue2a, A, ListPoints) → NumBlue2a
  29. CountIf(A ≟ Blue2b, A, ListPoints) → NumBlue2b
  30. CountIf(A ≟ Blue3a, A, ListPoints) → NumBlue3a
  31. CountIf(A ≟ Blue3b, A, ListPoints) → NumBlue3b
  32. CountIf(A ≟ Reda, A, ListPoints) → NumReda
  33. CountIf(A ≟ Redb, A, ListPoints) → NumRedb
  34. IndexOf(SelectedElement[ListQuadType], ListQuadType) → QuadType
  35. Sequence(Sequence[(s, t), s, 0, 23], t, -3, 15) → ListDots, {{(0, -3), (1, -3), (2, -3), (3, -3), (4, -3), (5, -3), (6, -3), (7, -3), (8, -3), (9, -3), (10, -3), (11, -3), (12, -3), (13, -3), (14, -3), (15, -3), (16, -3), (17, -3), (18, -3), (19, -3), (20, -3), (21, -3), (22, -3), (23, -3)}, {(0, -2), (1, -2), (2, -2), (3, -2), (4, -2), (5, -2), (6, -2), (7, -2), (8, -2), (9, -2), (10, -2), (11, -2), (12, -2), (13, -2), (14, -2), (15, -2), (16, -2), (17, -2), (18, -2), (19, -2), (20, -2), (21, -2), (22, -2), (23, -2)}, {(0, -1), (1, -1), (2, -1), (3, -1), (4, -1), (5, -1), (6, -1), (7, -1), (8, -1), (9, -1), (10, -1), (11, -1), (12, -1), (13, -1), (14, -1), (15, -1), (16, -1), (17, -1), (18, -1), (19, -1), (20, -1), (21, -1), (22, -1), (23, -1)}, {(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0), (11, 0), (12, 0), (13, 0), (14, 0), (15, 0), (16, 0), (17, 0), (18, 0), (19, 0), (20, 0), (21, 0), (22, 0), (23, 0)}, {(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (9, 1), (10, 1), (11, 1), (12, 1), (13, 1), (14, 1), (15, 1), (16, 1), (17, 1), (18, 1), (19, 1), (20, 1), (21, 1), (22, 1), (23, 1)}, {(0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2), (11, 2), (12, 2), (13, 2), (14, 2), (15, 2), (16, 2), (17, 2), (18, 2), (19, 2), (20, 2), (21, 2), (22, 2), (23, 2)}, {(0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3), (9, 3), (10, 3), (11, 3), (12, 3), (13, 3), (14, 3), (15, 3), (16, 3), (17, 3), (18, 3), (19, 3), (20, 3), (21, 3), (22, 3), (23, 3)}, {(0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (7, 4), (8, 4), (9, 4), (10, 4), (11, 4), (12, 4), (13, 4), (14, 4), (15, 4), (16, 4), (17, 4), (18, 4), (19, 4), (20, 4), (21, 4), (22, 4), (23, 4)}, {(0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (7, 5), (8, 5), (9, 5), (10, 5), (11, 5), (12, 5), (13, 5), (14, 5), (15, 5), (16, 5), (17, 5), (18, 5), (19, 5), (20, 5), (21, 5), (22, 5), (23, 5)}, {(0, 6), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 6), (8, 6), (9, 6), (10, 6), (11, 6), (12, 6), (13, 6), (14, 6), (15, 6), (16, 6), (17, 6), (18, 6), (19, 6), (20, 6), (21, 6), (22, 6), (23, 6)}, {(0, 7), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (7, 7), (8, 7), (9, 7), (10, 7), (11, 7), (12, 7), (13, 7), (14, 7), (15, 7), (16, 7), (17, 7), (18, 7), (19, 7), (20, 7), (21, 7), (22, 7), (23, 7)}, {(0, 8), (1, 8), (2, 8), (3, 8), (4, 8), (5, 8), (6, 8), (7, 8), (8, 8), (9, 8), (10, 8), (11, 8), (12, 8), (13, 8), (14, 8), (15, 8), (16, 8), (17, 8), (18, 8), (19, 8), (20, 8), (21, 8), (22, 8), (23, 8)}, {(0, 9), (1, 9), (2, 9), (3, 9), (4, 9), (5, 9), (6, 9), (7, 9), (8, 9), (9, 9), (10, 9), (11, 9), (12, 9), (13, 9), (14, 9), (15, 9), (16, 9), (17, 9), (18, 9), (19, 9), (20, 9), (21, 9), (22, 9), (23, 9)}, {(0, 10), (1, 10), (2, 10), (3, 10), (4, 10), (5, 10), (6, 10), (7, 10), (8, 10), (9, 10), (10, 10), (11, 10), (12, 10), (13, 10), (14, 10), (15, 10), (16, 10), (17, 10), (18, 10), (19, 10), (20, 10), (21, 10), (22, 10), (23, 10)}, {(0, 11), (1, 11), (2, 11), (3, 11), (4, 11), (5, 11), (6, 11), (7, 11), (8, 11), (9, 11), (10, 11), (11, 11), (12, 11), (13, 11), (14, 11), (15, 11), (16, 11), (17, 11), (18, 11), (19, 11), (20, 11), (21, 11), (22, 11), (23, 11)}, {(0, 12), (1, 12), (2, 12), (3, 12), (4, 12), (5, 12), (6, 12), (7, 12), (8, 12), (9, 12), (10, 12), (11, 12), (12, 12), (13, 12), (14, 12), (15, 12), (16, 12), (17, 12), (18, 12), (19, 12), (20, 12), (21, 12), (22, 12), (23, 12)}, {(0, 13), (1, 13), (2, 13), (3, 13), (4, 13), (5, 13), (6, 13), (7, 13), (8, 13), (9, 13), (10, 13), (11, 13), (12, 13), (13, 13), (14, 13), (15, 13), (16, 13), (17, 13), (18, 13), (19, 13), (20, 13), (21, 13), (22, 13), (23, 13)}, {(0, 14), (1, 14), (2, 14), (3, 14), (4, 14), (5, 14), (6, 14), (7, 14), (8, 14), (9, 14), (10, 14), (11, 14), (12, 14), (13, 14), (14, 14), (15, 14), (16, 14), (17, 14), (18, 14), (19, 14), (20, 14), (21, 14), (22, 14), (23, 14)}, {(0, 15), (1, 15), (2, 15), (3, 15), (4, 15), (5, 15), (6, 15), (7, 15), (8, 15), (9, 15), (10, 15), (11, 15), (12, 15), (13, 15), (14, 15), (15, 15), (16, 15), (17, 15), (18, 15), (19, 15), (20, 15), (21, 15), (22, 15), (23, 15)}}
  36. RandomElement(ListVec01) → RanVecRectOrSq
  37. RandomElement(ListVec02) → RanVecRhom

AI 绘图提示语参考

在平面中生成一组四边形族，并固定一条边的位置不变；先给出多组候选位移向量，决定其余顶点相对该固定边的平移方向与长度，再为每种配置画出四边形的边线。设置三条颜色不同的候选线段集合用于对比，分别得到多条蓝色线段与一条红色线段作为不同方案的边；根据选中的方案选择相应的位点组合，依次生成四个顶点并连成四边形。最后在一行参数位置上展示多组结果点与对应的四边形外形变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。