积极 AI Logo 积极 AI

“脐带”圆锥/椭圆几何作图与相似关系演示

“脐带”圆锥/椭圆几何作图与相似关系演示
几何画板及绘图内容加载中，请稍等……
2023-03-15 21:38
圆角解析几何几何作图函数图像
所属专题 geometriadinamica.es
本范例展示如何在坐标平面中构造一组与“脐带（umbilical）”相关的圆锥曲线关系，并通过交点与切线/角度约束体现几何对称与相似性质。核心数学概念包括：圆与椭圆的几何约束、二次函数与圆锥曲线的对应、相似（由角度/比例关系导出）、以及由参数/点序列驱动的构形过程。作图要点：利用 GeoGebra 的 Segment、Point、OrthogonalLine 构造基础线段与垂线；用 Circle 与 Ellipse 建立圆和椭圆；通过 Intersect 取得关键交点；用 Angle 计算并施加角度条件；通过 CurveCartesian 生成笛卡尔坐标曲线；再借助 Sequence 与包含参数的构造对象（如通过 v、u、p、Q、z/zz、ZZ、zElipse 等派生变量与对象链）实现连续位置更新。适合高中到大学初级的解析几何、圆锥曲线与几何作图课程，用于课堂演示、探究活动或竞赛训练中的“二次曲线几何理解”。

GeoGebra 命令及参数

  1. Segment(Q5, Q6) → p5
  2. Point(p5) → P5
  3. Segment(Q3, Q4) → p2
  4. Point(p2) → P3
  5. Segment(P3, Q4) → p4
  6. Point(p4) → P4
  7. Segment(Q1, Q2) → p1
  8. Point(p1) → P1
  9. Segment(P1, Q2) → p3
  10. Point(p3) → P2
  11. OrthogonalLine(C, xAxis) → zz2
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz2, 1) → ZZ4
  14. Intersect(zz0, zz2, 2) → ZZ3
  15. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  16. Point(zElipse2) → Z
  17. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  18. Angle(ZZ8, C, Z) → zAngInc
  19. OrthogonalLine(C, yAxis) → zz1
  20. Intersect(zz0, zz1, 2) → ZZ2
  21. Intersect(zz0, zz1, 1) → ZZ1
  22. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  23. Point(zElipse1) → XY
  24. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  25. Angle(ZZ6, C, XY) → zAngRot
  26. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 8 * (((sin((3 * α)) * (7 + cos(α - (2 * v)) + (2 * cos(α + v)))) * sin(zAngRot)) + ((cos((3 * α)) * (7 + cos(α - (2 * v)) + (2 * cos(α + v)))) * cos(zAngRot)))), (r / 8 * (((((-sin((3 * α))) * (7 + cos(α - (2 * v)) + (2 * cos(α + v)))) * cos(zAngRot)) * sin(zAngInc)) + (((cos((3 * α)) * (7 + cos(α - (2 * v)) + (2 * cos(α + v)))) * sin(zAngRot)) * sin(zAngInc)) + ((sin(α - (2 * v)) + (2 * sin(α + v))) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 8 * (((sin((3 * u)) * (7 + cos(u) + (2 * cos(u)))) * sin(zAngRot)) + ((cos((3 * u)) * (7 + cos(u) + (2 * cos(u)))) * cos(zAngRot)))), (r / 8 * (((((-sin((3 * u))) * (7 + cos(u) + (2 * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((cos((3 * u)) * (7 + cos(u) + (2 * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sin(u) + (2 * sin(u))) * cos(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r / 8 * (((sin(β) * (7 + cos(β / 3 - (2 * v)) + (2 * cos(β / 3 + v)))) * sin(zAngRot)) + ((cos(β) * (7 + cos(β / 3 - (2 * v)) + (2 * cos(β / 3 + v)))) * cos(zAngRot)))), (r / 8 * (((((-sin(β)) * (7 + cos(β / 3 - (2 * v)) + (2 * cos(β / 3 + v)))) * cos(zAngRot)) * sin(zAngInc)) + (((cos(β) * (7 + cos(β / 3 - (2 * v)) + (2 * cos(β / 3 + v)))) * sin(zAngRot)) * sin(zAngInc)) + ((sin(β / 3 - (2 * v)) + (2 * sin(β / 3 + v))) * cos(zAngInc)))), v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r / 8 * (((sin(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * sin(zAngRot)) + ((cos(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * cos(zAngRot)))), (r / 8 * (((((-sin(u)) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * cos(zAngRot)) * sin(zAngInc)) + (((cos(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * sin(zAngRot)) * sin(zAngInc)) + ((sin(u / 3 - (2 * v)) + (2 * sin(u / 3 + v))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 8 * (((sin(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * sin(zAngRot)) + ((cos(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * cos(zAngRot)))), (r / 8 * (((((-sin(u)) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * cos(zAngRot)) * sin(zAngInc)) + (((cos(u) * (7 + cos(u / 3 - (2 * v)) + (2 * cos(u / 3 + v)))) * sin(zAngRot)) * sin(zAngInc)) + ((sin(u / 3 - (2 * v)) + (2 * sin(u / 3 + v))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, (2 * paso)) → list1

AI 绘图提示语参考

该图形是一个脐形曲线，由圆与椭圆的相交点及角度约束生成。构造顺序为：先绘制圆和与坐标轴垂直的直线，通过相交得到椭圆和关键点；再利用角度测量确定旋转关系；最后使用参数化方程绘制曲线。图形包含圆、椭圆、相交点、角度和参数化曲线等元素，具有对称性，适合展示圆与椭圆在几何构造中的关联。
以上提示语为根据原始绘图命令整理而来，仅供参考。