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Dupin (ciclide anular de) 几何作图与积分概念

Dupin (ciclide anular de) 几何作图与积分概念
几何画板及绘图内容加载中，请稍等……
2023-03-15 19:43
圆积分几何作图
所属专题 geometriadinamica.es
该范例演示了通过几何构造和积分计算来理解Dupin曲线的几何性质。核心数学概念包括圆、角度、旋转和定积分。作图要点涉及使用GeoGebra命令如Segment、OrthogonalLine、Circle、Intersect、Ellipse、Angle和CurveCartesian等对象。适合高中生或大学低年级学生用于学习解析几何和微积分基础。

GeoGebra 命令及参数

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  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 / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(α))))) / (8 - ((5 * cos(v)) * cos(α))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(α)))) * sin(v) / (8 - ((5 * cos(v)) * cos(α)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(α))))))) / (8 - ((5 * cos(v)) * cos(α))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(α)))) * sin(v) / (8 - ((5 * cos(v)) * cos(α)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(α))) / (8 - ((5 * cos(v)) * cos(α)))) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 10 * ((((5 * (k - (5 * cos(β)))) + ((8 * cos(β)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(β)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(β) / (8 - ((5 * cos(β)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(β)))) + ((8 * cos(β)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(β)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(β) / (8 - ((5 * cos(β)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(β)) * sin(u))) / (8 - ((5 * cos(β)) * cos(u)))) * cos(zAngInc)))), u, u1, u2) → a1
  33. CurveCartesian(((r / 10 * ((5 * (k + 5)) - (8 * (8 - (k * cos(u))))) / (8 + (5 * cos(u)))) * sin(zAngRot)), (r / 10 * (((((-((5 * (k + 5)) - (8 * (8 - (k * cos(u))))))) / (8 + (5 * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k + (5 * sin(u))) / (8 + (5 * cos(u)))) * cos(zAngInc)))), u, u1, u2) → curva1
  34. Sequence(CurveCartesian[(r / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(v)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(v)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(u))) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(v)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(v)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(u))) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI 绘图提示语参考

在同一坐标系中先给定一个圆心C的圆，并取两条互相垂直的坐标轴分别相交得到关键点；利用圆与这两条正交直线的交点构成两个焦点约束，然后画出两条由同一组交点定义、共用相似缩放参数的椭圆。再从其中一条椭圆与其中一条直线的交点，测出与圆心和某交点相关的角度方向，用作后续曲线参数的倾角基准；对另一组交点同样得到用于增量的角度。最后把这些角、焦点关系与三角函数参数关系组合成两条以参数v与u生成的解析曲线，并再用序列采样在参数区间内得到曲线集合的点列，同时保留由两椭圆在不同参数处的交点所连成的线段作参考。
以上提示语为根据原始绘图命令整理而来，仅供参考。