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Klein (botella de) 几何作图与函数图像

Klein (botella de) 几何作图与函数图像
几何画板及绘图内容加载中，请稍等……
2023-03-15 20:39
函数解析几何变换几何作图
所属专题 geometriadinamica.es
这个范例展示了如何通过几何作图方法绘制函数图像，并探讨了二次函数与相似变换的关系。核心数学概念包括函数图像、二次函数、相似变换和几何作图。作图要点涉及使用GeoGebra命令如Segment、Point、OrthogonalLine、Circle、Intersect、Ellipse、Angle、CurveCartesian等来构造图形。适合高中生或大学低年级学生用于学习解析几何和函数图像的绘制方法。

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, xAxis) → zz2
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz2, 1) → ZZ4
  12. Intersect(zz0, zz2, 2) → ZZ3
  13. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  14. Point(zElipse2) → Z
  15. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  16. Angle(ZZ8, C, Z) → zAngInc
  17. OrthogonalLine(C, yAxis) → zz1
  18. Intersect(zz0, zz1, 2) → ZZ2
  19. Intersect(zz0, zz1, 1) → ZZ1
  20. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  21. Point(zElipse1) → XY
  22. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  23. Angle(ZZ6, C, XY) → zAngRot
  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 * (((((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)))), u, u1 + p, u2) → curva1B
  32. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2))) * sin(zAngRot)) + (((16 * sin(u)) + ((4 * (1 - cos(u) / 2)) * sin(u))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + ((4 * (1 - cos(u) / 2)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2 - p) → curva1A
  33. CurveCartesian((r / 10 * (((((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(α)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(α) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2) → curva2B
  34. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), u, u1, u2 - p], v, v1, v2, paso) → list2
  35. CurveCartesian((r / 10 * (((((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(α)) + (((4 * (1 - cos(α) / 2)) * sin(α)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(α)) + (((4 * (1 - cos(α) / 2)) * sin(α)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(α) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2) → curva2A
  36. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(β))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(β))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(β)) * cos(zAngInc)))), u, u1, u2 - p) → a1
  37. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(β)) * cos(zAngInc)))), u, u1 + p, u2) → a2
  38. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2], u, u1 + p, u2, paso) → list3
  39. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2], u, u1, u2 - p, paso) → list4
  40. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), u, u1 + p, u2], v, v1, v2, paso) → list1

AI 绘图提示语参考

绘制一个包含两个椭圆和两条曲线的图形。首先画出两条互相垂直的直线，然后以它们的交点为圆心画一个圆，与两条直线相交得到四个点。连接这些点形成四边形，再以相邻的两对顶点为圆心画两个椭圆。接着，根据角度参数绘制两条曲线，分别连接椭圆上的点和四边形上的点。最后，将这两条曲线与椭圆和四边形组合，展示出函数图形与几何变换的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。