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螺旋面变换演示

螺旋面变换演示
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2023-03-15 20:22
函数三角函数立体几何参数方程几何作图
所属专题 geometriadinamica.es
①本范例演示了螺旋面（Helicoide）的形成与变换过程；②涉及核心数学概念包括直线、余弦函数、有理数、正弦函数、线性规划；③作图要点：运用了线段、点、垂线、圆、交点、椭圆、角度、笛卡尔曲线、序列等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解空间几何与参数方程的应用。

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, yAxis) → zz1
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz1, 1) → ZZ1
  14. Intersect(zz0, zz1, 2) → ZZ2
  15. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  16. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  17. Point(zElipse1) → XY
  18. Angle(ZZ6, C, XY) → zAngRot
  19. OrthogonalLine(C, xAxis) → zz2
  20. Intersect(zz0, zz2, 2) → ZZ3
  21. Intersect(zz0, zz2, 1) → ZZ4
  22. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  23. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  24. Point(zElipse2) → Z
  25. Angle(ZZ8, C, Z) → zAngInc
  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 / 3 * (((((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (α * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2) → curva2
  32. CurveCartesian((r / 3 * (((((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * sin(zAngRot)) * sin(zAngInc)) + (((β * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → a1
  33. CurveCartesian((r / 3 * ((((-sin(a)) * cosh(v)) * sin(zAngRot)) - ((cos(a) * sinh(v)) * cos(zAngRot)))), (r / 3 * ((((sin(a) * cosh(v)) * cos(zAngRot)) * sin(zAngInc)) - (((cos(a) * sinh(v)) * sin(zAngRot)) * sin(zAngInc)) + (((3.14159 * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → curva1A
  34. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2], u, u1, u2, paso) → list1

AI 绘图提示语参考

绘制一个螺旋面，通过一系列椭圆和曲线来表示。首先，创建一系列点和线段，然后利用这些点和线段构造椭圆和曲线。接着，通过旋转和平移操作，生成螺旋面的多个截面，最后使用这些截面来描绘螺旋面的整体形态。
以上提示语为根据原始绘图命令整理而来，仅供参考。