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Rotating the Lemniscate of Bernoulli in 3D

Rotating the Lemniscate of Bernoulli in 3D
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2018-03-02 11:42
Solid GeometryTransformationParametric EquationSlider Animation
①This applet demonstrates the surface of revolution generated by rotating the Bernoulli lemniscate (a figure-eight curve) about any chosen axis in three-dimensional space, with interactive controls for the axis orientation and viewing angle. ②Core mathematical concepts include the parametric equation of the lemniscate, surfaces of revolution in 3D, the influence of the rotation axis's direction and position on the resulting surface, and the interplay between analytic geometry and solid geometry. ③Construction highlights: the applet uses basic geometric commands such as Circle, OrthogonalLine, Intersect, Ray, and Line to define the rotation axis and auxiliary elements; CurveCartesian plots the lemniscate; Surface generates the revolved surface; Rotate, Point, Angle, CircleSector, AngularBisector, and Segment manage rotation parameters and visual feedback; sliders (e.g., φ and θ) allow dynamic adjustment of the axis direction. ④This resource is suitable for high school to undergraduate mathematics education, particularly in analytic geometry, solid geometry, or mathematical experimentation, helping students visualize how a parametric curve sweeps out a surface when rotated about an arbitrary axis and explore how different axes alter the surface's shape.

GeoGebra Commands & Parameters

  1. Circle(zAxis, A) → q
  2. OrthogonalLine(XYZ, xOyPlane) → g
  3. Intersect(g, xOyPlane) → D
  4. Ray(O, D) → i
  5. Intersect(i, q) → G,
  6. Line(O, M) → f
  7. CurveCartesian(((a * sqrt(2)) * cos(t) / ((sin(t) * sin(t)) + 1)), (((a * sqrt(2)) * cos(t)) * sin(t) / ((sin(t) * sin(t)) + 1)), t, 0, (2 * π)) → b
  8. Surface(b, k, f) → c
  9. Rotate(b, k, f) → b'
  10. Point(yAxis) → B
  11. Point(zAxis) → C
  12. Angle(C, O, M) → φ
  13. Angle(A, O, D) → θ
  14. CircleSector(O, G, A) → e
  15. CircleSector(O, C, XYZ, xOyPlane) → p
  16. AngularBisector(A, O, D) → j
  17. AngularBisector(C, O, M) → m
  18. Intersect(m, p) → E,
  19. Intersect(j, e) → F,
  20. OrthogonalLine(G, i, xOyPlane) → n
  21. OrthogonalLine(G, xOyPlane) → r
  22. Intersect(r, f) → H
  23. Segment(H, G) → h

AI Drawing Prompt Reference

Visualize a rotating surface generated by revolving the Bernoulli lemniscate (a figure-eight curve) around any chosen axis in 3D space. Include interactive controls for the axis orientation and viewing angle. Highlight the parametric equation of the lemniscate, 3D surfaces of revolution, and how the rotation axis's direction and position affect the surface. Use sliders to dynamically adjust the axis direction and show how the curve sweeps out the surface.
The prompt above was compiled from the original drawing commands and is for reference only.