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Intersections of Gerono Lemniskate Surface with a Moving Plane

Intersections of Gerono Lemniskate Surface with a Moving Plane
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2016-10-14 22:57
Solid GeometryTransformationParametric Equation
This example demonstrates the intersections of the Gerono Lemniskate surface with a moving plane. Key mathematical concepts include parametric equations, solid geometry, and transformations. Using GeoGebra commands such as Surface, CurveCartesian, Sequence, RemoveUndefined, and Zip, the intersections can be dynamically observed. Suitable for high school students studying parametric equations and solid geometry.

GeoGebra Commands & Parameters

  1. Surface(fx(u, v), fy(u, v), fz(u, v), u, u_{min}, u_{max}, v, v_{min}, v_{max}) → f3
  2. Textfield(fx) → inputBox1
  3. Textfield(fy) → inputBox2
  4. Textfield(fz) → inputBox3
  5. Textfield(u_{min}) → inputBox4
  6. Textfield(u_{max}) → inputBox6
  7. Textfield(v_{min}) → inputBox7
  8. Textfield(v_{max}) → inputBox8
  9. CurveCartesian(fx(u, (3 * u) - acos(z0 / 5)), fy(u, (3 * u) - acos(z0 / 5)), z0, u, u_{min}, u_{max}) → g1
  10. CurveCartesian(fx(u, (3 * u) + acos(z0 / 5)), fy(u, (3 * u) + acos(z0 / 5)), z0, u, u_{min}, u_{max}) → g2
  11. Sequence(CurveCartesian[fx((acos(z0 / 5) + ((2 * π) * i)) / 3 + v / 3, v), fy((acos(z0 / 5) + ((2 * π) * i)) / 3 + v / 3, v), z0, v, 0, 15], i, n1, n2, 1) → l3
  12. Sequence(CurveCartesian[fx(((-acos(z0 / 5)) + ((2 * π) * i)) / 3 + v / 3, v), fy(((-acos(z0 / 5)) + ((2 * π) * i)) / 3 + v / 3, v), z0, v, 0, 15], i, n1, n2, 1) → l4
  13. RemoveUndefined(Join[Sequence[Zip[a + (0, v_{min} + (dv * (s0 - 1))), a, {Roots[fz(x, v_{min} + (dv * (s0 - 1))) - z0, u_{min}, u_{max}]}], s0, 1, Nn + 1]]) → Rootsxv
  14. Zip((fx(x(a), y(a)), fy(x(a), y(a)), z0), a, Rootsxv) → curvexv
  15. RemoveUndefined(Join[Sequence[Zip[a + (0, u_{min} + (du * (s0 - 1))), a, {Roots[fz(u_{min} + (du * (s0 - 1)), x) - z0, v_{min}, v_{max}]}], s0, 1, Nn + 1]]) → Rootsux
  16. Zip((fx(y(a), x(a)), fy(y(a), x(a)), z0), a, Rootsux) → curveux
  17. Textfield(Nn) → inputBox5

AI Drawing Prompt Reference

Draw a parametric surface shaped like a lemniscate surface. Add a horizontal plane that moves vertically. Show the intersection curves between the plane and the surface, including two main curves and a family of curves generated by varying a parameter. Also mark the traces of intersection points on the surface along two parameter directions. The whole scene should dynamically update as the plane moves, with input boxes to adjust surface parameters and plane height.
The prompt above was compiled from the original drawing commands and is for reference only.