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Level Curves by Patrick Clément

Level Curves by Patrick Clément
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2014-10-16 01:00
FunctionCoordinate GeometryParametric Equation
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1 surface 3D surface 3D
This example demonstrates the construction of level curves. It involves concepts of functions and coordinate geometry. Using GeoGebra commands such as CurveCartesian, Mirror, and Sequence, level curves can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. CurveCartesian(t, sqrt(a2^(2) + sqrt((h * b^(3)) - (t^(2) - a1^(2))^(2))), t, 0, 6.283185307179586) → a
  2. CurveCartesian(t, sqrt(a2^(2) - sqrt((h * b^(3)) - (t^(2) - a1^(2))^(2))), t, 0, 6.283185307179586) → c
  3. Mirror(a, yAxis) → a'
  4. Mirror(c, yAxis) → c'
  5. Mirror(a, xAxis) → a'_1
  6. Mirror(c, xAxis) → c'_1
  7. Mirror(a', xAxis) → a''
  8. Mirror(c', xAxis) → c''
  9. Sequence(CurveCartesian[t, sqrt(a2^(2) + sqrt((i * b^(3)) - (t^(2) - a1^(2))^(2))), h - i, t, 0, 6.28319], i, -1, h, inc) → liste1
  10. Sequence(CurveCartesian[t, sqrt(a2^(2) - sqrt((i * b^(3)) - (t^(2) - a1^(2))^(2))), h - i, t, 0, 6.28319], i, -1, h, inc) → liste2
  11. Mirror(liste1, Plane[(0, 0, 0), (1, 0, 0), (0, 0, 1)]) → liste1'
  12. Mirror(liste2, Plane[(0, 0, 0), (1, 0, 0), (0, 0, 1)]) → liste2'
  13. Mirror(liste1, Plane[(0, 0, 0), (0, 1, 0), (0, 0, 1)]) → liste1'_1
  14. Mirror(liste2, Plane[(0, 0, 0), (0, 1, 0), (0, 0, 1)]) → liste2'_1
  15. Mirror(liste1', Plane[(0, 0, 0), (0, 1, 0), (0, 0, 1)]) → liste1''
  16. Mirror(liste2', Plane[(0, 0, 0), (0, 1, 0), (0, 0, 1)]) → liste2''

AI Drawing Prompt Reference

This is a graphical construction of level curves. The figure includes two basic curves a and c, along with curves a' and c' obtained through mirror transformations. Further, multiple curves liste1 and liste2 are generated via sequences, and liste1'' and liste2'' are obtained through additional mirror transformations. The figure demonstrates the combination of functions and coordinate geometry, using parametric equations and mirror operations to form symmetric curve structures. The construction order is to first draw the basic curves, then perform mirror transformations, and finally generate sequence curves and mirror them.
The prompt above was compiled from the original drawing commands and is for reference only.