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Geometric Surface Fitting and Optimization

Geometric Surface Fitting and Optimization
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2015-01-31 22:47
Solid GeometryProbability & StatisticsOptimization
In collections surface 3D surface 3D
This example demonstrates how to construct a surface using polynomial fitting and involves optimization problems. Key mathematical concepts include function fitting, optimization methods, and probability distributions. Construction involves GeoGebra commands such as Point, FitPoly, and Surface. Suitable for high school or university mathematics courses to learn geometric modeling and optimization techniques.

GeoGebra Commands & Parameters

  1. Point(yAxis) → C
  2. Point(yAxis) → C'
  3. Point(x = 1) → D
  4. Point(x = 1) → D'
  5. Point(x = -2) → A
  6. Point(x = -2) → A'
  7. Point(x = -1) → B
  8. Point(x = -1) → B'
  9. Point(x = 2) → E
  10. Point(x = 2) → E'
  11. FitPoly({C, D, A, B, E}, 4) → fy1
  12. FitPoly({C', D', A', B', E'}, 4) → fy2
  13. Line((-2, my(-2), 0), zAxis) → dz1
  14. Line((-1, my(-1), 0), zAxis) → dz2
  15. Line((0, my(0), 0), zAxis) → dz3
  16. Line((1, my(1), 0), zAxis) → dz4
  17. Line((2, my(2), 0), zAxis) → dz5
  18. Point(dz1) → F
  19. Point(dz1) → F'
  20. Point(dz2) → G
  21. Point(dz2) → G'
  22. Point(dz3) → H
  23. Point(dz3) → H'
  24. Point(dz4) → I
  25. Point(dz4) → I'
  26. Point(dz5) → J
  27. Point(dz5) → J'
  28. Sequence((x(Element[Lz3d, i]), z(Element[Lz3d, i])), i, 1, 5) → Lz
  29. Sequence((x(Element[Lzz3d, i]), z(Element[Lzz3d, i])), i, 1, 5) → Lzz
  30. FitPoly(Lz, 4) → fz1
  31. FitPoly(Lzz, 4) → fz2
  32. Surface(u, my(u) + (ry(u) * cos(v)), mz(u) + (rz(u) * sin(v)), u, -2, 2, v, -3.14159, 3.141592653589793) → a
  33. Sequence(CurveCartesian[t, my(t) + (ry(t) * cos(t + i)), mz(t) + (rz(t) * sin(t + i)), t, -2, 2], i, 0, (2 * π) - (2 * π / n), (2 * π / n)) → liste1
  34. Sequence(CurveCartesian[t, my(t) + (ry(t) * cos((-t) + i)), mz(t) + (rz(t) * sin((-t) + i)), t, -2, 2], i, 0, (2 * π) - (2 * π / n), (2 * π / n)) → liste2

AI Drawing Prompt Reference

Create a 3D surface defined by two 4th-degree polynomial functions fz1 and fz2. The surface intersects with a series of lines dz1 to dz5 parallel to the z-axis at points F to J, which are then used to fit the surface.
The prompt above was compiled from the original drawing commands and is for reference only.