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GeoGebra Transformation Example: u^2-v^2 and 2uv

GeoGebra Transformation Example: u^2-v^2 and 2uv
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2022-09-01 17:18
Coordinate GeometryTransformationLocus
This example demonstrates the geometric significance of transformations using u^2-v^2 and 2uv, combining algebra and geometry. Key mathematical concepts include quadratic form transformations, coordinate system conversion, and locus analysis. Construction involves GeoGebra commands such as Polygon, Sequence, Point, Derivative, Locus, and Textfield. Suitable for high school students or college freshmen learning analytic geometry and calculus fundamentals.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → square, f, g, h, i, C, D
  2. Sequence(Segment[A + (i * (B - A) / n), D + (i * (C - D) / n)], i, 0, n) → vertical
  3. Point(Element[vertical, 1]) → A1
  4. Derivative(u, x) → ux
  5. Derivative(u, y) → uy
  6. Derivative(v, x) → vx
  7. Derivative(v, y) → vy
  8. Sequence(Segment[A + (i * (D - A) / n), B + (i * (C - B) / n)], i, 0, n) → horizontal
  9. Point(Element[vertical, 2]) → A2
  10. Point(Element[vertical, 3]) → A3
  11. Point(Element[vertical, 4]) → A4
  12. Point(Element[vertical, 5]) → A5
  13. Point(Element[vertical, 6]) → A6
  14. Point(Element[vertical, 7]) → A7
  15. Point(Element[vertical, 8]) → A8
  16. Point(Element[horizontal, 1]) → B1
  17. Point(Element[horizontal, 2]) → B2
  18. Point(Element[horizontal, 3]) → B3
  19. Point(Element[horizontal, 4]) → B4
  20. Point(Element[horizontal, 5]) → B5
  21. Point(Element[horizontal, 6]) → B6
  22. Point(Element[horizontal, 7]) → B7
  23. Point(Element[horizontal, 8]) → B8
  24. Locus(C1, A1) → E1
  25. Locus(C2, A2) → E2
  26. Locus(C3, A3) → E3
  27. Locus(C4, A4) → E4
  28. Locus(C5, A5) → E5
  29. Locus(C6, A6) → E6
  30. Locus(C7, A7) → E7
  31. Locus(C8, A8) → E8
  32. Locus(D1, B1) → F1
  33. Locus(D2, B2) → F2
  34. Locus(D3, B3) → F3
  35. Locus(D4, B4) → F4
  36. Locus(D5, B5) → F5
  37. Locus(D6, B6) → F6
  38. Locus(D7, B7) → F7
  39. Locus(D8, B8) → F8
  40. Textfield(u) → InputBox1
  41. Textfield(v) → InputBox2
  42. Point(square) → E

AI Drawing Prompt Reference

Draw a geometric transformation example formed by u^2-v^2 and 2uv, showcasing the combination of algebra and geometry. Start by drawing a square, then generate sequences of vertical and horizontal line segments inside the square, marking points along these segments, and use these points to create loci, ultimately presenting the transformation trajectories of u and v, reflecting the geometric significance of quadratic form transformations and coordinate system conversion.
The prompt above was compiled from the original drawing commands and is for reference only.