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Matrices - Invariant Lines

Matrices - Invariant Lines
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2020-10-18 21:56
MatrixTransformationGeometric Construction
This example demonstrates how matrices affect invariant lines in plane geometry. Key mathematical concepts include matrix transformations and invariant lines. Using GeoGebra commands such as If, Determinant, Vector, Point, etc., the effect of matrix transformations can be constructed. Suitable for high school mathematics education and students learning matrix transformations.

GeoGebra Commands & Parameters

  1. If(r ≠ 0, (t - q - sqrt((q - t)^(2) + ((4 * r) * s))) / ((2 * r)), s / (q - t)) → m_{-}
  2. Determinant(M) → determ
  3. If(abs(determ) ≤ 0.01, true, false) → detcheck
  4. Textfield(q) → inputBox2
  5. Textfield(r) → inputBox1
  6. Textfield(s) → inputBox3
  7. Textfield(t) → inputBox4
  8. Vector(O, S) → u
  9. Vector(Q, U) → v
  10. Vector(P, T) → a
  11. If(abs(((q - 1) * (t - 1)) - (r * s)) ≤ 0.01, 0, ((q - 1) * (t - 1)) - (r * s)) → det
  12. If(q ≟ 1 ∧ r ≟ 0, (s * x) + ((t - 1) * y) = 0, ((q - 1) * x) + (r * y) = 0) → eq1
  13. If(r ≠ 0, (t - q + sqrt((q - t)^(2) + ((4 * r) * s))) / ((2 * r)), s / (q - t)) → m_{+}
  14. If(abs(t - (r * m_{+}) - 1) ≤ 0.001, true, false) → kcheck_{+}
  15. If(abs(t - (r * m_{-}) - 1) ≤ 0.001, true, false) → kcheck_{-}
  16. Point(g) → A
  17. OrthogonalLine(A, g) → h
  18. Point(h) → B
  19. Vector(A, B) → b
  20. Translate(A, c) → A'
  21. Line(A', g) → i
  22. Mirror(g, i) → g'
  23. Mirror(i, g') → i'
  24. Mirror(g', i') → g''
  25. Mirror(i', g'') → i''
  26. Mirror(i, g) → i'_{1}
  27. Mirror(g, i'_{1}) → g'_{1}
  28. Mirror(i'_{1}, g'_{1}) → j
  29. Mirror(g'_{1}, j) → k
  30. Mirror(j, k) → j'
  31. OrthogonalLine(A, f) → l
  32. Point(l) → C
  33. Vector(A, C) → d
  34. Translate(A, e) → A'_{1}
  35. Translate(f, e) → f'
  36. Mirror(f, f') → f'_{1}
  37. Mirror(f', f'_{1}) → f''
  38. Mirror(f'_{1}, f'') → m
  39. Mirror(f'', m) → f'''
  40. Mirror(f', f) → f''_{1}
  41. Mirror(f, f''_{1}) → f'_{2}
  42. Mirror(f''_{1}, f'_{2}) → n
  43. Mirror(f'_{2}, n) → p
  44. Mirror(n, p) → n'
  45. If(r ≟ 0 ∧ q ≟ 1, true, false) → vertcheck
  46. Mirror(f'_{1}, f') → f_1
  47. Translate(eq2, (2, 0)) → eq2'
  48. Mirror(eq2, eq2') → eq2'_{1}
  49. Mirror(eq2', eq2'_{1}) → eq2''
  50. Mirror(eq2'_{1}, eq2'') → g_1
  51. Mirror(eq2'', g_1) → eq2'''
  52. Mirror(eq2', eq2) → eq2''_{1}
  53. Mirror(eq2, eq2''_{1}) → eq2'_{2}
  54. Mirror(eq2''_{1}, eq2'_{2}) → h_1
  55. Mirror(eq2'_{2}, h_1) → i_1
  56. Mirror(h_1, i_1) → j_1
  57. Rotate(p_1, π / 6) → k_1
  58. Rotate(p_1, π / 3) → l_1
  59. Rotate(p_1, π / 2) → m_1
  60. Rotate(p_1, ((2 * π)) / 3) → n_1
  61. Rotate(p_1, ((5 * π)) / 6) → q_1
  62. If(q ≟ t ∧ r ≟ 0 ∧ s ≟ 0, true, false) → encheck
  63. Defined(f) → fcheck
  64. Defined(g) → gcheck
  65. If(q ≟ 1 ∧ r ≟ 0, (s * x) + ((t - 1) * y) = 0, ((q - 1) * x) + (r * y) = 0) → eq3

AI Drawing Prompt Reference

On a coordinate plane, introduce adjustable parameters q, r, s, and t and use them to define two cases of lines via conditional formulas: depending on a computed determinant-like check, choose one of two equations to determine the relevant line(s). Construct orthogonal directions by drawing perpendicular lines through the key reference point(s), and build vectors from the given reference point to establish the base line used for symmetry. Then perform a sequence of reflections successively across the chosen base line and the generated perpendicular/related lines, producing multiple mirrored lines and corresponding points, so that two families of invariant lines under the parameter-driven transformation appear in symmetric pairs. Use the same scheme again with the second branch to show the counterpart invariant structure.
The prompt above was compiled from the original drawing commands and is for reference only.