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Classification of conics: general case

Classification of conics: general case
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2016-07-05 17:59
Coordinate GeometryMatrixTransformationGeometric Construction
This example demonstrates the classification of conic sections through rotation and translation transformations of quadratic equations. Key math concepts include transformations, quadratic curves, matrix operations. GeoGebra commands like Rotate, Translate, Matrix, Eigenvec1fixed, Eigenvec2fixed are used to construct and manipulate the graphs. Suitable for high school and above learners to understand the geometric properties and transformation methods of conics.

GeoGebra Commands & Parameters

  1. If(a ≟ c, π / 4, (1 / 2 * atan(b / (a - c)))) → theta
  2. Rotate(e_1, theta) → Eigenvec1fixed
  3. Rotate(e_2, theta) → Eigenvec2fixed
  4. Transpose(P) → PT
  5. Vertex(g) → Vertex2, D, E, F
  6. Center(g) → C
  7. Vertex(conic) → Vertex, G, H, I
  8. Center(conic) → B
  9. Rotate(e_2, (Signo * θ)) → Eigenvec2
  10. Textfield(a) → InputBox1
  11. Textfield(b) → InputBox2
  12. Textfield(c) → InputBox3
  13. Textfield(d) → InputBox4
  14. Textfield(e) → InputBox5
  15. Textfield(f) → InputBox6
  16. If((a - c)^(2) + (2 * b^(2)) > 0, 2, If[(a - c)^(2) + (2 * b^(2)) < 0, 0, 1]) → numEigenvalues
  17. Vector(If[C ≟ C, C, Vertex2], (0, 0)) → vectorauxiliar01
  18. Translate(g, vectorauxiliar01) → g'
  19. Coefficients(g') → coefficients
  20. Vector((0, 0), If[B ≟ B, B, Vertex]) → vectorauxiliar02
  21. Rotate(e_1, (Signo * θ)) → Eigenvec1
  22. Line((0, 0), Eigenvec1) → uaxis
  23. Line((0, 0), Eigenvec2) → vaxis
  24. Translate(uaxis, Vector[(t * vectorauxiliar02)]) → uaxis'
  25. Translate(vaxis, Vector[(t * vectorauxiliar02)]) → vaxis'
  26. Center(conic) → Centre
  27. Center(g) → Centre2
  28. Determinant(A) → DetA
  29. Determinant(AQ) → DetAQ
  30. Determinant(P) → DetP
  31. LaTeX($5) → text1
  32. Text(text1, If[Δ_2 ≟ 0, Vertex + 0.5, B + 0.5], true, true) → text5
  33. If(sgn(theta) ≟ -1, abs(α), abs(α)) → limittheta
  34. Element(PTAP, 1, 1) → TestEigen1
  35. Element(PTAP, 2, 2) → TestEigen2