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Graphical and Numerical CAS Solution of a System with Implicit Functions

Graphical and Numerical CAS Solution of a System with Implicit Functions
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2017-02-05 19:12
FunctionCoordinate GeometryGeometric Construction
This example demonstrates solving a system of two equations with implicit functions using graphical and numerical CAS methods. Key mathematical concepts include implicit functions, intersections, and coordinate geometry. Using GeoGebra commands such as ImplicitCurve, Intersect, and PolyLine, the solution is visually represented. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. ImplicitCurve(f1(x, y)) → a
  2. Intersect(Circle[D, r_a], OrthogonalLine[D, Segment[D, ClosestPoint[a, D]]], 2) → Z2_a
  3. Intersect(Circle[D, r_a], OrthogonalLine[D, Segment[D, ClosestPoint[a, D]]], 1) → Z1_a
  4. PolyLine(Zip[ClosestPoint[a, B], B, Z1_a + ((Z2_a - Z1_a) / n_a * Sequence[i, i, 0, n_a])]) → polZa
  5. ImplicitCurve(f2(x, y)) → b
  6. Intersect(Circle[D, r_b], OrthogonalLine[D, Segment[D, ClosestPoint[b, D]]], 1) → Z1_b
  7. Intersect(Circle[D, r_b], OrthogonalLine[D, Segment[D, ClosestPoint[b, D]]], 2) → Z2_b
  8. PolyLine(Zip[ClosestPoint[b, B], B, Z1_b + ((Z2_b - Z1_b) / n_b * Sequence[i, i, 0, n_b])]) → polZb
  9. Element({Intersect[polZb, polZa]}, 1) → PZ
  10. Textfield(Nn) → InputBox4
  11. Textfield(e) → InputBox5
  12. Textfield(x_{min}) → InputBox6
  13. Textfield(x_{max}) → InputBox7
  14. Textfield(y_{min}) → InputBox8
  15. Textfield(y_{max}) → InputBox9
  16. Polygon((x_{min}, y_{min}), (x_{max}, y_{min}), (x_{max}, y_{max}), (x_{min}, y_{max})) → poly1, g1_s, g2_s, g3_s, s4_s
  17. Length(lr1) → llr1
  18. Length(lf2) → llf2
  19. Textfield(r) → InputBox16
  20. Element(lr1, i_{manual}) → C_t
  21. Length(lf1) → llf1
  22. Textfield(n_a) → InputBox21
  23. Textfield(n_b) → InputBox22
  24. Textfield(r_a) → InputBox23
  25. Textfield(q) → InputBox24
  26. PolyLine(Z1_a + ((Z2_a - Z1_a) / n_a * Sequence[i, i, 0, n_a])) → poly2_aa
  27. PolyLine(Z1_b + ((Z2_b - Z1_b) / n_b * Sequence[i, i, 0, n_b])) → poly2_bb
  28. Point(poly2_aa, e1) → R_a
  29. Point(poly2_bb, e2) → R_b
  30. ClosestPoint(a, R_a) → R'_a
  31. ClosestPoint(b, R_b) → R'_b
  32. Length(R'_a - R'_b) → dmaxmins2
  33. If(Defined[PZ], If[fza2 < fzb2, (f1za2, f2za2), (f1zb2, f2zb2)], (NaN, NaN)) → errorz2
  34. If(Defined[PZ], If[fza2 < fzb2, R'_a, R'_b], (NaN, NaN)) → Pz2
  35. If(ca, PZ, Pz2) → PZ0
  36. If(ca, f11, x(errorz2)) → f110
  37. If(ca, f22, y(errorz2)) → f220
  38. Element(listxyres1, f) → B_t
  39. Textfield(r_b) → InputBox13
  40. Element(roots0, f) → A
  41. Textfield(f1) → InputBox1
  42. Textfield(f2) → InputBox3
  43. Corner(1) → Corner1
  44. Corner(2) → Corner2
  45. Corner(3) → Corner3
  46. ImplicitCurve(f01(x, y)) → c
  47. Textfield(α) → InputBox11
  48. Textfield(i_{manual}) → Eingabefeld1
  49. Textfield(f) → Eingabefeld2
  50. Length(lr2) → llr2
  51. If(Length[listxyres1] ≟ 0 ∨ (i > m1), E, Element[listxyres1, i]) → B
  52. Textfield(v_i) → InputBox2
  53. TableText({Join[{{"N"}, Sequence[i, i, 1, Length[roots0]]}], Join[{{""}, Sequence["→", i, 1, f]}], Join[{{" x:"}, x(roots0)}], Join[{{" y:"}, y(roots0)}], Join[{{"f1*10^-9"}, lf1_{Pr}2}], Join[{{"f2* 10^-9"}, lf2_{Pr}2}]}, "vr") → Text2
  54. TableText({Join[{{"N"}, Sequence[i, i, 1, llr1]}], Join[{{""}, Sequence["→", i, 1, i_{manual}]}], Join[{{"x:"}, x(lr1)}], Join[{{"y:"}, y(lr1)}], Join[{{"f1* 10^-9"}, lf1}], Join[{{"f2* 10^-9"}, lf2}]}, "vr") → text8