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Complex Linear Mapping Diagram f(z) = Bz; Parametric Curve

Complex Linear Mapping Diagram f(z) = Bz; Parametric Curve
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2016-01-14 08:31
FunctionCoordinate GeometryComplex NumberParametric Equation
This example demonstrates the effect of the complex linear mapping f(z) = Bz on a parametric curve, involving core concepts like complex numbers, linear transformations, parametric equations, and complex plane mappings. It utilizes GeoGebra commands such as Sequence, Zip, Textfield, Vector, CurveCartesian, Intersect, Point, Circle, ToPolar, and Locus to visually reveal the transformation properties, making it suitable for high school and above learners to understand the intuitive representation of complex function transformations and geometric mappings.

GeoGebra Commands & Parameters

  1. Sequence(t + ((Ax * t^(2)) * ί), t, -3, 3, 0.5) → z_*
  2. Zip((x(z), y(z), height_{image}), z, fz_*) → fpoint_*
  3. Textfield(fz_*) → inputBox1
  4. Vector(z_#, f_{(z#).}) → w
  5. Textfield(f_{(z#)}) → inputBox2
  6. CurveCartesian(t, 0, height_{image}, t, -5, 5) → Xh
  7. CurveCartesian(0, t, height_{image}, t, -5, 5) → Yh
  8. Textfield(z_*) → inputBox3
  9. Sequence(Line[Element[z_*, j], Element[fpoint_*, j]], j, 1, 12) → list1
  10. Intersect(xAxis, yAxis) → B_1
  11. Point(xAxis) → C
  12. Circle(B_1, C) → p
  13. ToPolar(B) → D
  14. Locus(f_{(z#)}, t_#) → loc1
  15. Locus(a, t_#) → loc2
  16. Locus(a, t_#) → c
  17. Locus(f_{(z#).}, t_#) → loc3
  18. Locus(A_1, t_#) → loc4
  19. Vector(A_1, z_#) → u
  20. CurveCartesian(t, (Ax * t^(2)), t, -5, 5) → γ
  21. Zip(Vector[A, B], A, z_*, B, fpoint_*) → fvec_*