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Mapping the Unit Circle to the Upper/Lower Half Plane

Mapping the Unit Circle to the Upper/Lower Half Plane
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2020-06-28 13:58
CircleComplex NumberTransformation
This example demonstrates the mapping of the unit circle to the upper and lower half planes. Key mathematical concepts include complex numbers and transformations. Using commands like rotation and locus, the dynamic mapping of points on the unit circle to the upper and lower half planes can be observed. Suitable for high school mathematics education and learning complex number transformations.

GeoGebra Commands & Parameters

  1. Circle(A, 1 / 4) → A1
  2. Point(A1) → C1
  3. Circle(A, 2 / 4) → A2
  4. Circle(A, 3 / 4) → A3
  5. Circle(A, 4 / 4) → A4
  6. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 1 / m), A) → B1
  7. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 2 / m), A) → B2
  8. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 3 / m), A) → B3
  9. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 4 / m), A) → B4
  10. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 5 / m), A) → B5
  11. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 6 / m), A) → B6
  12. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 7 / m), A) → B7
  13. Rotate(Segment[A, (x(A) + 1, y(A))], ((2 * pi) * 8 / m), A) → B8
  14. Point(A2) → C2
  15. Point(A3) → C3
  16. Point(A4) → C4
  17. Point(B1) → D1
  18. Point(B2) → D2
  19. Point(B3) → D3
  20. Point(B4) → D4
  21. Point(B5) → D5
  22. Point(B6) → D6
  23. Point(B7) → D7
  24. Point(B8) → D8
  25. Locus(E1, C1) → G1
  26. Locus(E2, C2) → G2
  27. Locus(E3, C3) → G3
  28. Locus(E4, C4) → G4
  29. Locus(F1, D1) → H1
  30. Locus(F2, D2) → H2
  31. Locus(F3, D3) → H3
  32. Locus(F4, D4) → H4
  33. Locus(F5, D5) → H5
  34. Locus(F6, D6) → H6
  35. Locus(F7, D7) → H7
  36. Locus(F8, D8) → H8
  37. Semicircle((x(A) - 1, y(A)), (x(A) + 1, y(A))) → up
  38. Semicircle((x(A) + 1, y(A)), (x(A) - 1, y(A))) → down
  39. Point(up) → B
  40. Point(down) → C
  41. Locus(B', B) → LB
  42. Locus(C', C) → LC