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Geometric Demonstration of the Complex Logarithm Function

Geometric Demonstration of the Complex Logarithm Function
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2016-04-09 06:00
FunctionComplex NumberTransformationGeometric ConstructionLocus
This example geometrically demonstrates the properties of the complex logarithm function. It involves core mathematical concepts such as complex numbers, logarithms, rotation, and the unit circle. The construction utilizes GeoGebra commands and objects like Line, Point, Circle, Polygon, Vector, OrthogonalLine, Translate, Intersect, and Locus. It is suitable for high school and above students to understand the geometric meaning and transformation characteristics of the complex logarithm function.

GeoGebra Commands & Parameters

  1. Line(Q, R) → j
  2. Point(j) → S
  3. Circle(D, E) → d_1
  4. Point(d_1) → J
  5. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  6. Vector(F, G) → u
  7. OrthogonalLine(F, u) → e_1
  8. Point(e_1) → H
  9. Translate(G, u) → G'_1
  10. Translate(H, u) → H'_1
  11. Polygon(F, H, H'_1, G) → poly2, f, h, h', g
  12. Vector(F, G'_1) → v
  13. Translate(F, v) → F'
  14. Translate(H, v) → H'
  15. Translate(H'_1, v) → I
  16. Translate(G, v) → G'
  17. Polygon(F', H', I, G') → poly2', f', h'_1, i, g'
  18. Translate(F', v) → F''
  19. Translate(H', v) → H''
  20. Translate(I, v) → I'
  21. Translate(G', v) → G''
  22. Polygon(F'', H'', I', G'') → poly2'', f'', h'', i', g''
  23. Point(poly1) → L
  24. Point(poly2) → M
  25. Point(poly2') → N
  26. Point(poly2'') → O
  27. Intersect(xAxis, yAxis) → P_1
  28. Circle(P_1, 1) → k
  29. Polygon(K, T, 5) → poly3, k_1, t, p, q, r, U, V, W
  30. Polygon(K, V, T, W, U) → poly4, k_2, v_1, t_1, w, u_1
  31. Point(poly4) → Z
  32. Locus(S', S) → loc6
  33. Locus(J', J) → loc1
  34. Locus(L', L) → loc2
  35. Locus(M', M) → loc3
  36. Locus(N', N) → loc4
  37. Locus(O', O) → loc5
  38. Locus(Z', Z) → loc7