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n次单位根 by moebiuscurve (2)

n次单位根 by moebiuscurve (2)
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2017-04-06 10:23
圆向量三角函数数列复数
①本范例演示了n次单位根在复平面上的分布及其几何意义；②涉及核心数学概念包括复数、三角函数、单位圆、数列等；③构造要点：使用GeoGebra命令Intersect, Circle, Vector, Sequence, Point, Text等绘制单位圆、向量、根点及标签；④适合高中及以上数学学习者，用于理解复数运算与几何表示的关联。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1) → c
  3. Vector(O, Z) → ZVector
  4. Circle(O, Z) → ZCircle
  5. Sequence((Z^(1 / n) * ℯ^((((ί * 2) * π) * k / n))), k, 0, n - 1) → Roots
  6. Circle(O, abs(ZVector)^(1 / n)) → WCircle
  7. Sequence(Text["W_{" + ((k * "}")), Element[Roots, k], n > 0, n > 0], k, 1, n) → RootsLabels
  8. Sequence(Vector[O, Element[Roots, k]], k, 1, n) → RootsVectors
  9. Point(WCircle) → W
  10. Sequence(Circle[O, abs(W^(k))], k, 2, n - 1) → WCircles
  11. Sequence(Vector[O, W^(k)], k, 1, n) → WVectors
  12. Vector(O, (1, 0)) → u
  13. Sequence(Text["W^{" + (k) + "}", W^(k), n > 0, n > 0], k, 2, n) → WLabels
  14. Text("W", W) → textW
  15. Sequence(Text["W^{" + (k) + "}, \angle" + (Angle[u, Vector[W^(k)]]), W^(k), n > 0, n > 0], k, 2, n) → WLabelsAngles
  16. Text("W, ∠" + (Angle[u, Vector[W]]), W) → textWAngles
  17. Text("Z, ∠" + (Angle[u, Vector[Z]]), Z) → textZAngles
  18. Text("Z", Z) → textZ
  19. Sequence(Text["W_{" + ((k * "}, \angle")) + (Angle[u, Vector[Element[Roots, k]]]), Element[Roots, k], n > 0, n > 0], k, 1, n) → RootsLabelsAngles