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复数根的绘制

复数根的绘制
几何画板及绘图内容加载中，请稍等……
2022-01-28 18:26
向量三角函数数列复数几何作图
①本范例演示了如何绘制复数的n次根及其几何表示；②涉及的核心数学概念包括复数、根的分布、极坐标表示、三角函数等；③作图要点提炼自构造摘要，使用了Sequence（序列）、Element（元素）、Intersect（交集）、Vector（向量）、Point（点）、Segment（线段）、If（条件判断）、TableText（表格文本）、Polygon（多边形）等GeoGebra命令和对象；④适合高中及以上数学学习者，用于理解复数运算和几何直观。

GeoGebra 命令及参数

  1. Sequence((j^(1 / n) * (cos((atan(b / a) + ((2 * q) * π)) / n) + (ί * sin((atan(b / a) + ((2 * q) * π)) / n)))), q, 0, k) → Sequence2, {1.0311581284881526 - 0.0997039294804784ί, 0.7208682482097111 + 0.7440275050185309ί, -0.13225012600310243 + 1.0274910528433818ί, -0.8857814579246445 + 0.5372328808793527ί, -0.9723012853800516 - 0.3575726079404533ί, -0.32665841361255604 - 0.9831186298291604ί, 0.5649649062224906 - 0.8683562714911719ί}
  2. Element(Sequence2, 1) → A2
  3. Element(Sequence2, 2) → A3
  4. Element(Sequence2, 3) → A4
  5. Element(Sequence2, 4) → A5
  6. Element(Sequence2, 5) → A6
  7. Element(Sequence2, 6) → A7
  8. Element(Sequence2, 7) → A8
  9. Element(Sequence2, 8) → A9
  10. Element(Sequence2, 9) → A10
  11. Element(Sequence2, 10) → A11
  12. Intersect(xAxis, yAxis) → A
  13. Vector(A, A2) → u
  14. Vector(A, A3) → v
  15. Vector(A, A4) → w
  16. Vector(A, A5) → a_{1}
  17. Vector(A, A6) → b_{1}
  18. Vector(A, A7) → c
  19. Vector(A, A8) → d
  20. Vector(A, A9) → e
  21. Vector(A, A10) → f
  22. Vector(A, A11) → g
  23. Point(yAxis) → B
  24. Point(yAxis) → C
  25. Segment(B, C) → h
  26. Point(xAxis) → D
  27. Point(xAxis) → E
  28. Segment(D, E) → i
  29. Vector((a, b)) → p
  30. Intersect(eq1, i) → F, G
  31. Intersect(eq1, h) → H, I
  32. Element(Sequence1, 1) → B2
  33. Element(Sequence1, 2) → B3
  34. Element(Sequence1, 3) → B4
  35. Element(Sequence1, 4) → B5
  36. Element(Sequence1, 5) → B6
  37. Element(Sequence1, 6) → B7
  38. Element(Sequence1, 7) → B8
  39. Element(Sequence1, 8) → B9
  40. Element(Sequence1, 9) → B10
  41. Element(Sequence1, 10) → B11
  42. If(k ≥ 0, B2 + "") → C14
  43. If(k ≥ 1, B3 + "") → C15
  44. If(k ≥ 2, B4 + "") → C16
  45. If(k ≥ 3, B5 + "") → C17
  46. If(k ≥ 4, B6 + "") → C18
  47. If(k ≥ 5, B7 + "") → C19
  48. If(k ≥ 6, B8 + "") → C20
  49. If(k ≥ 7, B9 + "") → C21
  50. If(k ≥ 8, B10 + "") → C22
  51. If(k ≥ 9, B11 + "") → C23
  52. Sequence(Element[Sequence3, α + 1], α, 0, k) → SequenceA, {"z_1", "z_2", "z_3", "z_4", "z_5", "z_6", "z_7"}
  53. Sequence(Element[Sequence2, α + 1], α, 0, k) → SequenceB, {1.0311581284881526 - 0.0997039294804784ί, 0.7208682482097111 + 0.7440275050185309ί, -0.13225012600310243 + 1.0274910528433818ί, -0.8857814579246445 + 0.5372328808793527ί, -0.9723012853800516 - 0.3575726079404533ί, -0.32665841361255604 - 0.9831186298291604ί, 0.5649649062224906 - 0.8683562714911719ί}
  54. Sequence(Element[Sequence4, α + 1], α, 0, k) → SequenceC, {" 1.04×\left [cos\left(-0.1\right)+i\;sin\left( -0.1 \right)\right ]\cr", " 1.04×\left [cos\left(0.8\right)+i\;sin\left( 0.8 \right)\right ]\cr", " 1.04×\left [cos\left(2.15\right)+i\;sin\left( 1.7 \right)\right ]\cr", " 1.04×\left [cos\left(2.6\right)+i\;sin\left( 2.6 \right)\right ]\cr", " 1.04×\left [cos\left(3.49\right)+i\;sin\left( 3.49 \right)\right ]\cr", " 1.04×\left [cos\left(4.39\right)+i\;sin\left( 4.39 \right)\right ]\cr", " 1.04×\left [cos\left(5.29\right)+i\;sin\left( 5.29 \right)\right ]\cr"}
  55. TableText({Append["\large\red{z}", SequenceA], Append["\large\red{a+bi}", SequenceB], Append["\large\red{rcis(θ)}", SequenceC]}, "vc|_") → text24
  56. Polygon(A2, A3, n) → poly1, t, r_1, K, L, M, N, O, P, Q, S, f_1, g_1, h_1, i_1, l_1, m_1, p_1, q_1

AI 绘图提示语参考

绘制复数的n次根的可视化图形，包括原点O和x轴、y轴的交点A，以及从A出发的10个向量，这些向量代表复数的n次根。使用三角函数和极坐标表示这些向量，通过交点和线段展示向量之间的关系，最终形成一个复数根的几何分布。
以上提示语为根据原始绘图命令整理而来，仅供参考。