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复线性映射图示II：f(z) = Az

复线性映射图示II：f(z) = Az
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2016-01-09 14:56
函数复数
所属专题 Mapping Diagrams to Visualize Complex Analysis
该范例演示了复数线性映射f(z) = Az的图形变换。核心数学概念包括复数、线性映射和相似三角形。通过使用GeoGebra命令如Sequence、Zip、Textfield等，构建了复数z的映射图像。适合高中数学教学和学习复数变换。

GeoGebra 命令及参数

  1. Sequence((r * (cos(t) + (sin(t) * ί))), t, -3.141592653589793, 3.141592653589793, 0.5235987755982988) → z_*
  2. Zip(f(z), z, z_*) → fz_*
  3. Zip((x(z), y(z), height_{image}), z, fz_*) → fpoint_*
  4. Sequence(Vector[Element[z_*, j], Element[fpoint_*, j]], j, 1, 12) → fvec_*
  5. Textfield(fz_*) → inputBox1
  6. Vector(z_#, f_{(z#).}) → w
  7. Textfield(f_{(z#)}) → inputBox2
  8. CurveCartesian(t, 0, height_{image}, t, -5, 5) → Xh
  9. CurveCartesian(0, t, height_{image}, t, -5, 5) → Yh
  10. Textfield(z_*) → inputBox3
  11. Sequence(Line[Element[z_*, j], Element[fpoint_*, j]], j, 1, 12) → list1
  12. Intersect(zAxis, f_1) → Vertex
  13. Intersect(xAxis, yAxis) → B
  14. Point(xAxis) → C
  15. Circle(B, C) → p
  16. ToPolar(A) → D
  17. CurveCartesian(((r * (1 - t / 3 + (ρ * t / 3))) * cos(Angle[z_#] + (t / 3 * Angle[A]))), ((r * (1 - t / 3 + (ρ * t / 3))) * sin(Angle[z_#] + (t / 3 * Angle[A]))), t, t, (-abs(z(Vertex))), abs(z(Vertex))) → j
  18. Zip(CurveCartesian[((r * (1 - t / 3 + (ρ * t / 3))) * cos(Angle[w] + (t / 3 * Angle[A]))), ((r * (1 - t / 3 + (ρ * t / 3))) * sin(Angle[w] + (t / 3 * Angle[A]))), (-t), t, 0, abs(height_{image})], w, z_*) → fgeodesic_*
  19. If(z(Vertex) ≥ 0, Cone[(0, 0, height_{image}), Vertex, (r * ρ)], Cone[(0, 0, 0), Vertex, r]) → cone
  20. Circle((0, 0), 1) → unit
  21. Circle((0, 0), r) → c
  22. CurveCartesian(((r * ρ) * cos(t)), ((r * ρ) * sin(t)), height_{image}, t, 0, 6.28319) → d