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Argument and Principal Argument

Argument and Principal Argument
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2020-04-21 10:01
VectorTrigonometrySolid GeometryComplex Number
This example visualizes the argument and principal argument of complex numbers in 3D space. It involves core math concepts like complex numbers, argument, principal argument, 3D vectors, and trigonometric functions. Constructed using commands like Surface, Vector, Angle, Segment in GeoGebra. Suitable for high school and above students to understand complex number geometry and trigonometric applications.

GeoGebra Commands & Parameters

  1. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (((2 * (-3)) * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A1
  2. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (((2 * (-2)) * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A2
  3. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (((2 * (-1)) * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A3
  4. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (0 * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A4
  5. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + ((2 * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A5
  6. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (((2 * 2) * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A6
  7. Surface((u * cos(v)), (u * sin(v)), a((u * cos(v)), (u * sin(v))) + (((2 * 3) * pi) * t_1) + (0 * (1 - t_1)), u, 0.01, r, v, (-pi) + 0.001, pi) → A7
  8. Vector((0, 0), B) → u
  9. Vector((0, 0), B) → v
  10. Angle((100, 0), (0, 0), B) → α
  11. Segment(B, A) → f

AI Drawing Prompt Reference

In 3D space, draw a complex plane with the origin at center, where the horizontal axes represent real and imaginary parts. From the origin, draw a vector along the positive real axis, and another vector pointing to a point in space. Mark the angle between these two vectors with an arc. Above the plane, create seven semi-transparent spiral surfaces representing the argument of a complex number, each corresponding to adding an integer multiple of 2π (from -3 to +3 times 2π). These surfaces rotate around the origin and extend vertically. Finally, connect the point to its projection on the plane with a segment to show the principal argument range.
The prompt above was compiled from the original drawing commands and is for reference only.