Loci on the Argand Plane

DEPY MAKRI, Jul 3, 2017

Table of Contents

  1. Fixed distance form the Axis origin or fixed argument
    1. Loci on the Argand Plane 1
    2. Loci on the Argand Plane 2
    3. Brief and analytic guidelines for visualising complex loci using Geogebra part 1
  2. fixed distance from Fixed distance from another complex number or fixed argument of the difference
    1. Loci on the Argand Plane 3
  3. fixed modulus or argument for the ratio of two complex numbers
    1. Loci on the Argand Plane part 5
    2. Loci on the Argand plane 4

1.

Fixed distance form the Axis origin or fixed argument

[list] [*]|zet|=constant [*] arg(zet)=constant [/list] We define zet=r(cosθ+isinθ) and we correspond sliders for the r and θ values.

  • 1. Loci on the Argand Plane 1
  • 2. Loci on the Argand Plane 2
  • 3. Brief and analytic guidelines for visualising complex loci using Geogebra part 1

1.1.

Loci on the Argand Plane 1

– DEPY MAKRI

1.2.

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1.3.

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2.

fixed distance from Fixed distance from another complex number or fixed argument of the difference

let w=z-z_1 and w=r(cosθ_+isinθ). We assign sliders to r and θ so that by changing their values we can 'trace' their places and visualise the corresponding loci We can choose upper and lower limits for the values of these variables and hence the loci could be sectors, arcs or annuli

  • 1. Loci on the Argand Plane 3

2.1.

Loci on the Argand Plane 3

Loci on the Argand Plane 3
– DEPY MAKRI

3.

fixed modulus or argument for the ratio of two complex numbers

If w=(zet-z_1)/(zet-z_2) then zet=(z_1-w*z_2)/(1-w). Let w=r*(cosθ+sinθ). Now we can assign sliders to change the values of r and θ and by turning on the 'trace' we can figure out the corresponding loci. Note that when r is constant and only θ changes, the produced loci is also called Appolonius circle.

  • 1. Loci on the Argand Plane part 5
  • 2. Loci on the Argand plane 4

3.1.

Loci on the Argand Plane part 5

Appolonius circle
– DEPY MAKRI

3.2.

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