Complex Visualiser

# Introduction

The Complex Visualiser is for you to explore complex numbers in
a geometrical setting. You can draw points (that
is, *complex numbers*) on an *Argand diagram*
and see how those points transform under various operations.
Various options allow you to see how the original points and
calculated points are related.

# Usage

The basic usage of this program is to draw points on the
complex plane (or *Argand diagram*).

Points can be
*labelled* and can have *guides* shown. The
style of the labels and guides can be
either *Cartesian* or *Polar*.
If *Cartesian* then the label is of the
form $x+\mathrm{i}y$
and the guides show
how it relates to $x$
and $y$ on the axes.
If *Polar* then the label is of the
form $(r,\theta )$
and the guides show how $r$ is the length and $\theta $
is the angle.
Lastly, when drawing points you can choose to show only the
current point or a trace of all points.

You can choose to apply an operation to your points. Some of
these operations require one or more extra points, which this
program calls *control points*. If needed, these are
displayed by default as slightly larger points. If more than
one is needed, they are ordered by brightness. These can be
moved around by dragging them. In addition, where deemed
useful then certain diagrams are shown to illustrate the
operation.

The operations are as follows. In the explanations, $z$ represents the point you make by clicking, $w$ represents the result of the operation, and the ${z}_{\mathrm{i}}$ are the control points.

**None**: No operation is applied, only the points you draw are shown.**Add**: Each point is added to the control point, So $w=z+{z}_{1}$. The diagram shows the addition parallelogram.**Subtraction**: The control point is subtracted from each point. So $w=z-{z}_{1}$. The diagram shows the subtraction parallelogram.**Multiplication**: Each point is multiplied by the control point. So $w=z\times {z}_{1}$. The diagram shows the two similar triangles: $(0,1,z)$ and $(0,{z}_{1},w)$.**Division**: Each point is divided by the control point. So $w=z/{z}_{1}$. The diagram shows the two similar triangles: $(0,1,w)$ and $(0,{z}_{1},z)$.**Power**: Each point is raised to the power of the control point. So $w={z}^{{z}_{1}}$. Note that for most values of ${z}_{1}$ this is*multi-valued*. Only the first value is shown.**Multi-Power**: Each point is raised to the power of the control point. So $w={z}^{{z}_{1}}$. Note that for most values of ${z}_{1}$ this is*multi-valued*. Five values are shown, with the*principal*value in the middle.**Conjugation**: Each point is conjugated, so $w=\overline{z}$.**Exponentiation**: The control point is raise to the power of each point. So $w={{z}_{1}}^{z}$. Note that for most values of ${z}_{1}$ this is*multi-valued*. Only the first value is shown.**Multi-Exponentiation**: The control point is raise to the power of each point. So $w={{z}_{1}}^{z}$. Note that for most values of ${z}_{1}$ this is*multi-valued*. Five values are shown, with the*principal*value in the middle.**Roots**: The roots of each point are calculated. You can choose which roots to calculate, in the range 2 to 12. All of the roots are shown, and the diagram shows the resulting regular polygon.**Möbius**: A Möbius transformation is applied to each point. This requires four control points. The transformation is $w=\frac{{z}_{1}z+{z}_{2}}{{z}_{3}z+{z}_{4}}$. By choosing the control points carefully, lots of different transformations can be viewed using this.