### About

# Conformal mapping

### Simulation window

In the *z*-plane (left) there is a **quadratic point array** of
predefined width and position. Its points are differentiated in color
according to their imaginary value. The lower left edge point is
accentuated in red; it is connected to the origin by a green vector
arrow. The array can be shifted in the plane, drawing the red point with
the **mouse.** Its coordinates can be varied independently by two **sliders
x, y**. Exact values can be written into the

**they may be beyond the slider range. The**

*x,y*number fields;**array width**can be changed by another slider; it can be contracted to a point.

In addition there is a **circular** color coded point array around
the origin. Its center is highlighted in magenta. The two symmetric
points at the real axis are accentuated in red and yellow. The circular
array can be shifted with the** mouse **by drawing at its center. The
diameter of the circular array can be varied by a **radius slider**
and can also be contracted to a point.

In both planes there are **black circles** with radii characteristic
for the special function (*1, e , π*). In the *z*-plane **red
lines **define the limits of periodic regions or of mapping strips

The arrays of the *z*-plane are mapped into the *w*-plane by
the complex function. The color coding is conserved in this process to
identify specific rows of points. When the arrays are contracted to a
point, one sees the mapping of this single point. Exact coordinates are
shown when a point is marked with the mouse.

The **play** button starts an animation that shifts the quadratic
array by a raster unit per second along a line that is interesting for
the special function (real axis, imaginary axis, unit circle). When the
array reaches the end of the scale, it jumps to the opposite limit. One
can shift the array with the mouse, sliders or coordinate number fields
while the animation is running. This way the plane can be quickly
rastered. The **pause** button stops the animation.

The color coding is most distinct when the window is blown up to **full
screen size**.

# Complex functions

#
Let *z = x + iy = e^(iθ) = cos(θ) + i sin(θ)*,

#
where *x,y* are real numbers and *i^2 = -1*

#
The graph on the left shows *z* as plotted in the complex plane.

#
The graph on the right shows a complex number *w*, which is mapped
from *z* using one of the functions provided.

#
The available functions provided in the **Combobox, along with their
equations, are as follow**s:

**- Exponential: w = e^(nz), where n
is a real number **

**- Sine: w = sin(z) **

**- Cosine: w = cos(z) **

**- Tangent: w = tan(z) **

**- Logarithmic: w = ln(z) **

**- Power: w = z^n, where n is a
real number**

this file was created by Dieter Roess in July 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

### end faq

### Sample Learning Goals

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### For Teachers

In the real numbers, we can express a relationship between two sets of numbers by mapping elements from one set to elements in another. We call this mapping a function.

To visualize this, we can plot out a graph of y against x, where y = f(x).

However, this is not possible with complex numbers, which are represented with a 2-dimensional plane. Thus, we map a complex number from one Argand plane to another Argand plane. By seeing how each function transforms a square array of points (red to blue) and the unit circle (black), we can reveal some very interesting properties of how complex numbers behave under a function.

The available functions are as follows:

- Exponential (** w = e^z**)

- Sine (

**)**

*w = sin(z)*- Cosine (

**)**

*w = cos(z)*- Tangent (

**)**

*w = tan(z)*- Logarithmic (

**)**

*w = ln(z)*- Power (

**)**

*w = z^n*Research

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### Video

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### Version:

- http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
- http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

### Other Resources

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