Authors: Ana Novak & KC Tan
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Title: Conformal Mapping
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Purpose:
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This model was designed as a tool for drawing conformal mappings.
Theory Of CONFORMAL MAPPING
Def: If f(z) is analytic at a, and if f'(a)!=0, the transformation w=f(z) is
said to be conformal at a.
The reason for, and justification of, the term conformal is contained in the
following theorem.
Theorem: If the mapping w=f(z) is conformal at a, then the mapping preserves
the angle of intersection of curves through a in both sense and magnitude.
The bilinear mapping:
The function (az+b)/(cz+d), ad-bc!=0, is simple (that is f(z)!=f(w)
for any two z,w) in the entire plane with the point z=-d/c omitted.
For if (az1+b)/(cz1+d)=(az2+b)/(cz2+d) then (ad-bc)(z1-z2)=0 and so,
since ad-bc!=0, we have z1=z2. Thus
w=(az+b)/(cz+d)
admits a unique solution, which in fact is
z=(dw-b)/(a-cw).
We may regard a functional relation w=f(z) either as a transformation
which moves a point z to a point f(z) in the same plane or a mapping of
one plane to one plane -the argument plane z- into the value plane, the
w-plane.
We see that w=(az+b)/(cz+d) may be obtained from the three functions:
1) w=z+a (translation) 2) w=az (scaling and rotation~w=e^i(fi))
3) w=1/z (inversion)
The general bilinear mapping:
w=(az+b)/(cz+d), ad-bc!=0,
is the conformal for all values of z except z=-d/c. For
dw/dz=(ad-bc)/(cz+d)(cz+d)!=0, z!=-d/c.
To conclude, all bilinear transformations are combinations of the three
simple types:w=z+a, w=az and w=1/z ; which can be then regarded as separe
simpler problems.
Eg.
EXAMPLE1:
Map the region {|z|<0, |z|<1, |z+2i|<1} with the function:
f(z)=(iz+1)/(z+i).
Solution:
f(z)=(iz+1)/(z+i)=i+2/(z+i)
We decompose the transformation into 4 simple ones:
w1=z+i
w2=1/w1
w3=2w2
w4=i+w3
In this example region consists of intersecting circles which should
be treated in the following way:
Circles:
Let x,y be real numbers; z=x+iy. Then z* called the conjugate of z,
denotes x-iy and zz*=(x+iy)(x-iy)=|z||z|.
When a,z are two complex numbers, the conjugate complex of z-a is z*-a*.
If a is represented by a fixed point A in the Argand diagram, the z of
any point on a circle with centre A and radius k satisfies the equation:
(z-a)(z*-a*)=|z-a||z-a|=k*k
then we may refer to it as the equation of the circle.
~Next step in solving this problem is drawing Argand maps...
(End of Example1)
Motivation For The Conformal Mapping Tool
Since the problems that appear in conformal mapping vary concerning the
complexity of the given function and conditions, the number of maps drawn
can be very high. Therefore, to cut the routine work and let students devote
time to understanding the concepts of conformal mapping (as a part of the
Complex Analysis course) it was necessary to make a suitable tool program.
Other transformations:
Relations between two complex variables w and z have developed into an
important branch of higher mathematics, the theory of conformal
representation. It would be easy to include a great deal more of this
theory into the existing program.
Features:
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The first thing to do is to maximise the window to view the whole model.
Example buttons - clicking on any of the example buttons will select one of the
functions with corresponding region (conditions) to do the mapping.
Help text box - currently not implemented. This text area is meant to provide
online help when clicking on the various function buttons and windows
Underneath the given conditions and function there are textfields to write each
of the corresponding decomposition (w1, w2, etc). Next to each of the textfields
is a button which when pressed should light up green or red depending on whether
the decomposition for that value is correct or not(not implemented). To enter in
a new decomposition, we press the NEW TRANSFORMATION button in the 2nd tool bar
to the right, which automatically creates another argand diagram to correspond
with that decomposition.
The idea is for the user to draw out the initial region with the given
conditions which appear in the first argand diagram.
To draw circles, we input the centre position of the circle in [x,y] format in
the first text field and the radius of the circle in the second text field and
then pressing the DRAW CIRCLE button to enter it into the diagram.
eg type in 1st textbox: [50,50]
and in 2nd textbox : 100
to draw a circle with its centre Point at (50,50) with radius of 100. This
would correspond to the condition |z+(50+50i)| < 100
To draw a straight lines parallel to the real or imaginary axis, we just enter
the position of the line on the axis and click on either the DRAW RE/IM
buttons.
eg type: 100
and click DRAW RE button to draw line at 100 on the real axis for the condition
Re < 100.
To define a line with a gradient, we input the line definition in the form:
[gradient,y-value of line through imaginary axis]
eg type: [1,0]
and click DRAW LINE to line a diagnol line with a gradient of 1 passing through
the origin
Shading of the region of conditions is implemented by a grid of points over the
diagram (by default 20x20). Each point is tested and if it falls within the given
conditions it is shaded red. The shading is updated automatically when new shapes
are added or transformations are changed.
Currently, by default the regions are chosen within the shapes (eg the conditions
z