# How to transform a set of lines to a set of circles using möbius transformation?

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### Accepted Answer

William Rose
on 12 Jan 2023

Edited: William Rose
on 12 Jan 2023

[edit: fix typos in my text; code is unchanged]

Your image does not include a scale or (more importantly) an indication of where the origin is. The lines in fig 1 and the circles in figure 2 represent sets of complex numbers. You can think of the lines as very large circles that pass thorugh the point at infinity. The Mobius transformation maps these "circles" in figure 1 to the circles in figure 2.

The general Mobius transform is

.

Notice that it appears that all three circles in fig.2 would intersect at a common point, if the circles were complete. Let's assume this common point is the origin. This point must map to infinity - the place where all three lines in fig.1 would "intersect". The general equation for f(z) will map z=0 (the origin) to the point at infinity, if a=d=0. Then we have

which simplifies to , up to a scaling factor. Let's try this:

z1=-ones(1,101)+i*[-5:.1:5];

z2=-2*ones(1,101)+i*[-5:.1:5];

z3=[-5:.1:5]+i*ones(1,101);

plot(z1,'-r.'); hold on; plot(z2,'-g.'); plot(z3,'-b.'); axis equal

Now transform the points:

fz1=1./z1; fz2=1./z2; fz3=1./z3;

Plot transformed points:

figure;

plot(fz1,'-r.'); hold on; plot(fz2,'-g.'); plot(fz3,'-b.');

axis equal

This looks like your figure 2. Note that the transform we used was f(z)=1/z, which corresponds to the Mobius transform, with a=d=0, and b=c=1.

##### 1 Comment

Jonathan Yahir
on 7 Mar 2023

Is this a conformal transformation in general terms? I am comfused by the change of lines to circles

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