# Computer pictures of Möbius transformations

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How to get a computer to calculate Möbius transformations

These are the three transformations in Exercise 3H Q.6 as calculated using the online Möbius transformation calculator.

The online calculator is at:

http://www.math.ucla.edu/~tao/java/Mobius.html

(Note: this calculator uses Java, so you’ll need to use a browser other than Chrome, and you will probably need to add the site to the “Exception List” in your Java settings).

The transformation is w=1/z for all three original z-paths.

(a) |z|=2

(b) arg z = π/4

(This is a half-line, not a line, and the online calculator only does lines, so I’ve marked points on the z half-line and the online calculator shows us where those points go in the w-plane. Not every half-line goes to a half-line! This one does, but most half-lines go to arcs of circles)

(c) y=2x+1

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The z-diameter in line with the pole ⟼ a w-diameter

The diameter of the z-circle in line with the pole (which is z=1 in the picture above) ⟼ a diameter of the w-circle.

The picture shows that in blue. (I’ve made the transformation w=5/(z−1) just to make the image a decent size. We’d get the same conclusions from w=1/(z−1), only the w-circle would be too small to see clearly.)

The points on the z-circle near the pole (orange points in the picture) ⟼ points on the w-circle further away

Points of the z-circle far from the pole (green points in the picture) ⟼ nearer points on the w-circle

Every other diameter of the z-circle (for example the blue line in the picture below) becomes an arc joining two points on the circumference of the w-circle. But as long as we know one diameter of the z-circle which ⟼ a diameter of the w-circle, then we know enough to find the w-circle.

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w = (z−i)(z+i)

If w = (z−i)/(z+i), the z-locus |z|=2 (left plane) produces the w-locus shown in the right plane, above.

the z-locus |z|=4 (left plane) produces the w-locus shown in the right plane, above.

the z-locus y=3x (where z=x+iy, so x=Re(z), y=Im(z)) produces the w-locus shown in the right plane, above.

the z-locus y=0 produces the w-locus shown in the right plane, above.

the z-locus x=0 produces the w-locus shown in the right plane, above.

the z-locus |z|=1 produces the w-locus shown in the right plane, above.

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w = (z+2)(z−2)

The transformation in all these examples is w = (z+2)(z−2)

1. |z|=3
• the z-locus is a circle, centre O, radius 3
• cartesian equation x2+y2=32

2. Re(z)=0
• the z-locus is a line along the imaginary axis
• cartesian equation x=0

3. Re(z)=Im(z)
• the z-locus is a line through the origin, π/4 anticlockwise from the positive real axis
• the z-locus is the half-line arg(z)=π/4 plus the half-line arg(z)=5π/4
• cartesian equation y=x

4. Re(z)=2
• the z-locus is a line parallel to the imaginary axis through z=2
• cartesian equation x=2

5. Re(z)=Im(z)+2
• the z-locus is a line through z=2, π/4 anticlockwise from the positive real axis
• cartesian equation x=y+2

6. |z|=2
• the z-locus is a circle, centre O, radius 2
• cartesian equation x2+y2=22

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What Molly Alice looks like, Möbius-transformed

This is Molly Alice as a stick figure transformed by w=4/(z−1). The stick legs, arms, and thorax ↦ arcs of circles, and the circle head ↦ a circle. The line segment which represents her left leg ↦ an infinitely long half-line because it goes through the pole (centre of inversion) z=1.