Computer pictures of Möbius transformations

Click here for pdf booklet for Möbius transformations lessons. Click here for exercises from the textbook, and answers.
Click here for worked answers to questions in textbook and all recent FP2 exam questions on this topic


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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

circle-radius-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)

argpi4

(c) y=2x+1

y2xplus1


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

diameter-points

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.

diameter


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w = (z−i)(z+i)

z2

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


z4

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


y3x

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.


y0

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


x0

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


z1

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

Screen Shot 2016-07-12 at 12.28.24


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

Screen Shot 2016-07-12 at 12.30.14



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

Screen Shot 2016-07-12 at 12.30.49



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

Screen Shot 2016-07-12 at 12.31.51



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

Screen Shot 2016-07-12 at 12.33.06



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

Screen Shot 2016-07-12 at 12.33.34


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What Molly Alice looks like, Möbius-transformed

mollyalice_mobius

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.