### Hawaiian earring

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• Fractal Friar
• Posts: 144

#### Hawaiian earring

« on: December 02, 2018, 09:52:08 PM »
Hi,

I have plane with Hawaiian earring
https://en.wikipedia.org/wiki/Hawaiian_earring
each circle has center = 1/r and radius =1/r
where  r is a real number greater then 1.

I have point z ( complex number)
How can I find r ?

I was trying:

abs(z-center) = abs(center - 0)
(z-c)^2 = c^2
z-2cz= 0

• 3f
• Posts: 1634

#### Re: Hawaiian earring

« Reply #1 on: December 03, 2018, 12:00:32 AM »
Center is at (c,0), so |z-c|=|c| so c = |z|^2/2Re(z) = 1/r

• Fractal Friar
• Posts: 144

#### Re: Hawaiian earring

« Reply #2 on: December 03, 2018, 06:35:07 PM »
Center is at (c,0), so |z-c|=|c| so c = |z|^2/2Re(z) = 1/r

so  r = (2Re(z))/|z|^2

If I take r = 1 then I have point z= 2

then using above equation

(2*2)/ 4 = 1

OK

and another point z = 1+1i

(2*1)/sqrt(2) = 0.7071067811865475

Wher is the error ?

• 3f
• Posts: 1634

#### Re: Hawaiian earring

« Reply #3 on: December 03, 2018, 10:11:42 PM »
$$|1+i|^2=2$$

• Fractal Friar
• Posts: 144

#### Re: Hawaiian earring

« Reply #4 on: December 03, 2018, 10:58:54 PM »

Thx.
The result

Code: [Select]
unsigned char ComputeColor(complex double z){ double r; //   r = (2Re(z))/|z|^2 r = 2.0*creal(z)/ (cabs(z)*cabs(z)); while  ( r>1.0) r -=1.0; return r*255; // 8- bit color = shades of gray}

• Fractal Friar
• Posts: 144

#### Re: Hawaiian earring

« Reply #5 on: December 03, 2018, 11:07:33 PM »
OK. Let's take more complex problem
Curves are not circles
- https://commons.wikimedia.org/wiki/File:Parabolic_orbits_insidse_upper_main_chessboard_box_for_f(z)_%3D_z%5E2_%2B0.25.svg

How one can colour curves  to get such result:

?

Look also :