Internal angle of components of Mandelbrot set
multiplier map of hyperbolic component converts component into unit circle
Main cardioid = component with attractive cycle of period length 1

c = F1(m) =m/2 - (m*m)/4
where m = r * ei *angle


On this image one can see internal rays of main cardioid of M-set for f(z)=z*z +c
for angles 1/20,2/20 .. 20/20.
they are lines: c= ( r * ei *angle)/2 - (sqr(r)*ei*2*angle)/4;
where r : 0 < r < 1

q:=20;
For p:=1 to q do
 begin
  angle:=p/q;
  DraweInternalRayOfMainCardioid(angle);
 end;

Here you can see internal rays as above and internal "curves" :
for i:=2 to 9 do
  begin
    radius :=i/10;
    DraweInternalCurveOfMainCardioid(radius);
  end;
Period 2 component


c = F2(m) =m/4 - 1

p,q: integer
rotation number = p / q ;
q = period of p/q bud under function f(z)= z*z + c
q = period of external angle a1,a2 under Doubling Map
p = the number of the attracting cycle jumps in "ears" in the filled Julia set in the counterclockwise direction at each iteration. Therefore bulb has rotation number p/q. See M-set explorer by Bob Devaney
IntAngle= internal angle ( or internal argument)= rotation_number * 2 * Pi [ radians ]
IntAngle= internal angle ( or internal argument)= rotation_number * 360 [ degrees ]
root point of (p/q) bulb ( component) = 0.5 exp (i*IntAngle) - 0.25 exp (2*i*IntAngle)
or
root ( IntAngle):= x+y*i
x = 0.5cos(IntAngle) - 0.25cos(2*IntAngle)
y = 0.5sin(IntAngle) - 0.25sin(2*IntAngle)
The disc attached to the main cardioid at internal angle p/q is an q-cycle disc
When you know rotation number = p/q of root you can compute external angles of 2 rays that land on that root. That external angles ar periodic with period= q under doubling map.
from: Parameter Ray Atlas Tables by Linas Vepstas  ( changed)
p,q:=positive integer
p/q=rotation number [rational number]
a1,a2 = external angles of 2 external rays landing on root point of p/q component ( bud)
external angles a1, a2 are periodic with period= q under doubling map.
period(a1)=period(a2)
q = period of p/q bud;
root point of p/q component = point in main cardioid ( border of M-set)
a1:=a1_numerator / a_denominator
a2:=(a1_numerator +1) / a_denominator


n p / q a1 a2 period of a1 under doubling map
1 0 / 1 0/1 1/1 1
1 1 / 1 0/1 1/1 1
2 1 / 2 1/3 2/3 2
3 1 / 3 1/7 2/7 3
4 1 / 4 1/15 2/15 4
5 1 / 5 1/31 2/31 5
... ... ... ... ...
n 1/n 1/(2n-1) 2/(2n-1) n
... ... ...
1 2 / 3 5/7 6/7 3
2 2 / 5 9/31 10/31 5
3 2 / 7 17/127 18/127 7
4 2 / 9 33/511 34/511 9
5 2 / 11 65/2047 66/2047 11
... ... ... ...
n 2/(2n+1) (2n+1+1)/(22n+1-1) (2n+1+2)/(22n+1-1) 2n+1
13 / 4
23 / 7 41/127 42/127
33 / 10 145/1023 145/1023
43 / 13 545/8191 546/8191
n3/(3n+1) (22n+1+2n+1+1)/(23n+1-1) (22n+1+2n+1+2)/(23n+1-1)
... ... ... ...
1 3 / 5
2 3 / 8 73/255 74/255
3 3 / 11 273/2047 274/2047
4 3 / 14 1057/16383 1058/16383
n3/(3n+2) (22n+2+2n+1+1)/(23n+2-1) (22n+2+2n+1+2)/(23n+2-1)
... ... ... ...
n4/(4n+1) (23n+1+22n+1+2n+1+1)/(24n+1-1) ...
... ... ... ...
1
2
3 4/15 4369/32767 4370/32767
n4/(4n+3) (23n+3+22n+2+2n+1+1)/(24n+3-1)

Algorithm for drawing internal ray for :
- angle t ,
- period p
- component Hp

* for period 1 and 2 use explicit functions
* for period >2 :
** start with center of component
** trace internal ray ( find next point c with similar t ) until abs(m(z))>1


How to find internal angle for p,c,center:
* compute period z points ( number of point = p )
* find multiplier
* compute arg(m(z)) and abs(m(z)) for z


source:
Main page

Feel free to e-mail me!

Author: Adam Majewski adammaj1-at-o2-dot-pl

http://fraktal.republika.pl

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