Unique property of center of gravity normal Houigh transform of a Mandelbrot plateau]

I had mentioned in a previous post that a Hough transform of the Mandelbrot set might be interesting.
Well in terms of graphics it is an entire flop!
In terms of the properties of the Plateau values it is a very strange success.
The transform I used was c=0.2867683 as the center of gravity (-c,0):
X0=Angle(X+c,Y)
Y0=((X+c)*Cos(X0)+Y*Sin(X0))
What I found was the the values of the Y0
term are very low, most of them are actually zero
and all seem to be less than 2. There is a gap between the rest of the output
which overlaps badly and has just about zero redeeming features as a graphic.
This program allows me to actually sum the Y0 values of the plateau where
K has reached it's limit for the program ( 258 in this case).
This is not the Mandelbrot version of the program where the R limit is
R=x^2+y^2
but
the measure version where
R=Sqrt[x^2+y^2]=(x^2+y^2)^(1/2)
There seems to be no difference in the plateau of the two,
but the colors of the output are more interesting ( usually) in this version
and Mathematically better as a dependence on a measured plane distance R.

I thought it important to report this result as it makes discrimination of the Mandelbrot set's
points very specific and seems to show a gap between the other points.

True Basic program:
999 PRINT " Gives a Hough transform and the area of a Mandelbrot set plateau with measure limit"
1000 PRINT "input 1920 for slow 120 for fast"
1010 DECLARE DEF parity_b, cmod
1020 INPUT m
1030 SET MODE "color"
1040 SET WINDOW 0,1920,0,1024
1050 FOR i=0 to 63 step 4
1060 LET j=i/64
1070 SET COLOR MIX (i) j,1-j, 1-j
1080 NEXT i
1090 FOR i=64 to 127 step 4
1100 LET j=(64-i)/64
1110 SET COLOR MIX (i) 1-j, j, 1-j
1120 NEXT i
1130 FOR i=128 to 191 step 4
1140 LET j=1-(128-i)/64
1150 SET COLOR MIX (i) j,1-j, 1-j
1160 NEXT i
1170 FOR i=192 to 255 step 4
1180 LET j=1-(192-i)/64
1190 SET COLOR MIX (i) 1- j, j, 1-j
1200 NEXT i
1210 REM rainbow LUT(look up table)
1220 FOR j=31 to 0 step -2
1230 LET k=j/31
1240 REM intensity levels
1250 FOR red0=0 to 1
1260 FOR green0=0 to 1
1270 FOR blue0=0 to 1
1280 REM colors of eight different kinds in 32 intensities
1290 SET COLOR MIX (count) k*red0,k*green0, k*blue0
1300 LET count=count+1
1310 NEXT blue0
1320 NEXT green0
1330 NEXT red0
1340 REM color bits
1350 NEXT j
1360 SET BACKGROUND COLOR "white"
1370 LET n=m*1024/1920
1380 REM by R.L.Bagula 15 Nov 2005© copy rights reserved
1390 PRINT "input power of z'=z^s +c"
1400 INPUT s
1410 CLEAR
1420 LET E=Sqr(256)
1430 LET E1=E/7
1440 FOR I=-E1-0.75 To E1-0.75 Step 2*E1/(N+0.5)
1450 LET D=D+1
1460 FOR J=-E1 To E1 Step 2*E1 /(M+0.5)
1470 LET C=C+1
1480 LET A=I
1490 LET B=J
1500 LET K=0
1510 LET D0=Sqr(A2+B2)
1520 LET R=D0
1530 DO
1540 IF K=0 Then
1550 LET F=A
1560 LET G=B
1570 LET R=D0
1580 END IF
1590 LET K=K+1
1600 LET L=R
1610 LET X=F
1620 LET Y=G
1630 LET w=angle(x,y)
1640 LET F=(R^S)*Cos(S*W)+A
1645 LET G=(R^S)*Sin(S*W)+B
1650 rem Barycenter normal Hough transform of Mandelbrot
1651 LET F0=Angle(F+0.2867683,G)
1652 LET G0=((F+0.2867683)*Cos(F0)+G*Sin(F0))
1660 LET R=Sqr(F*F+G*G)
1670 LOOP Until K>Int(E*E)+1 or R>=E
1680
1690 LET count=count +1
1700 LET KK=Mod(259+cmod(K,256),256)
1710 SET COLOR mod(KK,256)
1715 IF K>Int(E*E)+1 then
1716 LET kcount=kcount+1
1720 Let g0count=g0count+g0
1721 PLOT 275*f0+1920/2,3*g0+1024/8
1722 1723 1729 END IF
1730 IF C>M Then LET C=0
1740 NEXT J
1750 NEXT I
1760 LET area=(2*e1)2
1770 LET areaM=area*kcount/count
1771 SET COLOR 255
1780 PRINT area, kcount,count,g0count
1790 PRINT "estimate of Mandelbrot plateau area", areaM
1800 END
1810 DEF parity_b(i)
1820 REM 100 digit binary OR function/16 bit integer
1830 LET C=0
1840 LET a=i
1850 REM modulo power subtraction
1860 REM binary parity
1870 FOR j=0 to 16
1880 LET ab=0
1890 LET ab=int(mod(a,2^(j+1)))/2^j
1900 LET a=a-ab*2^j
1910 LET C=C+ab
1920 NEXT j
1930 LET parity_B=mod(C,2)
1940 END DEF
1950 DEF cmod(i,k)
1960 LET C=0
1970 LET a=mod(i,k)
1980 DECLARE DEF parity_b
1990 LET b=parity_b(i)
2000 IF b=1 then LET c=-a else LET c=a
2010 LET cmod=c
2020 END DEF

Roger L. Bagula { email: rlbagula@xxxxxxxxxxxxx or rlbagulatftn@xxxxxxxxx }

11759 Waterhill Road, Lakeside, Ca. 92040 telephone: 619-561-0814

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