Print Page - Dumb Down Triangle Inequaltiy / Stripe Average / Curvature avg coloring methods

Please help me understand the formulas to generate the band colorings of a fractal that use the formulas Triangle Inequality / Stripe Average / Average Curvature...etc as explained by JUssie Harkonen.

I am not a math genius so it needs to be dumbed down.

It seems like when I read his thesis... i think I understand it... but when I go to implement it, It's not working. It's close..... It seems like I'm not averaging far enough ahead or behind.... and when I try to average, it always loooks way worse.... all one color..... Not sure if i'm averaging incorrectly.... or using the value I obtain incorrectly.

I have read the two threads that talk about this subject and still do not understand.

I have tried many ways and i can't seem to get it to eliminate the banding:

All attempts had tested bailout values at 4, 1000, 1000000000 (thats the max value i can use without causing some overflow error)
Dividing the arcTanSum by the number of iterations.
Totaling only the last four values of the ArcTanSum and dividing by 4
Calculating 1 to 4 iterations past the bailout of Z
Using the arcTanSum value as a direct color value
Using the arcTanSum value as part of the smooth coloring formula - this is the closest I've got but it gets messy towards the fractal "edge"
I've watched videos about the catmull-Rom spline interpolation..... have no idea what numbers to use where or what to do with them if i even got them.

I am not sure if the things I have tried WERE correct idea but implemented INCORRECTLY...... or just plain wrong.

Here's my code:
I've left any code out that I figured was unimportant/ didn't affect calculations

Code:

 For Y = 1 To H ' Image Height
            For x = 1 To W ' Image Width
                
                cRE = (((cX + x) - W / 2) * 4 / W) * Zoom 
                cIM = (((cY + Y) - H / 2) * 4 / W) * Zoom

                Rx = 0
                Ry = 0
                

                inside = False
                arcTanSum= 0

                For I = 0 To MaxIterations

                    lastZ = Z
                    orX = Rx
                    orY = Ry

                    newX = Rx * Rx - Ry * Ry + cRE
                    Ry = (Rx + Rx) * Ry + cIM
                    Rx = newX
                    Z = Rx * Rx + Ry * Ry

                    arcTanSum = arcTanSum + 0.5 * Math.Sin(Math.Atan2(Ry, Rx)) + 0.5 'Formula I got from another thread from this forumn

                    If Z > EscapeVal Then
                        c = I
                        inside = True
                        Exit For
                    End If
                Next

                If inside Then

                    For ix = 0 To 3

                        newX = Rx * Rx - Ry * Ry + cRE
                        Ry = (Rx + Rx) * Ry + cIM
                        Rx = newX
                        Z = Rx * Rx + Ry * Ry
                        arcTanSum = arcTanSum+ 0.5 * Math.Sin(Math.Atan2(Ry, Rx)) + 0.5

                    Next

                    mu = ((arcTanSum)) + 1 - Math.Log(Math.Log((Math.Sqrt(Z)))) / Math.Log(EscapeVal)  ' This formula was initially used with variable 'C' (the iteration count at escape) to generate fraction for smooth iteration coloring

                    clr1 = Int(mu)
                    t2 = mu - clr1

                    t1 = 1 - t2
                    clr1 = clr1
                    clr2 = (clr1 + 1)

                    r = (Math.Sin(FreqR * (clr1) + PhaseR) * 127 + 128) * t1 + (Math.Sin(FreqR * (clr2) + PhaseR) * 127 + 128) * t2 'SMOOTH COLOR
                    g = (Math.Sin(FreqG * (clr1) + PhaseG) * 127 + 128) * t1 + (Math.Sin(FreqG * (clr2) + PhaseG) * 127 + 128) * t2 'SMOOTH COLOR
                    b = (Math.Sin(FreqB * (clr1) + PhaseB) * 127 + 128) * t1 + (Math.Sin(FreqB * (clr2) + PhaseB) * 127 + 128) * t2 'SMOOTH COLOR


                Else
                    r = 0 
                    g = 0 
                    b = 0 
                End If

            Next

        Next

Here's one of the last fractals I generated attached.

I have visited many, many....many bulletin boards in my days seeking answers....and have always found. I have never posted a question myself because I've always found the answer and understood it enough to continue on. Please excuse any newbie mistakes I may have made in making this post.

Nevermind! After a lot more trial and error, I finally figured out what I was doing wrong. One thing was I was using the wrong value of Z. I should have been using the last value of Z before the escape. I included the calculation for the upper and lower bounds correctly. I thought just using the previous Z value was the same as subtracting the real and imaginary parts from the Z value... ... Also I realized the the value that it generated for the color is a number less than 1 and multiplying that number by the total colors gave me a color in an indexed palette..... but you can use that number directly if you use the math.sin and freguency phase method to calculate the actual color at run time. I got it to work doing the T.I.A. method and when that worked I went back and applied the same part to my angle calculations and it made the smooth lines like described by Jussi Harkonen.

Here's what it looks like now.... all smooth and no bands! Hooray!

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Fractal Software => Help & Support => Topic started by: MandelBRO on July 08, 2017, 03:15:39 AM