I have uploaded new Distance Coloring plug-in files to jlb.ulb. These are similar in concept to Orbit Traps: for each iteration, as the z associated with a pixel moves in the complex plane, information can be accumulated about distances. Afterwards this information is used to calculate an index into the gradient for coloring. See a sample image and a upr in the following post.
Load the upr in the following post into UltraFractal to follow along. This upr has 16 layers, each with the same divergent formula (a simple Julia), the same smooth gradient (gold to brown), and the same location and magnification. Each layer colors the formula differently. The top layer is the standard Smooth (Mandelbrot) coloring for comparison.
Experiment by making changes in these examples.
As with all the UF formulas, you don’t need to understand the details to make interesting fractals.

In the Accumulation Phase, zz is chosen from among z, z-z_prev, z/z_prev, z-pixel, and z/pixel. Here z_prev is the previous value of z. Then, optionally, zz may be replaced by F(azz), where F is one of the built-in functions (like sin or log) and “a” is a parameter. This zz is used for distance calculations.
Next is the distance method. It calculates a distance, d.
All of the iterations may be used, or just some of them. I describe various distance methods below.
Note that in the Final phase, “How to color,” the first six possibilities only use specific distances: First, Last, Smallest, Largest, Distance Range (Largest-Smallest), and Distance Ratio (Largest/Smallest). These are floats rather than complex values.
The remaining six possibilities use averages, and there are more possibilities. The averages may be weighted. Positive values of p give more weight to later iterations; negative values of p give more weight to earlier iterations; p of zero weights all iterations the same.
For these final six possibilities, each distance d may be used, or the multiplied values d
zz or dG(bzz), where G is one of the built-in functions (like sin or log) and “b” is a parameter. If the multipliers are used, the result is a complex value and other averages are possible. See below for the averages
After the averages, two final smooths are possible to mimic the standard colorings of Smooth (Mandelbrot) and Triangle Inequality Average, though they may give interesting shapes for other formulas.
Finally, after an answer is calculated, it or its reciprocal may be used for the index into the gradient.

Averages:
(Note: It’s not necessary to know the mathematics to use the formulas!)
Assume that N of the iterations were used.
Arithmetic Average: Add up all the distances and divide by N.
Geometric Average: Multiply all the distances together and take the Nth root.
Harmonic Average: Add up the reciprocals of all the distances, divide by N, and take the reciprocal.
Standard Deviation: Add up the squares of (distance minus Arithmetic Average) and divide by N.
Coefficient of Variation: Divide the standard deviation by the Arithmetic Average.
Fractal Dimension: Divide the standard deviation by the Distance Range.

Distance methods:
Circles: Use from one to five circles, each defined by its center and size (radius). Color using only iterations with zz inside one or more circles, or outside one or more circles, as specified. If inside, use the smallest distance to a circle center. If outside, use the largest distance. A circle shape of 2 gives normal round circles; shape less than 2 makes them pointy; shape more than 2 makes them squarer. For inside, a positive power emphasizes iterations closer to the center(s); negative power emphasizes iterations farther from the center(s). For outside, the reverse is true.
Exponential smoothing: Distance is calculated as in the standard versions in standard.ucl and standard.ulb, except for the scale parameter.
Lines: Use one or two lines. Each line is defined by a point on the line and an angle. Color using iterations only on the positive or negative side of the line or lines, as specified. Distances from the line(s) are calculated using a scale and a power. A positive power emphasizes iterations farther from the line(s); a negative power emphasizes iterations closer to the line(s).
Quadrant: The plane is divided into four quadrants depending on the center point and an angle. Color using only iterations with zz in the specified quadrant. Distances are calculated based on one of many combinations of the real and imaginary parts of zz, relative to the quadrant center and angle.
Simple: Distances are calculated based on one of many combinations of the real and imaginary parts of zz.
Square: A square is defined by its center and its side length. Color using only iterations with zz inside or outside the square, as specified. For inside, a positive power emphasizes iterations closer to the center; a negative power emphasizes iterations farther from the center. For outside, the reverse is true.
TIA Distance: Distance is calculated as in the standard versions in Standard.ucl and Standard.ulb. Sometimes the TIA final smooth is useful.
Trap: This is a wrapper for using one of the TrapShape distance calculating plug-ins. These can be found in ck.ulb, dmj5.ulb, kcc5.ulb, mhb.ulb, mmf.ulb, mt.ulb, reb.ulb, rkb.ulb, sam.ulb, and standard.ulb.

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One of 16 images in the upr:

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One of 16 images in the upr: ![61157ba1424ae.jpg](serve/attachment&amp;path=61157ba1424ae.jpg) DistanceExamples { ::FcN8qhn2t31SzJOOQ47pq8fQlPlszC4nQyslOkJ7MZnpmsVtTStXpE2CQ76XrsJPoyP+tlMk kJILIEgQC6GY9ob3ud/9pWNi+cSYJJ+j7vHCVyKjpYrfnVUSSDpI6Nkk8YahF6aWU5QcnAb0 QKbwwS5HjJ3S5FYn2ixGypRsyCs13YJoPFPi+bH1yxtlrtrjFikWyIxMSBLdA+Wax+7JHrUo hk8SWWK26ikssyhozJp04e8sSLUWOJkVeL2x2GdFrg1D0u0MUCtcYWEOZUcJLnUAzWCJPHm6 q5jmWS5Y7W2oEygUsz+71PjDdmIbOhcDT0uYOzp8wh0w/Fn1vPqPLmmSSg7/Lg7+ICPq5o+J WIY643C3WjA93SMD5dLo0IsdzgW2NhJpbe21wE61S85eEWc2oSsD1f/9YpFsIalplTSL6DdL NLliKyiZRYf3jC8b7H0BMgwYmtvxsUKh/oefcgX7AflqaYsU5motnRTpcW4ZcSEDu0pZxZcw ANR/Dn81TjBrH26fi71cUcvP+tv/puTf2rbENvq7gq5vyKOTrT/uo5aG/EF7x9cmuVdfnmpa KGHNGPemr2Hu4YU/nc1vgjzGoaSIwDRH4hozsi+eF78sIVDNCsTPxydBT8yiVd9+ntaa610H BYXH7OuHpqnhPPxHuYyPUtCMj98MMJMrYoqpoH2plqHkJ0uk4YsF6k4Ylq4wsrTIp3iPSlfQ JhXidVNsrLvNXGT4pNkfdp4NUlmZRgDc0s3+y5y67kiS0UTgSdd88cO6zSJwNLEJLqKUGSM1 WzqkVBOUaIfIOyMtxphsc57FTfNSG2oQG/UEeFBikIDq67hYgScD232DJnF4h/0L1xe6lOyv jjvMMeVTQYmpN1utnv/kL7+wk4csrrTngpN0R6QpYIePMTTFyko6PWnhHgVd3++Y+uBBKRJO lxDBIJ0BekD3ZRIiow4gbbQCPCtIbgBq45AVY9lDI/y4xHKV8ZBNAjdapSMjOH3Spy+Aqh15 s0D+BlE/rfFcxO0aBhQm4cvyxQU3t0RJqD+IbtYIBCH6qux8sYCvG7tsDcHsj6WiCLhGtbVv 6DPwdqTwk0Bwgrfou1NQhYhG1LWXNi1F3WzY90JXPcj5IYPNC2D74qZw+6M0+gh2Vng91IYf cDtSOQnkDAJ3QroD0I6AcDNWbWaXRE0vKDeWnD61YH14/vdJPdB8xtc2T/BhnklyCRncFlTG sAEoqh90fCogqGbFUqb9Em+saX2KSTy1euTzaqnh1kh10iyaqRO7Gas6lZ/cJNFszwZyRHnJ ZEo6JN5u2IN1+Vi1kzRvW0m0Rhg7rTyrZaTub30mMkmWPkmsigVbq2cVRe6iEQlpPr0PZYPt JZP5aIPZIP9qkyp3PsncnD7J7mwr4LdSn0yfqRwaLrTOvSZdyxk1JTWn25z60JcW5QAB2k3p tMmTfuIHsnyq/YyoFo21whSyEaVQjyN4NYxdccHbvjCCM0nWe6T0byVSeyOoVDn5Qf6OB3p7 acng90dLK7pPfTeVVLth4PJLYltR0lCW6aCcxZFCu4sEQL3XMccRM65ArI8haJ8gWKwFv6BX sttNoLziuczrB6ybgFp301sI9FDlpVt7wxcwaEW1ZxacXTLTfzD0sdCz8f8S1wMwLtNd9fRI NB1i04vyzD8ZH07efLzOp/WCz57At9CTppbAVWtLdRNcCETTEXbtApU5Iv5y7rFytepd9QW4 QRQa5Ogxrvj5Zs0S5OhXbW8ipOYv6VZhvMIpPYpRAua3qdpMcxN8srXCg7LqBIk6VjQSsmyr 0sNsmaXzg4aQcfovCuvGIXDkrByVDk7HQNMQuvhgc/CjbK3pdQA3t5dobaSU9rZ1umNq7FD9 KOIAUg96M3snuU7SnB29FC72euouv7Wo7acDND2QpZeV9Lc3sXmbCQ3/aEJCgbKhYH76lYsr B4d1e8nUXwMi6gZv0l5Kcl3IHDKTwX4Uaa3Ju/aqO4oJdxRLiqu6t9/gbMIEvj1OXJ4UHkln pAP3thtq8wRHg+Jm5oD3prxTzKI32AyWq1NuDdget+APC2N2UPz6dWSgjTEkuO0ktxADWxid kUMubOne1z5MpQEcbZy2o010gVsBxKe/UCIWX+1TqO1HNYGr8ckZyQmBswkhslMDZ6OAiie0 rIaSRmJDZLeGyM/Eo31XoD8OBnagsM/2BWXVTRtFTh6AJvY0rJe0bmCqQWED+bCstf689QTZ ZslWxCQzrlTk/X7E4daGtfTUWf0fT4MJIhBYbrAYDWh7OPqmvBV7FeQyuknIaC0Ng09cOVPW 8/8WuEitIWbj5vvFzZB1qKJeU/6BN2R/3b5SOJHdQSJqB6TjKhg89jv9Jb+zuC0h3DjiwDBl yge8iTjnEto284JBMWRoFk8NY+7knLmWJlSF48L7KE/FiLe/7QW6H8CqST7L4YmP5oMXNC7j 6Npujjxqmh3QnzB+ZTh06mkBLz7HgzyoCDi571irbJxK3JWg1/zoVNSN }

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Thank you for this, Jim, your examples are very pretty!

I struggle terribly with the maths terminology as I never did anything like this stuff at school but usually get by with working visually. Mercifully, and as you say, understanding is not a requirement with UF. These formulas can still be used by the maths-blind like me.

Thank you for this, Jim, your examples are very pretty! I struggle terribly with the maths terminology as I never did anything like this stuff at school but usually get by with working visually. Mercifully, and as you say, understanding is not a requirement with UF. These formulas can still be used by the maths-blind like me. :D

Chris Martin
Gallery: Velvet--Glove.deviantart.com

Currently using UF6.05 on Windows 11 Professional 64-bit

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Great stuff Jim, thanks for adding it.

Great stuff Jim, thanks for adding it.

Ultra Fractal author

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