Ducky Plus
Ducky Plus extends the capabilities of the existing Ducky-formulas. Per pixel it can in one run follow various basic codes. These successive iterations take place in three steps:
From start to n1 ['End of First Step'] the first series of iterations,
from n1 to n2 ['End of Last Step'] the second series,
and finally from [n2 + 1] to 'Maximum iterations' the third and last series.
If n2 is set lower than n1, or if both n1 and n2 are equal, then automatically n2 is increased so that in any case, step 2 will execute one single iteration.
The possibilities of step 1 and 3 are the same, but step 2 has somewhat more options. It is not necessary to use all three steps. If we want to leave out step 3, then n2 can be set to a value at least 1 higher than that in 'Maximum iterations'. To leave out step 1, n1 has to be set to zero. It is also possible to utilize only one single step. That must always be step 2. In that case n1 must be 0 and n2 one higher than 'Maximum iterations'.
All three steps [but not the 'Burning Ship' codes] use functions as available in the standard list complex functions in UF. Each step can use different functions. On these basic functions in all three cases, a correction is possible, with 'Function correction'. Further, they are currently equipped with a 'modifier' which may affect the shape of both the real and the imaginary branch of the formula separately, and a correction to the Julia-vector. With this last parameter for each step an adjustment can be made to the overall Julia vector c, the 'seed' value [at least in the Julia mode, not in the Mandelbrot mode. In the second step a more parameters are available. They will be discussed in more detail later on.
Codes 4, 5 and 6, the so-called 'Burning Ship' options, available in step 2, do not make direct use of the UF list of complex basic functions. These functions are not complex but 'floats'. Again, here, all aforementioned parameters are available, the 'Function correction' separately for the real and the imaginary function.
At the start of 'Ducky Plus' in most cases the image will be seen, as a little inviting black plane. By now chosing an appropriate 'Inside Coloring' this is quickly corrected. We can therefore choose a Lyapunov [dmj.ucl, em.ucl] or take one of the many 'Orbit Trap' related colorings, but also other colorings are possible. Unlike most other 'Ducky' formulas we do'nt get both the M- and the J-image, we only see the not very attractive Mandelbrot-image. With e.g. F7 and a click on the 'Explorer' it's possible to get a Julia image of your choice.
For each of the steps, after it is completed, a final operation is available in the form of a last 'End Function', together with a 'Function correction'. The function may be applied for all iterations or only for iterations starting at a certain iter value. In the latter case we have to click on button 'Limited End Functions' and to set the desired start value with parameter 'Start End Function'.
Steps 1 and 3 have only the original 'Ducky' option, with only complex numbers and functions. Step 2 on the other hand offers several codes, three with complex functions as UF standard available ['Ducky', 'Ducks', and 'Anatra'] and three which use various 'float' functions [the 'Burning Ship' codes]. 'Ducks' is based on the same UF formula of Sam Monnier [sam.ufm], 'Burning Ship' uses the formula of Michael Mischelitsch and Otto E. Rössler from 1992 [https://en.wikipedia.org/wiki/ Burning_Ship_fractal], as also has been done earlier in UF by Pasi Piitulainen [jcx.ufm] and Kerry Mitchell [lkm3.ufm].
In addition, there are in step 2 four different settings: '[Sam] Monnier's preparation', 'Hokum', 'Double Clutch' and 'Log Transition'.
Monnier Preparation
This code is described by Monnier was originally intended for 'Ducks' formula and also seems to work for other options. Within this preparation are three parameters: 'Order', 'Angle' and 'Twist'. Unlike the name suggests, 'Angle' does not give rotations but changes, just like the other two parameters, the fractal image.
Hokum
The base code for this option is mathmatically spoken doubtful but it gives unexpected fractal images.
Double Clutch
You don't need to 'double clutch', this option works for you automatically. This part of the code can be regarded as a function of a function, to write it easy: z = function [function [z]]. In addition, the exponents in the internal and external functions can be set separately. Also 'Double Clutch' [with different exponents] can change the image dramatically.
Logarithmic Transition
This function, applied at the end of the formula in Step 2, also comes from Sam Monnier [sam.ufm, 'Ducks']. As with his 'Preparation'', it is possible to apply it in other options than 'Ducks'. Again, the fractal image is changed radically.
Of the three "Burning Ship" options usually two give the same image. This is unavoidable but not disturbing. The options 'BS Ducks' and 'BS Anatra' basically have the same features.
Note.
Sometimes it happens that a change in 'Inside Coloring' has no effect on the image. Then you are arrived in the outside regio, so change the 'Outside Coloring'. This is of course not specific for 'Ducky Plus', but with fractals of the Ducky type it happens quiet often.
Ducky Plus
Ducky Plus extends the capabilities of the existing Ducky-formulas. Per pixel it can in one run follow various basic codes. These successive iterations take place in three steps:
From start to n1 ['End of First Step'] the first series of iterations,
from n1 to n2 ['End of Last Step'] the second series,
and finally from [n2 + 1] to 'Maximum iterations' the third and last series.
If n2 is set lower than n1, or if both n1 and n2 are equal, then automatically n2 is increased so that in any case, step 2 will execute one single iteration.
The possibilities of step 1 and 3 are the same, but step 2 has somewhat more options. It is not necessary to use all three steps. If we want to leave out step 3, then n2 can be set to a value at least 1 higher than that in 'Maximum iterations'. To leave out step 1, n1 has to be set to zero. It is also possible to utilize only one single step. That must always be step 2. In that case n1 must be 0 and n2 one higher than 'Maximum iterations'.
All three steps [but not the 'Burning Ship' codes] use functions as available in the standard list complex functions in UF. Each step can use different functions. On these basic functions in all three cases, a correction is possible, with 'Function correction'. Further, they are currently equipped with a 'modifier' which may affect the shape of both the real and the imaginary branch of the formula separately, and a correction to the Julia-vector. With this last parameter for each step an adjustment can be made to the overall Julia vector c, the 'seed' value [at least in the Julia mode, not in the Mandelbrot mode. In the second step a more parameters are available. They will be discussed in more detail later on.
Codes 4, 5 and 6, the so-called 'Burning Ship' options, available in step 2, do not make direct use of the UF list of complex basic functions. These functions are not complex but 'floats'. Again, here, all aforementioned parameters are available, the 'Function correction' separately for the real and the imaginary function.
At the start of 'Ducky Plus' in most cases the image will be seen, as a little inviting black plane. By now chosing an appropriate 'Inside Coloring' this is quickly corrected. We can therefore choose a Lyapunov [dmj.ucl, em.ucl] or take one of the many 'Orbit Trap' related colorings, but also other colorings are possible. Unlike most other 'Ducky' formulas we do'nt get both the M- and the J-image, we only see the not very attractive Mandelbrot-image. With e.g. F7 and a click on the 'Explorer' it's possible to get a Julia image of your choice.
For each of the steps, after it is completed, a final operation is available in the form of a last 'End Function', together with a 'Function correction'. The function may be applied for all iterations or only for iterations starting at a certain iter value. In the latter case we have to click on button 'Limited End Functions' and to set the desired start value with parameter 'Start End Function'.
Steps 1 and 3 have only the original 'Ducky' option, with only complex numbers and functions. Step 2 on the other hand offers several codes, three with complex functions as UF standard available ['Ducky', 'Ducks', and 'Anatra'] and three which use various 'float' functions [the 'Burning Ship' codes]. 'Ducks' is based on the same UF formula of Sam Monnier [sam.ufm], 'Burning Ship' uses the formula of Michael Mischelitsch and Otto E. Rössler from 1992 [https://en.wikipedia.org/wiki/ Burning_Ship_fractal], as also has been done earlier in UF by Pasi Piitulainen [jcx.ufm] and Kerry Mitchell [lkm3.ufm].
In addition, there are in step 2 four different settings: '[Sam] Monnier's preparation', 'Hokum', 'Double Clutch' and 'Log Transition'.
Monnier Preparation
This code is described by Monnier was originally intended for 'Ducks' formula and also seems to work for other options. Within this preparation are three parameters: 'Order', 'Angle' and 'Twist'. Unlike the name suggests, 'Angle' does not give rotations but changes, just like the other two parameters, the fractal image.
Hokum
The base code for this option is mathmatically spoken doubtful but it gives unexpected fractal images.
Double Clutch
You don't need to 'double clutch', this option works for you automatically. This part of the code can be regarded as a function of a function, to write it easy: z = function [function [z]]. In addition, the exponents in the internal and external functions can be set separately. Also 'Double Clutch' [with different exponents] can change the image dramatically.
Logarithmic Transition
This function, applied at the end of the formula in Step 2, also comes from Sam Monnier [sam.ufm, 'Ducks']. As with his 'Preparation'', it is possible to apply it in other options than 'Ducks'. Again, the fractal image is changed radically.
Of the three "Burning Ship" options usually two give the same image. This is unavoidable but not disturbing. The options 'BS Ducks' and 'BS Anatra' basically have the same features.
Note.
Sometimes it happens that a change in 'Inside Coloring' has no effect on the image. Then you are arrived in the outside regio, so change the 'Outside Coloring'. This is of course not specific for 'Ducky Plus', but with fractals of the Ducky type it happens quiet often.
