Toolbox Parameters for
MDE formulas


This HTML page describes the different toolbox parameters you can find in MDE formulas. For further explanations or just suggestions, you may contact me at mdessur@hotmail.com.

I would also mention that English is not my first language. So, I would appreciate comments on how to better this text from that point of view. Thanks.


Table of contents


1. Bailout Parameters

Fractal formulas are computed in such a way that for each dot of an image, you may do it until the end of times. They are always repeating a set of steps different in each formula, each repetition being  called iteration. So, to get an image in a shorter time, we just stop after some conditions happens, one being the maximum number of  iterations (maximum iterations box of the formula tab).

The other condition to consider is bailout. This is any condition related to the mathematical behaviour of the formula that we may use. When this condition is reached before the maximum number of iterations, the formula has bailed out. Any dot calculated before bailing out is said to be inside, otherwise it is outside, which are refered to in the formula tabs.

In the toolbox, I give three parameters to play with the bailout conditions:

They are all related.
 
Bailout Test This is a choice of possible behaviours for the formula.
Choices mod (default), real, imag, or, and, manh, manr, iter, angle, path, diff, minus, product, division, real^imag, imag^real
Comments Some of these choices may seem to give the same visual results. However, one can see the difference with a zoom of less than 1. Some formulas can give ribbons, crosses, squares, etc... according to the choice. Some (like iter)  can also give more interesting results by boosting the Bailout Value. Also, for each of these choices, one may apply a Bailout Function to make some variation.

 
Bailout Function This choice of functions is applied to the Bailout Test. It makes some variation on the result.
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.

 
Bailout Value This is a number you must supply for the bailout condition. This is when this value is reached for your choice of Bailout Test and Bailout Function that the formula bails out.
Choices You must supply a real number not smaller than 1.0. The default value is 4.0.
Comments Some Bailout Test may have more interesting visual results by boosting this value.


2. Odd-even parameters

The principle behind the Odd-Even parameters is the following: Actually, if the result is even, a function is applied to Z that you may choose among the parameters. If it is odd, nothing special is done.

With colouring methods, this set of parameters has easily a ribbon texture effect.

There are four related parameters:

Odd-even Type This is where we choose which part or property of Z will be verified if even or odd.
Choices No (default), Real, Imag, Real*Imag, Real+Imag, distance, angle, magnitude, path, trap real, trap imag, trap r*i, trap r+i, trap distance, trap angle, trap magnitude, trap path, iteration, trap iter
Comments
  • No - the odd-even parameters won't be applied
  • Real - the real part of Z will be used.
  • Imag - the imaginary part of Z will be used.
  • Real*Imag - the real part of Z times the imaginary part will be used.
  • Real+Imag - the real part of Z plus the imaginary part will be used.
  • Distance - the distance between Z and the previous Z
  • Angle - the angle of Z to the origin
  • Magnitude - the magnitude of Z
  • Path - the approximate length of the orbit at Z
  • Iteration - the number of iterations at Z
The other choices are not available in some formulas/colouring methods.
When a choice is related to the trap, it means the current trap.
The Odd-even Factor, the Apply odd-even sign? and the Odd-even Function parameters will never work if this parameter is set to No.

 
Odd-even Factor This is a factor that must be applied to determine if we have an odd or even number.
Choices Ceil (default), Floor, Trunc, Round
Comments Only integer numbers may be odd or even and with fractals, numbers have always a decimal part. So, we must remove this one and there are four ways to do it: ceiling, flooring, truncating and rounding. 
  • By ceiling, you take the higher next integer (ex: 3.1416 will give 4, -3.1416 will give -2);
  • by flooring, you take the smaller next integer (ex: 3.1416 will give 3, -3.1416 will give -4);
  • by truncating, you just truncate the decimal part (ex: 3.1416 will give 3, -3.1416 will give -3);
  • by rounding, you just round up (ex: 3.1416 will give 3, 3.5416 will give 4, -3.1416 gives -3, -3.5416 gives -2).

 
Apply odd-even sign? By this option, we may ask to consider or not the sign of the number to be turned as an integer
Choices Enabled, Not enabled (default)
Comments Results of removing the decimal part are different if we take care of the sign or not. See above the examples for the four different ways to get an integer number.

 
Odd-even Function This is the function that is applied if the number is even.
Choices All the standard functions you may use in UF parameters. Default is flip, which corresponds to switching the real and imaginary part. If you choose ident, this means no function applied.


3. Barnsley Parameters

The principle behind the Barnsley parameters is the following: Actually, if the result is zero or positive (you can change 0 to another offset value), a function is applied to Z that you may choose among the parameters. If it is negative, nothing special is done.

There are three related parameters:

Barnsley Type This is where we choose which part of Z will be verified if positive, zero or negative.
Choices No (default), Real, Imag, Real*Imag, Real+Imag, Diff, Angle, Magnitude, Path, Iteration, trap real, trap imag, trap r*i, trap r+i, trap distance, trap angle, trap magnitude, trap path, trap iter
Comments
  • No - the Barsnley parameters won't be applied.
  • Real - the real part of Z will be used.
  • Imag - the imaginary part of Z will be used.
  • Real*Imag - the real part of Z times the imaginary part will be used.
  • Real+Imag - the real part of Z plus the imaginary part will be used.
  • Diff - the distance between Z and the previous Z
  • Angle - the angle of Z to the origin
  • Magnitude - the magnitude of Z
  • Path - the approximate length of the orbit at Z
  • Iteration - the number of iterations at Z
The other choices are not available in some formulas/colouring methods.
When a choice is related to the trap, it means the current trap.
The Barnsley Function parameters will never work if this parameter is set to No.

 
Barnsley Offset This parameter lets you change the 0 of the result of calculation for another value.
Choices You may use any number. Default is 0 for backward compatibility with previous version.
Comments Instead of having the result of calculation being >= 0 (zero or positive) or < 0 (negative), you can change it to >= Offset or < Offset
For Barnsley types parameters Diff, Angle, Magnitude, Path and Iteration, you can use only positive values for the Offset.

 
Barnsley Function This is the function that is applied if the number is zero or positive.
Choices All the standard functions you may use in UF parameters. Default is flip, which corresponds to switching the real and imaginary part. If you choose ident, this means no function applied.


4. Coordinates parameter

The most known system of coordinates is the cartesian one. This is the one we learn at school. You have a grid of numbered straight lines where the position of each dot can be identified by a set of two numbers: (x,y) where x is the number of the horizontal line and y the one of the vertical one that meet at the dot.

Another popular system is the polar one. In the polar one, instead of a square grid, you have a set of concentric wheels with their spokes. Each dot may be identified by a set of two numbers called (rho, theta) where rho is the number of the wheel and theta the number of the spoke that meet at the dot. It is more used by mathematicians and engineers and implies the use of trigonometry to switch between both systems.

Finally, system of coordinates can also be artificially invented as tricks of arithmetic.

There is only one parameter.
 
Coordinates Choice of the system of coordinates
Choices When French: Cartésien, Polaire, Hyperbolique
When English: Cartesian, Polar, Hyperbolic
Comments The default is usually Cartesian. However when Polar curves (or Courbes polaires) are implied, Polar becomes the default. Hyperbolic is artificially invented and can be missing in some colouring methods.


5. Function Application (new)

Some functions may be applied at some step of the calculations. In some older colouring methods, it is done in a different way.

There are two parameters:

Apply F(x) This is the function that will be applied to the real part of Z
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.

 
Apply F(y) This is the function that will be applied to the imaginary part of Z
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.


6. Colouring Mode Parameters (old)

With this toolbox, one can use some formula properties to change the way gradients are handled.

There are two related parameters:

Colouring Mode This is the formula property that is used to make the handling
Choices When French: minimum distance (default), iteration@min, angle@min, maximum distance, iteration@max, angle@max, average distance, opérations
When English: minimum distance (default), iteration@min, angle@min, maximum distance, iteration@max, angle@max, average distance, operations
Comments All these properties are based on properties related to the distance, mostly the minimum (@min) and the maximum (@max) ones. Operations has a different philosophy and is handled by its own parameter.

 
Operations This is a choice of arithmetic operations on the real/imaginary part of the last Z that can be used as colouring mode.
Choices When French: réel (default), imaginaire, real+imag, real*imag, real-imag, imag-real, real/imag, imag/real, real^imag, imag^real
When English: real (default), imaginaire, real+imag, real*imag, real-imag, imag-real, real/imag, imag/real, real^imag, imag^real
Comments This parameter works only if the operations option has been chosen as colouring mode. For those unfamiliar with the operations symbols:
  • * means the multiplication
  • / means the division
  • ^ means the power


7. Function Application (old)

Some functions may be applied at some step of the calculations.

There are three related parameters:

Function Application This is the parameter used to apply up to two functions, one on the real and/or one on the imaginary part of Z.
Choices When French: aucune (default), f(x)->réel, f(x)->imaginaire, f(y)->réel, f(y)->imaginaire, f(x)->ré et f(y)->im, f(x)->im et f(y)->ré
When English: none (default), f(x)->real, f(x)->imag, f(y)->real, f(y)->imag, f(x)->re et f(y)->im, f(x)->im et f(y)->re
Comments The options are read this way: f(x)->real means that the function chosen at the F(x) function parameter will be applied to the real part of Z. When the option has a f(y) instead, this the the F(y) function parameter that will be used. The same can be said if this is an imag instead of a real, where the function will be applied to the imaginary part of Z.

 
F(x) This is the first choice of function that may be applied
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.
Comments The function will be applied to the real or imaginary part of Z in relation with the option chosen in the Function Application parameter (f(x)->something). Keeping this function to ident has the same effect as Function Application to none.

 
F(y) This is the second choice of  function that may be applied
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.
Comments The function will be applied to the real or imaginary part of Z in relation with the option chosen in the Function Application parameter (f(y)->something). Keeping this function to ident has the same effect as Function Application to none.


8. Masking Parameters

These parameters allow you to make some masking. This means that some parts of the layer can become transparent  based on a set of conditions determined by the parameters. I say they can because some manual handling is further needed. People who wants to know more about this handling have to go to http://www.parkenet.org/jp/ufopacitytutorial.html where informations are given (the little black Mandelbrot button).

There are five related parameters:

Mask Type This parameter let you choose a mask shape based on some formula properties.
Choices Real pixel (default), imag pixel, real Z, imag Z, distance, angle, iteration, magnitude, path, trap distance, trap angle, real trap, imag trap, trap magnitude, trap iter, trap path, shape, Barnsley, odd-even.
Comments Some choices are not available in some colouring methods.
Those related to the trap refer to the current trap
Barnsley and odd-even refers to the current values of the Barnsley toolbox or the Odd-even toolbox.

 
Masking This is a range for masking
Choices None (default), lower, upper, between, outside
Comments By default, there is no masking. If you choose lower, all the values for the Mask Type under the Lower Mask Threshold value are masked. If you choose upper, these are the same above the Upper Mask Threshold value that are masked. You can combine lower and upper at the same time by choosing outside. Finally, Between is the opposite of outside.

 
Lower Mask Threshold This is the value of the lower limit in some options of the Masking parameter
Choices Any value is OK. The default value is 0.5.
Comments It has no effect if you choose upper as Masking parameter.

 
Upper Mask Threshold This is the value of the upper limit in some options of the Masking parameter
Choices Any value is OK. The default value is 1.0.
Comments It has no effect if you choose lower as Masking parameter.

 
Masking Function This is an additional way to modify the mask shape.
Choices All the standard functions you may use in UF parameters. Default is ident, which corresponds to no function applied.


9. fBm Texture Parameters

With this toolbox, you can get a fBm texture. fBm (stands for fractional Brownian motion, from Robert Brown, an English botanist who identified it in 1828 with the movement of pollen particles) is a noise based on some randomizing.

As there are actually many ways of calculating fBm, I will just try to explain it coarsely.

One starts from a straight line and then calculates a third point in its middle. This is the way this third point is calculated that is interesting as some randomizing is used. Thus, it rarely falls on the straight line. It is most often shifted above or below it and it can also be slightly shifted left or right. By linking this third point to the two initial ones, one gets two lines. The process can be repeated on these two new lines, with a result of four lines then repeated again with these four lines, etc. Each repetition is an iteration (or octave).

There are nine related parameters:

fBm Texture Amount This is the strenghth of the texture
Choices Any number is OK. 
Comments By default, value is 0, which means no texturing.

 
fBm Initialization This is a parameter to set an initial value based on some formula properties.
Choices Pixel (default), z, distance, angle, magnitude, path, trap z, trap distance, trap angle, trap magnitude, trap path, Z, handover, #Z
Comments With this parameter, the texture adapts to the overall shapes and colouring of the image.Pixel gives a smoky effect.
z gives an effect that is the closest to a random texture.
Distance gives an effect of contour line.
Angle gives an effect of rays perpendicular to the shapes.
Magnitude and path have some line effects.
Z (and/or #Z) is close to z but with some difference. When both options are available, use #Z instead, as it corrects a first unnoticed limitation of UF, which doesn't make any difference between upper and lowercase letters in some parts of the parameters.
Handover uses the value of all the previous used textures in the colouring method (ex: Random) and thus adds to them.
Some choices are not available in some formulas/colouring methods.
For the choices related to the trap, it means the current trap.

 
fBm Scale With this parameter you can set the texture to a bigger or smaller overall density. 
Choices Any number is OK. Default is 1.
Comments This parameter is applied to the initialization parameter before starting calculations. So, it affects the overall scale.
A smaller number gives bigger shapes and a looser density. A bigger number gives smaller shapes and a tighter density.

 
fBm Offset This is a parameter that let you get some overall shift on the texture
Choices Any complex number. A value can be chosen with the eyedropper. Default is 0.
Comments This parameter is applied to the initialization parameter before starting calculations. So, it affects the overall position.

 
fBm Rotation With this parameter you can give some angle to the overall texture.
Choices Any number is OK. Default is 0.
Comments This parameter is applied to the initialization parameter before starting calculations. So, it affects the overall rotation.

 
fBm Scale Step The initialization parameter is shifted with this value and the rotation step at each iteration. 
Choices Any number is OK. Default is 0.5.
Comments The overall texture is neater with bigger numbers and more blurry with smaller numbers. 
Randomizing is based on this parameter and the rotation step.

 
fBm Rotation Step The initialization parameter is shifted with this value and the scale step at each iteration. 
Choices Any number is OK. Default is 37.
Comments Randomizing is based on this parameter and the scale step.

 
fBm Octaves This is  the number of iteration 
Choices Any positive integer number is OK. Default is 7.
Comments After a certain number of iterations, the texture doesn't visibly change anymore or turns monochrome.

 
fBm Exponent This is a parameter used to scramble numbers
Choices Any number is OK. Default is 2.0.
Comments When using smaller number, patterns are more and more obvious. A bigger number blurs them


10. Colouring Mode Parameters (new)

With this toolbox, one can use some formula properties to change the way gradients are handled.

There are two related parameters:

Colouring Mode This is the formula property that is used to make the handling
Choices distance (default), iteration, angle, magnitude, path, arithmetic mean, geometric mean, real, imag, real+imag, real*imag, real-imag, imag-real, real/imag, imag/real, real^imag, imag^real, 1/real, 1/imag
Comments As a recall, 
  • * means the multiplication
  • / means the division
  • ^ means the power

 
Colouring Fix This is where the formula property is handled.
Choices @minimum (default), @maximum, average, @last Z, @trap
Comments
  • @minimum means that the Colouring Mode parameter is handled where the distance is the smallest.
  • @maximum means that the Colouring Mode parameter is handled where the distance is the largest.
  • average means the average of both the previous.
  • @last Z means at the last iterated value of Z.
  • @trap means at the current trap.


11. Trap Parameters

These parameters allow you to make some trapping. A trap is a place meeting some conditions and where a colouring method wants to stay in some way. It wants to keep and use some values related to that place until the end. The conditions and what to do once falled into are decided by it (the method not the place).

In my colouring methods, a trap is the first value of Z meeting some conditions that you can set from parameters. Once there, it keeps some values related to this Z and this is what is called trap something or @trap in some other pararameters.

To set a trap condition, you must follow a certain pattern:

property - lower/equal/over than - some value
For example: So, the first value of Z meeting these chosen conditions is the trap. It notes some other values related to this Z and these values can be used until the last iterated Z. The current parameter settings of this toolbox are anyway the current trap. There is always a current trap as the other parameters using its values must work.

Properties, bounds and values are set by three related parameters:

Trapping This parameter is used to set the bound type.
Choices lower (default), upper
Comments If you choose lower, the first Z where the value for the Trap Type is lower or equal to the Trap Threshold is the current trap. If you choose upper, this is the first Z where the Trap Type is over the Trap Threshold
You can get the same results with different conditions if the first met value of Z is the same in both cases. This happens more frequently when you choose lower and less when you choose upper.

 
Trap Type This parameter lets you choose the property.
Choices Iteration (default), magnitude, distance, angle, real, imag, path
Comments Real stands for the real part of Z and imag stands for its imaginary part

 
Trap Threshold This is the numerical value of the condition
Choices Any value is OK. The default value is 1.
Comments By default, the current trap is the first Z where the iteration is lower or equal to 1. So the very first Z met.


12. Random Texture Parameters

There is only one parameter, related to the strength.
 
Random Texture Amount This is the strenghth of the texture
Choices Any number is OK. 
Comments By default, value is 0, which means no texturing.


13. Decimal Texture Parameters

With this toolbox, you can get a decimal texture. This texture is my own idea. The principle behind it  very simple: taking the decimal part of some value related to the last iterated Z and using it as a texture.

Decimal is a strong texture. To soften its effect or the colour shift, use the Decimal Limit parameter, the Apply Decimal Sign parameter or both.

There are seven related parameters:

Decimal Texture Amount This is the strenghth of the texture
Choices Any number is OK. 
Comments By default, value is 0, which means no texturing.

 
Decimal Initialization This is the value related to Z that will be used for the decimal part.
Choices Real Z (default), imag Z, distance, angle, magnitude, path, real trap, imag trap, trap distance, trap angle, trap magnitude, trap path, handover
Comments Real trap means the real part of the current trap will be used and imag trap, its imaginary part.
Handover uses the value of all the previous used textures in the colouring method (ex: Random, fBm) and thus add to them.
Some choices are not available in some formulas/colouring methods.
For the choices related to the trap, it means the current trap.

 
Apply Decimal Sign? By this option, you may ask to consider or not the sign in the extraction of the decimal part
Choices Enabled (default), not enabled.
Comments This parameter is used with the Decimal Type parameter as the result of the subtraction can be a negative number and in such case the texture effect can be strong enough to get an important colour shift. When you enable it, all the resulting numbers are changed to positive and you avoid this shift. 

 
Decimal Type This parameter relates to the way you can get the decimal part or some variations of it.
Choices Ceil (default), Floor, Trunc, Round
Comments The decimal part cannot be directly extracted. The only way to get the decimal part is to subtract the number with its integer part and there are four ways to get this one: ceiling, flooring, truncating and rounding. 
  • By ceiling, you take the higher next integer (ex: 3.1416 will give 4, -3.1416 will give -2);
  • by flooring, you take the smaller next integer (ex: 3.1416 will give 3, -3.1416 will give -4);
  • by truncating, you just truncate the decimal part (ex: 3.1416 will give 3, -3.1416 will give -3);
  • by rounding, you just round up (ex: 3.1416 will give 3, 3.5416 will give 4, -3.1416 gives -3, -3.5416 gives -2).
So, you get the real decimal part by using Trunc. With the other ones, you can get some variations.

 
Decimal Scale With this parameter you can get larger or smaller patterns.
Choices Any number is OK. Default is 1.
Comments With larger numbers you get smaller and softer patterns and with smaller numbers you get larger and coarser patterns. There is also a possible colour shift. You can use the Decimal Limit and Apply Decimal Sign? to reduce colour shifting.
The default value corresponds to the real dimensions/colours of the pattern.

 
Decimal Size This parameter has the opposite effect of the previous
Choices Any number is OK. Default is 1.
Comments With smaller numbers you get smaller and coarser patterns and with larger numbers you get larger and softer patterns. There is also a possible colour shift. You can use the Decimal Limit and Apply Decimal Sign? to reduce colour shifting.
The default value corresponds to the real dimensions/colours of the pattern.

 
Decimal Limit This parameter sets a decimal value you cannot trespass
Choices Any number is OK. Default is 1.
Comments This parameter is very useful to make colour shift correction or soften colours if needed.


14. Popgnarl Texture Parameters

This texture is a combination of Andreas Lober's Popcorn and Gnarl textures. Basically, they are the same, outside they use a different formula. These formulas are also iterated like the fBm texture can be.

Popgnarl is a strong texture. To soften its effect or the colour shift, use the Popgnarl Limit parameter.

There are nine related parameters:

Popgnarl Texture Amount This is the strenghth of the texture
Choices Any number is OK. 
Comments By default, value is 0, which means no texturing.

 
Popgnarl Initialization This is the value related to Z that will be used in the formula.
Choices Pixel, z (default), distance, angle, magnitude, path, trap Z, trap distance, trap angle, trap magnitude, trap path, Z, handover, #Z
Comments For the choices related to the trap, it means the current trap.
Handover uses the value of all the previous used textures in the colouring method (ex: Random, fBm, Decimal) and thus add to them.
Some choices are not available in some formulas/colouring methods.
Z (and/or #Z) is close to z but with some difference. When both options are available, use #Z instead, as it corrects a first unnoticed limitation of UF, which doesn't make any difference between upper and lowercase letters in some parts of the parameters.
The difference between this parameter and the next one is that this one is the starting value for the formula and the next one, what will be used of its result to plot the texture.

 
Popgnarl Mode This parameter means the result of the formula that will be used.
Choices Real (default), imag, real+imag, real*imag, real-imag, real/imag, imag/real, real^imag, imag^real, 1/real, 1/imag, angle, magnitude, arithmetic mean, geometric mean
Comments The difference between this parameter and the previous one is that this last one is the starting value for the formula and this current one, what will be used of its result to plot the texture.

 
Popgnarl Formula This is where you can choose which texture you want from Andreas's orginal ones.
Choices Gnarl (default), popcorn, popsicle, tannous, narlog
Comments Gnarl and popcorn are from Andreas's and correspond to these textures formulas. The three other ones have been inspired by them.

 
Popgnarl Scale With this parameter you can get larger or smaller patterns.
Choices Any number is OK. Default is 1.
Comments With larger numbers you get smaller and softer patterns and with smaller numbers you get larger and coarser patterns. There is also a possible colour shift. You can use the Popgnarl Limit to reduce colour shifting.
The default value corresponds to the real dimensions/colours of the pattern.

 
Popgnarl Scaling This parameter gives some further variations on the texture patterns.
Choices Ceil (default), Floor, Trunc, Round

 
Popgnarl Size This parameter has the opposite effect of the previous
Choices Any number is OK. Default is 1.
Comments With smaller numbers you get smaller and coarser patterns and with larger numbers you get larger and softer patterns. There is also a possible colour shift. You can use the Popgnarl Limit to reduce colour shifting.
The default value corresponds to the real dimensions/colours of the pattern.

 
Popgnarl Limit This parameter sets a decimal value you cannot trespass
Choices Any number is OK. Default is 1.
Comments This parameter is very useful to make colour shift correction or soften colours if needed.

 
Pognarl Octaves This is the number of iteration 
Choices Any positive integer number is OK. Default is 5.
Comments After a certain number of iterations, the texture doesn't visibly change anymore.


15. Epsilon Range Parameter

This parameter gives you some control on the formula bailout. By using it, you force the formula to escape when its value falls inside a certain range instead of reaching the regular bailout conditions. Its overall effect is the appearance of ribbons that cross each other. It is inspired from Wayne Kiely's Epsilon_Rot.
 
 
Epsilon Range This is the parameter for setting the range values
Choices This is a complex number where you use the real part for the lower value of the range and the imaginary part for the upper value of the range. Default values depend on the formula.
Comments Use very small numbers, like between -1 and 1 or even smaller. The real value must be smaller than the imag value.