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.
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:
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. |
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 |
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.
|
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. |
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 |
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. |
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. |
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. |
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:
|
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. |
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. |
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 |
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,
|
Colouring Fix | This is where the formula property is handled. |
Choices | @minimum (default), @maximum, average, @last Z, @trap |
Comments |
|
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 valueFor example:
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. |
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. |
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.
|
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. |
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. |
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. |