Some suggestions for "Trap the Light Fantastic" 5 Sept 2007

I am a great lover orbit of trap colorings, especially Dennis Magar's "Thingamajigs" and "Doodads", which is why I created the extended "II" versions. Those two colorings especially are full of fantastic dimensional-appearing trap shapes, and are easy to use and friendly to boot. It has always been my hope to someday write an orbit trap coloring that creates such dimensional trap shapes, but goes even farther in terms of allowing the user space to explore and to experiment.

I believe that to some extent I have succeeded with "Trap the Light Fantastic". This coloring is not without its flaws, but in my weeks of explorations with it I have (almost) always found unique and interesting shapes that retain that dimensionality and lighting that I so appreciate in orbit trap colorings.

However at first glance I'm sure that it seems tremendously confusing with all its user-selectable parameters and functions. Nor is it as friendly as the colorings mentioned above in terms of consistently offering good results "right out of the box". The default set has limitations, and many of the trap shapes appear at first glance to be similar or even identical, and many others do not seem to offer much at all that is pleasing. This is, unfortunately, the trade-off for range and flexibility.

I grappled with the idea of reducing options and trap shapes to give the coloring a neater appearance, but in the end I decided that flexibility is more important than presentation. But in that case I thought it necessary to write up this little dissertation to give users some pointers and ideas of what is offered here and how to use it to best advantage (at least insofar as I am able to do so).

First a very quick and somewhat inaccurate description of orbit trap colorings,  in my limited understanding. The point of any coloring is to assign every pixel an index value--the point on the gradient from which it will take its value. With orbit trap colorings, (as far as I know), we have a complex variable '#z', which is split into real and imaginary components (which translate into horizontal and vertical values). Once it has been divided into these two 'x' and 'y' components, they are processed in various ways and then recombined, and it is this processing that essentially creates the shapes of the traps.

This coloring, henceforth referred to as TTLF, has a few more steps than that. #Z is shaken and stirred before being split, so to speak: being modified by various functions and parameters, then it is split and the trap shapes formed, where more functions and params are added. 

But afterwards, instead of leaving well enough alone and letting the combinations of x's and y's which form the trap shape be the final determinants of that shape, they are used only as another factor, being combined in various ways with even more permutations of our friend #z, which goes through several more cycles of various machinations before being allowed to be the arbiter of what appears on your screen.

At each step, I have introduced more function and parameter choices, and these can be found on the user interface with prefixes like "Z style, Z flavor, Distance, Point, Trap, Trap Shape,  X, Y and Z, Shape, etc. Many may seem to have similar effects as you use them, but because they all operate at different points in the process, each to some extent has a unique effect.

TRAP SHAPES
This brings me to the first important point: the effect of the "Trap Modifiers". Here you will see familiar-sounding names of trap shapes used in a number of colorings such as Plane Curve Traps, Thin Orbit Traps, Painter's Traps and others. But their effect is quite different here than in those colorings, in which they create well-defined trap shapes. In TTLF they are just another modifier, and while they introduce very interesting possibilites, they do not create those familiar shapes. In fact many look totally redundant; others seem to create  only lines and blobs and noise. But this is not the whole story.

While this parameter is an important one, it is only one, which is why I have called it "Trap Modifiers" and not "Trap Shapes". Basically in this area of the loop #z is split into a number of real and imaginary components (x's and y's). Many of these have only a small effect in some scenarios, but suddenly become much more prominent in others, such as when new function and parameter value choices are made. It is definitely worthwhile to occasionally scroll through trap choices as you work, as things that appeared similar to others or useless in themselves at the beginning might have taken on a different character down the road. Often I like to choose a trap shape at random and begin working from there: each has potential. But after finding something interesting I take a stroll through the trap modifiers and often find interesting variations on what I have created. In any case the choices are discrete, so it is not hard to get back to where you were.

This brings me to an important point. It is easy to go far afield and not be able to find your way back to your starting point (at least this always happens to me). So I *strongly* suggest that if you have something you like, make a copy of the layer before you go cruising through new functions and parameters. That way you can always get back.

WORKING WITH PARAMETERS
I think that one of the  keys to success with this coloring is the exploration of parameters, and for this the "explore tool" of UF4 is of great value, in fact it is critical for optimizing your results. What I have found is that as I change things using the explore tool, certain values are reached at which the trap shapes "peak". Before and after this value only lines or amorphous forms can be found, but as one approaches the critical point things arrange themselves into very definite dimensional-appearing trap shapes. I cannot tell you which parameters to explore, as all the parameters and functions are interactive. Some of the beta testers were asking me to indicate which were the "important" params. I have found that all are important at one time or another, and conversely are unimportant at other times. This is one of the "love it/hate it" aspects of TTLF. You can find nice stuff without ever touching the parameters, but you are missing huge opportunities if you work like that. Every time you change a trap shape or a function you are presented with manifold opportunities to optimize what is in front of you by exploring the various parameters.

OVERVIEW:
Basically speaking, this is an orbit trap coloring with two independent trap sections. Either a single trap may be used, in which case the entire second section is inactive, or the traps can be blended, in which case both sections contribute to the final result. In addition there is a "Modulated" setting, in which the two traps alternate by iteration. To make matters worse, or better as the case may be, each trap section allows the user to choose either a single version of the trap, or to double it and combine the two versions in various ways. These options, along with the wealth of parameters and functions, makes this coloring very flexible.

I will try to go through things more or less from the beginning and explain what they do. Let's start at the top of the user tab.

HEADING: BASIC TOOLS
Here are the most elemental aspects of the coloring, which should mostly be familiar.

TRAP MODE: This determines the relationship of the elements. "Closest" is smooth and dimensional. "First" puts sharp edges on the shapes--like cutouts--and puts the smaller iterations on top of the larger ones. "Last" is the same but puts the largest iterations over the smaller one. The cutoff of the edges can be varied using the parameter "Edge Threshold" when either of these modes are chosen. "Average" does what it says, and averages the traps, so that things happen where they meet. I've added a bunch of styles of averaging, controlled by fixed, hidden functions, which can be chosen using the param "Avg. Flavor".  Each of these has two different modes, which can be selected using the bool param "Change Avg. Flavor". "Farthest" and "Sum" are the remaining choices, which are also controlled by the "Edge Threshold" value.

COLORING MODE: Different ways of coloring the elements. You will notice that parameters appear with each different choice: these affect the way that the gradient is mapped onto the elements.

SHADING: A very important parameter! This will change the opacity of the traps. If you find that you are getting banding in the area outside the figure, you should be able to turn this back into a solid color by choosing a value under 1. You will need to adjust the gradient when changing this parameter.

HEADING: TRAP MIXING
Here are the controls for choosing the trap shape which isn't really a trap shape (as explained above), plus those for choosing one trap section or the other or both. 

BLEND TRAPS: If this bool param is checked a number of new params will appear---
>>>>>>>>>>
 --BLEND MODE: Chooses the operator to blend the traps (multiplication or addition). You may have to adjust your gradient depending on the choice. If "+" is chosen another two params appear:
    >-- % TRAPs: Control the proportion of one trap to the other in the addition mode. You are not limited to 100%. You will have to adjust the gradient depending on your choices.
--BLEND FNs: Change how the traps blend together--different functions give quite different effects.
>>>>>>>>>>

***NOTE*** From here many parameters and headings are doubled, since there are two complete trap sections. The headings and parameters that deal with the first trap section always have the suffix "I", and those for the second have the suffix "II". They appear only when they are active.

If  the bool param "BLEND TRAPS" is left unchecked the parameters above are not active; instead one parameter appears:
--FORMULA SCENARIO: Three choices here: "Trap I, Modulated, Trap II". The first and third choices allow you to work only with the first or second trap sections. If you choose "Modulated" a certain number of iterations of one trap are followed by a certain number of iterations of the other trap. With this option two more parameters appear:

>--MODULATION REPEAT: This is not my code so I cannot vouch for it, but the hint reads: "At what iteration the modulation starts. If '0' all iters are of the modulation formula".

>--# OF MODULATED ITERS: Hint says: "How many iterations of 2nd formula are present".

It is important to remember that iterations proceed around the various arms of spirals, so this is not so predictable as it seems it should be, but it is still a very cool effect and you should definitely play with it!

TRAP MODIFIER I and/or II: These show the trap shape(s) in play. As mentioned, these do not create the trap shapes indicated; rather that trap is used as a modifier that affects the final shape, which usually bears no resemblance to the shapes created by these equations in other orbit trap colorings.

HEADING: GLOBAL PARAMS AND FUNCTIONS
There is only one set of these, which affect both trap sections.

ITERATIONS TO SKIP: Does what it says.

MAX ITERATIONS: The total number of iterations allowed. The default is set high so as not to limit this in normal use.

ROOT FN: This is the only global shaping function (apart from the Morph section coming up) so it goes here. It affects #z at the very top of the stream.

FINAL INDEX FN: Determines how the gradient will be mapped in the final image.

HEADING: MORPH PARAMETERS

These Morph parameters also affect #z at the top of the stream. Anyone who has used my colorings should be basically familiar with them so I will not describe them in detail, but I will say that using the Morph II params will greatly widen the range of trap shapes available and I heartily recommend playing trying it, especially if you have shapes with horns that go out to infinity getting smaller and smaller. One note that I think is useful: As you scroll through the "MORPH" choices, more parameters and functions come into play. With Morph II, you will see that Morph Functions Z1 and Z2 are set to "zero" by default. Set these to any other function to make the Z1 and Z2 parameters active.

HEADING: SHAPING STUFF (I/II)
There are two headings and two full sets of params, which appear depending on your choices above with traps. I will only describe them generically:

DOUBLE TRAP: If this box is unchecked a single instance of the trap modifier chosen in this section is in play. The next parameter is:

TRAP STYLE: This is the most basic of all the shaping parameters, and describes what "modifier" #z is put through before being split and entering the trap shape section.

>>>>>>>>>>
If you check the DOUBLE TRAP box, the trap is doubled, and new parameters appear:

--TRAP A & B STYLE(s): Since we have two separate instances of the trap running, we can choose their modifying parameters and functions independently before combining them. These two params allow us to choose the modifier for the traps separately as described above.

--OPERATOR: This allows us to choose which arithmetic operator mediates the combination of the two instances of the trap.

--COMBINATION FN: A separate function is applied to Trap B, apart from those which appear later in the Function heading. I've put it here for convenience.

SHAPE and SHAPE VARIATION: Some fixed functions are applied that change the appearance of the traps

SHADING/SHAPE: This parameter normally changes the shading on the elements, however in some cases the shape is changed as well.

CHANGE SHADING/SHAPE: This bool param changes the way the preceding param is applied, changing its effect.

HEADING: GENERAL PARAMETERS (I/II)
Here are some general parameters, again under two discrete headings for the two trap sections:

COMPLEXIFY DISTANCE: This bool param, when checked, adds another layer of processing to the traps, and new params appear just under the bool box. *In addition* a new heading appears below with DISTANCE PARAMETERS, to be discussed later. 

Basically all the parameters and functions in this section affect the appearance and shape of the traps, and I will not describe them separately, except to say that it is worthwhile to scroll through and explore them to see what effects they have

HEADING: DISTANCE PARAMETERS
This heading only appears when "COMPLEXIFY DISTANCE" is checked, as described above. When DISTANCE MODE in "General Parameters" above is checked, a second set of distance parameters appears.

***A GENERAL NOTE ABOUT PARAMETERS***
Essentially at every point where a variable is in play, there are three parameters to modify it. "Offset" adds or subtracts a value, "Strength" is a multiplier, and "Power" is exponential.
Many will seem similar in some instances, but all affect different points in the process and do have different effects. Also, the effects depend to an extent on other parameter, function and trap choices.

HEADING(s) Z and SHAPE, TRAP, POINT, X&Y and TRAP TYPE PARAMETERS
I have divided the great number of params into collapsing sections for convenience, although there is no real reason to separate them. These sections are very similar in structure except for the SHAPE parameters, which I will describe here:

The SHAPE parameter section looks like the other sections with one exception: When the bool box  DOUBLE TRAPS (in the "Shaping Stuff" section) is checked, a bool box appears at the bottom of the shape params captioned "SPLIT SHAPE PARAMS". When this is checked a second set of "SHAPE B" parameters opens below it. Here is the reason: By default, both traps (when the traps are doubled) share a common set of parameters. This generally gives the most pleasing effects, but there are times when it is valuable to be able to set the params for the doubled traps independently. If you wish to be able to set the params separately for the two traps, check this box, and the "Shape" params control only the A trap, while the "Shape B" params control only the B trap. 

Now here is an *important note*: if you have set the Shape params in the default mode (both traps using the same set of params), and now wish to experiment with independent params, you will need to copy and paste the complex values (those you have changed from the defaults)  from the "Shape" params to the "Shape B" params--if you wish to keep that appearance that you have. This can be accomplished by right clicking on the param fields and choosing "Copy Complex Value" and then "Paste Complex Value" from the context menu. 

HEADING: A FLOOD OF FUNCTIONS (I/II)
Here are all the functions grouped together, like the parameters above. I heartily suggest you experiment with these, as many wonderful effects can be achieved. Often the effects of a function choice can be optimized by going back and playing with the parameters.

********************************************

This is as far as I will go in explaining this coloring, as the texture choices are rather straightforward, or at least not much different from those in my other colorings. In closing I wish you many happy hours (days, weeks) going through the voluminous choices that this coloring offers, and I hope you will get as much delight from finding unexpected, unusual and interesting shapes and shadings as I. Happy Fractalling!


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Naru's Gnarly Potpourri Help and Tips

OVERVIEW

This coloring is based on Mark Townsend's "Gnarly Orbit Traps". Gilles Nadeau
wrote an original adaptation including fBm texturing and adding a feature found
in Mark's "Advanced Gnarl" ufm, which allows the user to choose between two
different "scenarios"--sets of formulae--and combine them in various ways.
Susan D. Chambless nicely elaborated Gilles' work, and then I was invited to
"do my thing". This is the result.

I have added my usual plethora of bells and whistles, including the same set of
textures found in my other ucls, Morph parameters, etc. In addition I have added
the plane curve trap shapes found in "Painter's Traps". It is a monster. I
counted 1180 user-selectable parameters and functions. Luckily you will only see
the ones usable at any given time.

First the bad news: this ucl is agonizingly slow. Some of my beta testers gave
up on it for that reason. I have done my best to optimize the code and eliminate
unnecessary flops, but at the end of the day it is still very
calculation-intensive. I tried making a stripped down version but the rendering
time remains the same, so you're going to have to live with it. This is an
explorer's coloring. It is designed for those who don't mind spending some time
mining parameters and functions for the gems that can be uncovered. I have found
it not-unfriendly, but the range of possibilities can be daunting. However the
rewards, in terms of dynamic and interesting new element shapes and textures,
can be great.

WHAT IT DOES

Those who know "Painter's Traps" will generally find this familiar, and in fact
they are closely related. In both colorings there are two sets of interacting
"shape determinants"--the formulae such as "Gnarl" and "Glyph", and a set of
Plane Curve Traps, which can be used independent of each other or combined.

The difference lies in the fact that in "Painter's Traps" the gnarl-type
formulae are applied as pixel patterns: they are "spread" across the image
without regard to the iterating loop, and therefore appear more as a wash or
background than as a part of the fractal image, although they do somewhat affect
and distort the fractal elements. In Naru, by contrast, the gnarl-type formulae
actually form the fractal elements themselves: they become the trap shapes that
iterate and repeat in spirals, for instance. If one then inserts the Plane Curve
Trap shapes there is an interaction in which the resulting shapes are hybrids of
both kinds of traps.

So this is the first main point. Naru can be used purely in "Formula" mode or
purely in "PCT" mode or with a varying combination of the two interacting. When
used in PCT mode only the resulting shapes are identical in shape and size to
those obtained with "Painter's Traps" (and, I believe, "Plane Curve Traps" in
kcc3.ucl and my "Plane Curve Traps II"), albeit with some slightly different
shading options, so these colorings can work very well together.

CONTROLS

Following are descriptions of the controls you will find on the tab from the top
down (more or less).

USER MODE
Here you have two choices, SIMPLE and ADVANCED. Since this coloring has so many
parameters--way too many for the tab--I chose to hide some of the less-used ones
in the simple setting. Basically what get hidden are a bunch of numerical
parameters for the individual pattern formulae and a few of the more esoteric
PCT parameters.

HEADING: GLOBAL PARAMETERS

TRAPPING MODE
This parameter appears only when "Plane Curve Traps" is chosen. "Closest" is the
look you are probably used to in orbit trapping. "First" creates a discrete edge
around the elements, with the earlier, smaller elements overlying later, larger
ones. "Last" is the opposite, overlaying the larger, later elements over the
smaller, earlier ones. "First Corrected" and "Last Corrected" are like their
namesakes, except that there is some variance in the element shading. You will
have to adjust the gradient if you use them.

ELEMENT WIDTH
The "masking" of the element edges--the effective element width is controlled by
this parameter, which appears if either First or Last is chosen.

COLORING MODE
Here are all the usual suspects, plus some choices that reference the formula
patterns specifically.

DISTANCE: This is the standard mode which colors the pixels according to the
distance they fall from the trap center. It gives the characteristic 3D or
shaded look of orbit trapping. In Soft Gnarly, since there are no specific trap
shapes, it tends to produce elements that appear spherical, although that can be
changed to some extent depending on other functions/parameters. With pixel
patterns present the patterns will change both the shading and to some extent
the shape of the elements.

ITERATION: This colors the pixels by what iteration they fall into. The elements
look flat but are of different colors  or tints. If pixel patterns are present
they will sometimes be incorporated, changing the apparent shape of the elements
according to how they lie.

MODULATED ITERATION: This is similar to Iteration, but it repeats the color
every given number of iterations. That number can be set with the parameter that
appears directly below when Mod Iter is chosen:

--COLOR RANGES:
(Appears when "Modulated Iter" is chosen).  This sets the number of color ranges
that repeat in the elements. You can set a decimal for some interesting effects:
setting a value of 4.1, for example, gives four repeating colors which shift
down the iterations. This can be quite interesting with careful setting of the
gradient.

MAGNITUDE, REAL, IMAGINARY ANGLE and ANGLE TO TRAP: These are other standard
ways of determining how the pixels are  colored in a trap coloring which I won't
go into here.

PATTERN MAGNITUDE, PATTERN REAL, PATTERN IMAG and PATTERN ANGLE are various ways
of coloring the pixel pattern only.

ITERATIONS TO SKIP
This parameter determines how many iterations are skipped. It affects the
largest, outermost elements first.

ITERATIONS TO SHOW
This parameter determines how many iterations are shown in total. Any number
over the maximum number of iterations in your particular fractal returns the
maximum; the number is set high only as "insurance". Once the number set becomes
smaller than the maximum number of iterations in your image, the smallest,
innermost iterations (such as those in the center of a spiral) start to be
skipped.

*TIP* It is possible to show one iteration at a time, or any group of contiguous
iterations by setting these two parameters in concert. Setting "Skip" to 12 and
"Show" to 13 gives you only the 13th iteration, for instance.

ELEMENT CONTRAST
Varying this parameter changes the internal shading of the elements. Higher
values tend to broaden the highlights and reduce the luminance difference
between highlights and darker areas. High values tend to produce bands in
background areas. This can be eliminated with distance masking.

ELEMENT SHADING
This parameter changes the final index value, with a corresponding variance in
the tonal relationships of the elements. Generally "normal" should be used with
coloring modes other than distance.

NORMAL: This is the standard "look". With a standard "shaping" gradient it gives
the look of highlighted elements on a black background.

NORMAL + FUNCTION: This is essentially the "normal" choice above, but when it is
chose a function block appears (see below) that allows for some choice in how
the gradient is mapped to the image.

WASH + FUNCTION: This setting varies the gradient mapping in the normally "dark"
part of the image, with the same function block as described above.

ELEMENT SHADING FN
This function block appears with either the second or third choice above. Many
of the function choices are not particularly useful, but some do create
interesting effects.

HEADING: GLOBAL FORMULA PARAMETERS

ALL FORMULA PARAMETERS
This is a simple Boolean parameter that, when unchecked, hides all the formula
parameters. This is useful when you want to access parameters lower on the tab
that are hidden by UF's unavoidable parameter overflow.

AMOUNT
This parameter determines the amount of formula pattern that will be applied. It
is a complex parameter, so there are both real and imaginary components. Default
is (100,100). Only if both are set to 0 does the effect of the chosen formula
completely disappear. In that case all the formula parameters will also be
hidden. Note that it is also possible to use negative numbers here.

FORMULA SCENARIO
For me this is the magic part of this coloring. As I mentioned, there is a
double set of formula patterns contained in this coloring, two each of  Gnarl,
Popcorn, Glyph, Vine, Hopalong, Martin, Twirl, Driftwood, Dynamic and Tanglewood
(a few are creations new to this coloring). Most simply a single formula from
this set can be chosen, but there are two other modes where elements of two
different formulae (one from each set) can be combined:

SINGLE: In this mode one of the ten formula patterns can be chosen to form the
elements. The parameters and functions unique to that formula will appear on the
tab.

MODULATED: In this mode the discrete elements of the first formula are replaced
by the discrete elements of the second formula. A pattern of some number of
formula 1 elements followed by some number of formula 2 elements is created.
Below are two parameters related to this mode that appear when it is selected.

--MODULATION REPEAT
(Appears when "Modulated" is chosen.) This determines how many iterations are
formed with the first pattern before the second pattern becomes active.

--# OF MODULATED ITERS
(Appears when "Modulated" is chosen.) This determines how many iterations are
formed with the second pattern before the first pattern becomes active again
(for the number of iterations specified in modulation repeat.)

BLEND: In this mode the shapes created by formula 1 and formula 2 are blended
together to create a new shape with elements of both. There are several
parameters (directly below) related to this.

--RANDOM SEED
(Appears when "Blend" is chosen.) Altering this value changes how the elements
are blended.

--BLEND PROPORTION
(Appears when "Blend" is chosen.) This determines the proportion of each of the
two formula shapes that is melded into the final shape. The range is 0-1. "0"
is formula 2 only, while "1" is exclusively formula one.

--BLENDING FUNCTION
(Appears when "Blend" is chosen.) Susan could give a precise mathematical
explanation for this, but I will only say that different functions blend the
elements in different ways. The default is "abs", which gives a 50% probability
for each formula when the blend proportion is .5. This is not the case with the
other functions, but I have adjusted values so that the result still seems to
include both formulae when .5 is used above. Note that several of the choices
have the qualifier "(fixed value)" appended. This means that changing the blend
proportion value does not change the final appearance of the elements at all.

# OF FORMULA ITERS
This parameter determines how many times the chosen formula is iterated to form
the elements. Lower values have a smoother look, while higher values give more
complex shapes. You can set "0" iters, which gives a different look to images
using PCT traps only, even when the formula amount is set to (0,0).

EMPHASIS
This parameter affects the sizes and shapes of the elements.

DEFORM 1 & 2, X & Y INTENSITY (PRE & POST), OFFSET & OFFSET 2
(These parameters are visible only in "Advanced" mode.) They all affect the
sizes, shapes and intensities of the elements.

EDGE 1, 2 and 3
(These parameters are visible only in "Advanced" mode). They all create edges
around (or at higher values truncate) the elements in different ways.

HEADING: MORE FORMULA SHAPING PARAMETERS

All the params and functions in this section are based on the principle that
orbit-trapping depends on a complex variable (z) being split into two float
variables (x and y) representing the real and imaginary values of the original
complex variable, which are manipulated in various ways before being recombined
back into a complex variable. What I have done here is to add a bunch of
parameters and functions that act on these x and y variables. I will not
describe the params and functions individually; suffice it to say that playing
with them can change the shape and appearance of the elements in quite
wide-ranging ways.

Towards the end you will find a param called EXTRA X/Y. This actually adds an
extra set of x and y variables which are combined with the original x and y
variables in different ways. If you change the default "none" it will open a
set of parameters and functions which act specifically on the extra x/y set.

HEADING: SINGLE FORMULA PARAMETERS

This lengthy section contains all the parameters specific to the individual
formula patterns. Only those that pertain to the chosen formula will be visible.
There are many choices, which I will not list specifically. In addition to the
traditional controls found in other formulae and colorings that use these
patterns (such as "Step Size", "Alpha", "Flavor" etc.) I have added a number of
others. Variable types can be changed, as well as arithmetic operators within
the formulae themselves. Each instance of an X or Y variable has three
parameters and a function attached, and in addition the "flavor" of each formula
type can be changed.  These controls are generally straightforward; please
explore them yourselves.

**NOTE** The numerical parameters attached to each variable appear only in
"Advanced" mode. Since there are up to 16 variables in the formulae they take up
an inordinate amount of space on the tab otherwise.

**NOTE and TIP** Switching variable types will sometimes have a major effect on
the shapes produced. It will also sometimes affect things so that changing other
variables will have no effect.

**NOTE** Some operator choices will appear redundant, but this will often change
if the functions associated with the variables are set differently.

HEADING: FORMULA 2 PARAMETERS

This section is an exact duplicate of the preceding 'Single' Formula section, for
use with the second formula when "Modulated" or "Blend" is chosen as the
scenario.

HEADING: GLOBAL FORMULA FUNCTIONS
These are functions that apply globally to all the formula patterns.

HEADING: PLANE CURVE TRAPS

ACTIVATE TRAPS
If this box is checked the final "z" value output by the formula section becomes
the initial "z" of the plane curve trap section. If the formula amount (see
above) is set to (0,0) the result will be clean plane curve trap shapes. As the
formula amount is increased progressively there is more and more a mix of the
two, until at (100,100)--generally--the formula shapes predominate over the trap
shapes, though they are modified by them.

TRAP TYPE
There are 135 trap shapes to choose from here. At the top--the default--is a
non-trap shape, labeled "None". This runs the formula "z" through the loop where
it can be modified by global parameters and functions, but does not add a
specific shape (though it does change the formula shapes somewhat when
inserted.)

HEADING: PCT GLOBAL PARAMETERS

**NOTE** The descriptions of the effects of the parameters in this section
applies when using PC Traps only, with no formula added. Adding formula makes
their behavior unpredictable (but is often very interesting.)

PCT GLOBAL PARAMETERS
This checkbox collapses the parameter group if you need more space or want to
navigate more easily on the tab.

LIGHTING
If only the PC Traps are used, this parameter varies the shading of the elements
without changing their shapes; however if the formula percent is greater than
(0,0) the choices here can have a large effect on the final appearance of the
image, changing both the shape, size relationships and shading of the element
components.

Further exploration of these effects can be done by varying the LIGHTING
FUNCTION 1 parameter immediately below this one. Going ever further, the final
choice in this "Lighting" parameter is "Function". Choosing this adds another
parameter described immediately below:

INVERT LIGHTING
This Boolean parameter gives the elements a completely different look. It also
opens new parameters as described below:

CHANGE INVERT LIGHT STYLE
This Boolean parameter offers a different look by changing the way Invert
Lighting is accomplished.

LIGHTING FUNCTION 1
In earlier versions this function was called "Final Z Function" and was grouped
with the functions below; however since it directly affects the lighting look I
have renamed it and moved it here. Basically this function and #2 below change
the shading of the elements without (generally) changing their shape, as do the
other trap functions.

FILL LIGHT FUNCTION 1 & 2
These functions appear when "Invert Lighting" is chosen and affect the way the
elements render in that case.

FILL LIGHT INTENSITY
This parameters appears when "Invert Lighting" is chosen. It changes the
highlights on the elements, and at higher values the shape and appearance of the
elements themselves.

--LIGHTING FUNCTION 2
(Appears when "Function" is chosen.) This parameter adds yet more choices to the
look of the "Lighting".

TRAP ROTATION
This parameter rotates the trapped elements around their centers. The setting is
in degrees.

TRAP SIZE
This setting changes the size of the elements but leaves their interrelationship
unchanged.

RATIO WIDTH/HEIGHT
This setting changes the relative height and width of the elements. Values below
1 vary it one way and those above 1 vary it the other way.

TRAP FLAVOR
This applies functions to the initial definition of the trap z variable,
resulting in different dimensions and relations of the elements, while retaining
their basic shape. (Visible in Advanced mode only.)

ELEMENT DIMENSION
This changes the way the gradient is mapped onto the elements and what is
outside the elements. Lower values tend to give sharper highlights to the
elements.

ELEMENT FOCUS
Higher values with this parameter make the element shapes thinner and more
focused, often defining features that are not present when values are low.

ELEMENT FISSION
Very low values with this parameter can give interesting shading effects inside
the elements. Larger values tend to cut out the center completely, as well as
change the size somewhat. (Visible in Advanced mode only.)

MOVE/SPREAD
Changing these values moves and spreads the elements in the X and Y planes.
(Visible in Advanced mode only.)

KALEIDO/SPLIT
Changing the real value of this parameter by integers adds extra element shapes
equidistant around the center of the element. For instance if you have a heart
shape at the real value 1, changing it to 4 will create an element with four
hearts spaced evenly around the center point of the former one heart. Decimal
values create partially formed elements. Changing the imaginary value creates a
fracture at some point in the element, which gets more pronounced as the value
is increased. (Visible in Advanced mode only.)

X EXPONENT, Y EXPONENT
These parameters change the shapes of the elements. Try positive and negative
numbers. (Visible in Advanced mode only and with a trap shape other than
"None".)

SUPER SHAPER 1-4
These parameters change the shapes of the elements. Try positive and negative
numbers. (Visible in Advanced mode only and with a trap shape other than
"None".)

HEADING: GLOBAL TRAP FUNCTIONS

All of these functions change the appearance of the elements, and apply to all
trap shapes.

HEADING: INDIVIDUAL TRAP PARAMETERS

All of these parameters either change the value of variables in the different
trap shapes or apply a function thereto. They offer yet more ways to change the
shape of the traps. It is especially important when using these that "Mode
Recipe" be set to "Straight" if you want the trapped elements in other coloring
modes to match those of Distance.

HEADING: MASKING

MASK TYPE
Masking can be performed in one of three ways, by DISTANCE (the usual), by
ITERATION, or by RANGE.

MASK THRESHOLD
In "Distance" masking this determines how far from the center of the orbit the
masking begins. In "Iteration" masking discrete iterations are masked based on
the value entered. I'm really not sure how "Range" masking is determined, as
this is one of Mark Townsend's creations.

MASK MODULATION
Another of Mark's creations. An interesting concept, though I haven't yet found
a use for it. Here is Mark's description: "With the Modulation Mask Type the
iteration that the orbit was caught by the trap is modulated by this value, and
if the Mask Threshold is smaller the pixel is made solid." (Not visible with
Distance masking.)

HEADING: MORPH PARAMETERS

MORPH PARAMETERS
Another Boolean parameter that hides the whole group to make navigation on the
tab easier and prevent parameter overflow.

MORPH -> MORPH II
Checking the box puts you into Morph II mode, which adds a number of options,
which I will go into a bit later.

MORPH
There are 11 choices of how #z is treated in Morph mode. As you scroll through
these more parameters and functions become visible reflecting the more complex
definition of this root variable and allowing you to alter aspects of it. Each
new #z can consist of up to three old #z's (depending on the "Morph" choice),
and each has some user-settable parameters associated with it. BIAS and TWIST are
complex parameters that vary the value of #z, and MORPH FUNCTION applies a
function to the #z in question. As all Morph choices past one contain either two
or three instances of #z, you will find up to three instances of these three
settings. Morph choices with three variables include an OPERATOR parameter as
well.

"Morph II" is an elaboration of the original Morph code, which adds an extra
instance of some variant of #z to each instance of #z already existing in
whichever Morph choice you set. For example in Morph choice "1" there is a
single #z with two associated parameters and a function. In Morph II choice "1"
there are two #z's each with two parameters and a function. This adds to your
choices. Here are the extra parameters that appear when "Morph II" is chosen:

Z1 MODE: This determines the relationship between the two #z variables. In Morph
"1" there was only one #z variable originally, so you will only see Z1 settings
added. In later Morph choices Z2 and Z3 choices become active and visible. Each
of these determines the kind of arithmetic operator used to define the
relationship between the two variables. Z+Z1 adds them, Z-Z1 subtracts Z1 from
the original #z and Z1-Z subtracts the original #z from Z1. Z2 and Z3 choices
also include Z*Zn, Z/Zn and Zn/Z.

Z1 (or Z2 or Z3) TYPE: This chooses the kind of #z variable which is paired with
the original #z. Choices are |Z|, ATAN2(Z), REAL(Z) and IMAG(Z).

In addition each Zn includes an appropriately labeled BIAS (such as BIAS Z1)
parameter, TWIST "Zn" parameter and MORPH "Zn" FUNCTION.

*NOTE* any given #z block" can be turned off by setting its function to "zero".
In Morph II Z2 and Z3 are off by default (their functions are set to "zero"). To
activate Z2 and Z3 set their associated functions to "Ident" or some other
function choice.

HEADING: TEXTURES

This section contains a number of different textures that can be added to the
image. Choices include RANDOM, FBM, DECIMAL, POPGNARL, GEOMETRIX and ADDITIONAL.
Each has a group of (mostly similar) parameters and functions, headed by a
Boolean parameter to allow you to hide the group in question. These textures are
additive, and one or more can be applied to any given layer.

OVERALL TX PERCENT
This allows you to vary the percentage of all the textures at once. Each texture
also has a parameter for varying only the amount of that texture as well.

RANDOM TX AMOUNT
This is the only control for the "Random" texture.

As for the rest, I will let you play with them by yourself. There are a number
of hints which should help you understand the meaning and use of the parameters.
The behavior of the different types is fairly consistent, although you will find
that different textures sometimes require different amounts to create a similar
intensity of effect (depending on lost of circumstances). Only "Additional"
texturing varies significantly, and will require a readjustment of the gradient
when applied.

Toby Marshall,  2005.01.28,  rev 2005.10.19

================================================================================

PAINTER'S TRAPS GUIDE
(revised 05.10.20)

INTRODUCTION

The genesis of this coloring was a rather casual attempt to add some
functionality to Mark Townsend’s "Soft Gnarly". Both Gilles Nadeau and Susan
Chambless added to it originally, and then I also got into the act. Through many
additions and betas "Painter’s Traps" took shape.

The original "Soft Gnarly" is a unique coloring in which patterns such as
"Gnarl", "Popcorn" and "Glyph" among others, are overlaid on elements created by
rudimentary orbit trapping. These are actually "pixel patterns" which are not
part of the loop that creates the elements; but unlike normal textures, these
patterns actually influence the element shapes "beneath" them. It is a very
clever setup for which Mark deserves a lot of credit.

However Mark's original was missing many features that I feel a good coloring
deserves: coloring modes, masking, and textures, as well as the ability to skip
iterations, among other things.

I managed to add those features, but it seemed to me a coloring as good as this
would benefit from more orbit shapes. To my surprise I managed to add the plane
curve trap shapes first introduced to UF by Samuel Monnier in "Thin Orbit Traps"
and later adapted by Ken Childress for "Plane Curve Traps" I also added some
shapes that were incorporated in "Plane Curve Traps II".  All of the shapes
together can be found in "Thin Orbit Traps II".

In addition I added a number of textures, borrowed from various places, and my
"Morph" parameters, which affect the trapping by altering the definition of #z.
Finally I have added a number of parameters and functions not found in the
original, as did Gilles, and Susan added her "Hopalong" pattern.

As I worked and refined the characteristics of the coloring, I endeavored not to
lose some of the interesting effects that my earlier, less polished efforts
created. Therefore there are several parameters that change important
characteristics of this coloring between "Straight" and "Funky" mode.

All of these additions have made Painter’s Traps quite complex, so I have
endeavored to write a guide for the beginner to help you try to get an overview
of the coloring as a whole, and understand the significance and use of the
various features of the coloring.

GETTING STARTED

I think the easiest way to get going is just to consider the controls you see on
the Properties tab when you open this coloring one by one. Some of them are
duplicated in the two modes, and if I treat them in the first mode (Soft Gnarly)
I will not mention them again when discussing the Plane Curve Traps mode.

MODE: Here there are two choices, Basic and Advanced. "Basic" simplifies the
interface and shortens the list of possibilities considerably, for easier
navigation. I've left out many parameters and functions, as well as a few
textures that I have found less useful in general. Please note that while those
certain parameters/functions are hidden in "Basic" mode they are still active.
So if you have set them in "Advanced" mode they will still have the same effect
when you switch to "Basic"—but you won't be able to see them to reset them.

COLORING TYPE
This determines the main mode of the coloring—there are two: "Soft Gnarly" is
pretty much identical to Mark Townsend's original (although users of that
coloring will notice some small differences in the controls.) There are a number
of added features not included in the original, including different coloring
modes, masking, textures, etc. First I will describe two parameters that appear
directly below this one when "Plane Curve Traps" is chosen, then I will describe
the controls you will find  using the "Soft Gnarly" mode before moving on to
talk about controls unique to "Plane Curve Traps. It needs to be understood that
the pixel pattern—if it is present—will affect many of the things I am going to
talk about here. I will try to include an explanation where applicable.

TRAPPING MODE
This parameter appears only when "Plane Curve Traps" is chosen. "Closest" is the
look you are probably used to in orbit trapping. "First" creates a discrete edge
around the elements, with the earlier, smaller elements overlying later, larger
ones. "Last" is the opposite, overlaying the larger, later elements over the
smaller, earlier ones.

ELEMENT WIDTH
The "masking" of the element edges--the effective element width is controlled by
this parameter, which appears if either First or Last is chosen.

**IMPORTANT NOTE** To use "Last" you must set the parameter "No. of Test Points"
(described later) to 1 and uncheck "Include Root Trap".

--SOFT GNARLY--

Here are the controls you will see when you switch to Soft Gnarly mode. Please
note that some parameters appear only when certain choices are made in other
parameters. I will "nest" those explanations with the parameter choice that
activates them.

COLORING MODE
This section applies to both Soft Gnarly and Plane Curve Traps mode. There are a
number of choices here. Note that pixel patterns, if present, actually become
part of the elements, so that the pixel patterns will affect the look of all
these modes in one way or another:

DISTANCE: This is the standard mode which colors the pixels according to the
distance they fall from the trap center. It gives the characteristic 3D or
shaded look of orbit trapping. In Soft Gnarly, since there are no specific trap
shapes, it tends to produce elements that appear spherical, although that can be
changed to some extent depending on other functions/parameters. With pixel
patterns present the patterns will change both the shading and to some extent
the shape of the elements.

ITERATION: This colors the pixels by what iteration they fall into. The elements
look flat but are of different colors or tints. If pixel patterns are present
they will sometimes be incorporated, changing the apparent shape of the elements
according to how they lie.

MODULATED ITERATION: This is similar to Iteration, but it repeats the color
every given number of iterations. That number can be set with the parameter that
appears directly below when Mod Iter is chosen:

COLOR RANGES: (Appears when "Modulated Iter" is chosen).  This sets the number
of color ranges that repeat in the elements. You can set a decimal for some
interesting effects—setting a value of 4.1, for example, gives four repeating
colors which shift down the iterations. This can be quite interesting with
careful setting of the gradient.

MAGNITUDE, REAL, IMAGINARY and ANGLE: These are other standard ways of
determining how the pixels are colored in a trap coloring which I won't go into
here.

MARTIN: This mode colors the pixel pattern only. This is adapted from another
coloring, so I am not sure about how or why the parameters and function that
appear with it work. Please experiment to get a feel for how it all works.
Martin Iterations, Martin Param, Martin Color Scale and Martin Color Function
all appear when "Martin" above is chosen, and affect the way it appears.

PATTERN MAGNITUDE, PATTERN REAL, PATTERN IMAG and PATTERN ANGLE are various ways
of coloring the pixel pattern only.

HEADING: GLOBAL PARAMETERS

SPREAD
There are nine traps in this coloring, and "Spread" spreads the elements created
by each in different directions. You will need to adjust the gradient when you
change the value of this parameter.

NO. OF TEST POINTS
This determines how many traps are used, and again if you change the value here
you will have to adjust the gradient. If you choose 1 only the first trap is
used, choosing 5, for instance, uses traps 1-5.

INCLUDE ROOT TRAP
Checking this box activates the "center" trap—the basic trap created by #z. It
has the effect of making the elements look more "filled out" If this is
unchecked the root trap parameter disappears in the next parameter
(Trap Config. = Separate).

TRAP CONFIG.
This is one of my additions, which allows the way the traps are determined to be
chosen. There are three basic configurations:

SEPARATE: This allows each trap type to be set independently, and you will see
nine new fields appear when you choose this option: ROOT TRAP and TRAP 1 -
TRAP 8, all with the same choices:

NORMAL: This is the standard.

MODULUS: This returns the modulus value of z. The shapes are the same but have
more "contrast".

REAL: This returns the real value of z only. It creates trap shapes that look
like tentacles.

IMAG: This returns the imag value of z only. It too creates line shapes but
different from those made by "Real".

ZERO: This "turns off" the trap in question.

FUNCTION: This adds a function block for each trap individually. Changing this
can lead to quite different looks while retaining the basic element shapes. Try
varying "Final Z Function" as well for yet more possibilities.

*TIP* Different traps with the same setting give different effects if "Spread"
is not set to 0. At 0 "real" for instance, will look the same for each trap—they
will add directly on top of one another. If "Spread" is not 0 "real" in each trap
will be slightly different.

*TIP* Try using only one trap for the most "solid" appearance.

CHAINED
This setting is basically the same as Separate, above, but allows the user to
set all the values at the same time. This parameter opens the following one:

CHAINED TYPE: The traps are all set to the same value at one time: NORMAL,
MODULUS, REAL, IMAG or FUNCTION. Function acts on all traps at once.

TONALITY
This parameter changes the look of the elements. In NORMAL mode no hidden
parameters appear. In either FLAVOR 1 or FLAVOR 2 mode a hidden parameter
appears called:

TONALITY FUNCTION:  The default function is set to "recip” It is important to
note that while this gives good results in "Plane Curve Traps" mode (which is
why it is the default)—it creates a blank screen when in "Soft Gnarly" In fact
many of the functions do not give good results in Soft Gnarly mode, but it is
worth checking out a few. I especially like "sqr" which gives edges to the
elements. The difference between Flavor 1 & 2 is only apparent when the
following two parameters (Edge 1 and Edge 2) are not set to their default.

EDGE 1 & EDGE 2
These two parameters are only visible when "Tonality" is not set to normal. They
create "edge" effects around the elements, setting them apart from the
background.

*TIP* Try setting the Tonality Function to "ident" and playing with these two
parameters: once their values are set differently than their default, the
difference between Flavor 1 and Flavor 2 in the "Tonality" parameter become
apparent.

OFFSET
This parameter changes the value of z and thus affects the pattern of the
trapped elements. It does not affect the pixel patterns.

ITERATIONS TO SKIP
This parameter determines how many iterations (trapped elements) are skipped. It
affects the largest, outermost elements first.

*NOTE* Pattern iterations precede trap iterations, even when pattern percent is
0. So if you have 20 pattern iterations the first iteration that will be skipped
in trap shapes is number 20. This does not affect the patterns

ITERATIONS TO SHOW
This parameter determines how many iterations are shown in total. Any number
over the maximum number of iterations in your particular fractal returns the
maximum; the number is set high only as "insurance" Once the number set becomes
smaller than the maximum number of iterations in your image, the smallest,
innermost iterations (such as those in the center of a spiral) start to be
skipped.

*TIP* It is possible to show one iteration at a time, or any group of contiguous
iterations by setting these two parameters in concert. Setting "Skip" to 12 and
"Show" to 13 gives you only the 13th iteration, for instance.

HEADING: SOFT GNARLY FUNCTIONS

OVERALL FUNCTION, REAL FUNCTION and IMAGINARY FUNCTION
All three of these change the shape of the traps at some settings. They are
interactive, and it is worth scrolling through and using them with some of the
other parameters listed above such as "Offset" or the real or imaginary trap
configuration settings if you are not satisfied with simple spherical trap
shapes.

CHANGE RE/IM FN FLAVOR
 Changes the effect of the functions when the Real and/or Imaginary Functions
 described above are not at "ident".

HEADING: MASKING

This is just standard distance masking. You can often do the same with a masking
layer but this is sometimes easier to use or can be used in conjunction with a
masking layer. INVERT MASK does just what it says.

HEADING: PIXEL PATTERNS

Here is the section that determines the pattern that overlays the elements in
either "Soft Gnarly" or "Plane Curve Traps" mode. This pattern is not affected by
the formula, being dependent only on #pixel for its genesis. There are a number
of different patterns based on well known formulae, and many controls that
determine the appearance, size and intensity of the pattern.

PIXEL PATTERN TYPE
Here you have the choices MARTIN, POPCORN, VINE, GLYPH, GNARL and HOPALONG. Each
choice makes visible only the parameters and functions that apply to that
particular formula. I am not going to get into the individual parameter choices
for each pattern, as they can be explored through trial and error, and also they
are not my creations. You will find, though, if you are familiar with these
formulae, that I have expanded parameters and functions beyond what are in the
originals. Please refer to the hints in these parameters and functions, when
present, for more information.

PATTERN PERCENT
This controls the intensity of pixel pattern that appears. 0% means no pattern
at all, and 100% means only the default as it existed in the original Soft
Gnarly formula. You can set percentages higher than 100, and you can also set
negative percentages.

PATTERN PERCENT REAL & IMAGINARY
These control the percentages (as above) of the X and Y variables which
determine the patterns. Setting lower values in one will reduce some of the
pattern shapes, while leaving others untouched.

*NOTE* these function correctly only when using "Normal" or "Modulus" in the
"Trap Config" parameter above. Set to "Real" only the "Pattern Percent Real"
parameter has an effect, controlling the overall percentage like "Pattern
Percent" above. Set to "Imag" only "Pattern Percent Imaginary" has any effect.

HEADING: GLOBAL PATTERN PARAMETERS

GLOBAL PATTERN PARAMETERS
This Boolean parameter allows you to collapse the whole group of parameters
under this heading. This can be very useful if you run out of parameter spaces
at the bottom of the tab and find a last field called "More Parameters", which
is a drop-down menu containing any further parameters that can't be displayed.
Collapsing this or other parameter groups with a similar Boolean parameter allow
more parameters to be displayed at the bottom of the tab.

# OF PATTERN ITERS
This parameter determines the number of times the pattern is iterated. Higher
values draw the patterns more fully, but very high values give too much contrast
or elements that are too busy.

*NOTE* the number of iterations set here must be added to the number in
"Iterations to Skip" before element iterations start to be skipped. See
"Iterations to Skip" above.

OFFSET
This shifts the whole pattern to the right or left or up and down (if rotation
is at 0), or anyway always in planes at right angles to each other.

OVERALL SCALE, X SCALE, Y SCALE
Overall Scale determines the scale of the pattern. A setting of .5 for example
reduces the size of the pattern elements by half. X and Y Scale control the
height and width of the elements. But the two types of scales are not exactly
analogous. For example, setting overall scale to 2 and X and Y Scale each to .5
gives the same overall pattern of elements, but with different shading.

ALPHA, BETA
Parameters contained in the original coloring. Check them out for their effects.
Beta appears only in a few patterns/flavors.

DEFORM 1, DEFORM 2
Exponential functions that stretch and deform the elements. Use both positive
and negative numbers.

PATTERN SYMMETRY
This parameter has three choices: NORMAL creates no symmetry in the pattern.
SYMMETRY creates bilateral symmetry, and FLIP SYMMETRY creates another kind of
bilateral symmetry.

HEADING: PATTERN FUNCTIONS

OVERALL PATTERN FN, PATTERN R & I FUNCTIONS
These functions all alter the appearance and disposition of the patterns. Not
all functions give pleasing results.

HEADING: MORE FORMULA PARAMETERS

This section is analogous to "More PCT Shaping Parameters" described in the
PCT section, but modifies the pixel patterns (primarily) and not the traps.

All the params and functions in this section are based on the principle that
orbit-trapping depends on a complex variable (z) being split into two float
variables (x and y) representing the real and imaginary values of the original
complex variable, which are manipulated in various ways before being recombined
back into a complex variable. What I have done here is to add a bunch of
parameters and functions that act on these x and y variables. I will not
describe the params and functions individually; suffice it to say that playing
with them can change the shape and appearance of the elements in quite
wide-ranging ways.

Towards the end you will find a param called EXTRA X/Y. This actually adds an
extra set of x and y variables which are combined with the original x and y
variables in different ways. If you change the default "none" it will open a
set of parameters and functions which act specifically on the extra x/y set.

**NOTE** Because some of these params and functions act on the PC Traps even if
the pixel pattern amount is 0, I have left this section independent of the
pattern section.


**********************

--PLANE CURVE TRAPS--

Before considering other global settings let's turn explore the options
available in the "Plane Curve Traps" mode.

I will list parameters only if they are different from what you find in "Soft
Gnarly" mode.

HEADING:  PLANE CURVE TRAPS

TRAP TYPE
With this parameter you can choose any of 135 trap shapes. Many were first
included in Samuel Monnier’s "Thin Orbit Traps" and I threw together some
others. You will find that there are substantial differences in the way the
gradient works between them, and so you may have to adjust your gradient to get
the same shading effects when you move from one to another.

HEADING: PCT GLOBAL PARAMETERS

PCT GLOBAL PARAMETERS
This Boolean parameter allows you to collapse the whole group of parameters
under this heading. You will find a similar parameter for many of the groups
under headings.

LIGHTING
This changes functions associated with the trap z parameter, changing the
appearance of the elements. This is easiest to see if there is no pixel pattern.
SIDELIGHT makes the elements appear to be lit around the edge, whereas TOPLIGHT
gives the elements the appearance of being lit from the top. WIDELIGHT is like
Toplight but the element edges are less defined. INNERLIGHT is something like
Sidelight but the interior of the traps appear more solid. OUTERLIGHT has the
opposite effect, making the outside of the trap shapes appear dimensional. GLOW
is similar to Sidelight but with wider borders. FUNCTION inserts a function
block below, called "Lighting Function 2" with which you can achieve more
permutations of the element appearance.

Generally Sidelight (and to some extent Innerlight, Outerlight and Glow) are
versatile than Toplight and Widelight, lending themselves to more variation. In
fact later there is a group of parameters labeled "Advanced Trapping and
Textures" that give unpredictable and often unpleasant results when used with
Toplight and Widelight. The effect of Function depends on the function chosen.

*TIP* Try varying "Lighting Function" 1 and 2 for more lighting effects (see
below).

*TIP* for another nice type of lighting try setting this parameter to either
"Widelight" or "Sidelight" distance functions (either "Chained Function" or the
individual trap functions) to "log" and Lighting Function 1 to "acos." Another
interesting effect can be achieved with "Chained Function"(or individual trap
functions using Separate)  set to "Normal". Lighting set to Outerlight and
Lighting Function 1 set to "flip" If you use an iteration layer you can match
these traps by setting Mode Recipe to "Funky" Trapping Mode to "None" and Mode
Flavor to either "Trap Z" or "Sum Z".

INVERT LIGHTING
This Boolean parameter gives the elements a completely different look. The
elements take on a very solid, 3D appearance when using TOPLIGHT, EDGELIGHT or
WIDELIGHT. A number of new params will appear if you make this choice, as
described below.

**IMPORTANT NOTE** If you choose "Invert Lighting" most of what I have written
under the category "Advanced Trapping and Texture" later does not apply. You
will want to keep both the "Trapping Recipe" and "Mode Recipe" there at
"Straight" unless you are looking for weird effects when using coloring modes
other than "Distance".

LIGHTING FUNCTION 1
In earlier versions this function was called "Final Z Function" and was grouped
with the functions below; however since it directly affects the lighting look I
have renamed it and moved it here. Basically this function and #2 below change
the shading of the elements without (generally) changing their shape, as do the
other trap functions.

FILL LIGHT INTENSITY, OFFSET, STRENGTH, POWER
These parameters appear only when "Invert Lighting" is chosen. All change the
highlights on the elements, and sometimes the shape and appearance of the
elements themselves.

LIGHTING FUNCTION 2
This function appears only when "Function" is chosen under "Lighting". Scroll
through for more lighting possibilities.

TRAP ROTATION
This parameter rotates the trapped elements around their centers. The setting is
in degrees.

TRAP SIZE
This setting changes the size of the elements but leaves their interrelationship
unchanged.

RATIO WIDTH/HEIGHT
This setting changes the relative height and width of the elements. Values below
1 vary it one way and those above 1 vary it the other way.

TRAP FLAVOR
This applies functions to the initial definition of the trap z variable,
resulting in different dimensions and relations of the elements, while retaining
their basic shape.

ELEMENT DIMENSION
This changes the way the gradient is mapped onto the elements and what is
outside the elements. Lower values tend to give sharper highlights to the
elements.

ELEMENT FOCUS
Higher values with this parameter make the element shapes thinner and more
focused, often defining features that are not present when values are low.

ELEMENT FISSION
Very low values with this parameter can give interesting shading effects inside
the elements. Larger values tend to cut out the center completely, as well as
change the size somewhat.

MOVE/SPREAD
Changing these values moves and spreads the elements in the X and Y planes .

KALEIDO/SPLIT
Changing the real value of this parameter by integers adds extra element shapes
equidistant around the center of the element. For instance if you have a heart
shape at the real value 1, changing it to 4 will create an element with four
hearts spaced evenly around the center point of the former one heart. Decimal
values create partially formed elements. Changing the imaginary value creates a
fracture at some point in the element, which gets more pronounced as the value
is increased.

X EXPONENT, Y EXPONENT
These parameters change the shapes of the elements. Try positive and negative
numbers.

SUPER SHAPER 1-4
These parameters change the shapes of the elements. Try positive and negative
numbers.

HEADING: MORE PCT SHAPING PARAMETERS


This section is analogous to "More Formula Parameters" described in the
earlier, but modifies the PC Traps and not the pixel patterns. Here is the
description again:

All the params and functions in this section are based on the principle that
orbit-trapping depends on a complex variable (z) being split into two float
variables (x and y) representing the real and imaginary values of the original
complex variable, which are manipulated in various ways before being recombined
back into a complex variable. What I have done here is to add a bunch of
parameters and functions that act on these x and y variables. I will not
describe the params and functions individually; suffice it to say that playing
with them can change the shape and appearance of the elements in quite
wide-ranging ways.

Towards the end you will find a param called EXTRA X/Y. This actually adds an
extra set of x and y variables which are combined with the original x and y
variables in different ways. If you change the default "none" it will open a
set of parameters and functions which act specifically on the extra x/y set.


HEADING: ADVANCED TRAPPING AND TEXTURE

The following parameters and functions generally work better with Sidelight than
Toplight, though most have an effect with both. In addition most of them have an
effect with textures whenever the initializing variable is trapped (such as
"Trap Z"). They are very interactive with each other, so it is quite difficult
to accurately describe their effects separately and in combination.
Basically you will not want to play with the params in this section at all if
you have checked "Invert Lighting".

TRAPPING RECIPE
This parameter offers two choices:
STRAIGHT is the classic distance trapping which will overlay layers created with
coloring modes other than Distance (such as Iteration, Angle, etc.) exactly over
elements drawn with Distance. STRAIGHT can be used with both "Sidelight" and
"Toplight". Use this setting if you have chosen "Invert Lighting".

FUNKY, in conjunction with the next parameters ("Trapping Mode" and "Mode
Flavor" uses different definitions of "Distance" to arrive at trap shapes with
other coloring modes. Not all the combinations overlay the Distance shapes
accurately.

*TIP* Using Mode Flavor "Trap Z" or "Sum Z" gives an interesting result for
Iteration or other coloring mode layers overlaying a Distance layer (in Mode
Recipe "Funky"—see immediately below.)

MODE RECIPE
This is a very important parameter to know about when using layers composed in
Distance mode with layers made with any other mode. For predictable results,
with (for instance) an Iteration layer exactly overlapping a Distance layer,
you should use the STRAIGHT mode, however very interesting results can be
obtained using FUNKY mode, in combination with Trapping Shift, Strength and
Power, Trapping Function and a few other parameters in this section. Use
STRAIGHT if you have chosen "Invert Lighting".

TRAPPING MODE
This parameter determines which variable will be used to determine the distance
for trapping the elements. The choices  are NONE, SUM Z, TRAP Z, MODULUS Z,
MORPH Z and #Z. They give different effects and it is worth checking them out
in Distance and also the other coloring modes.

*TIP* "Sum Z"(with "Sum Z" or "Trap Z" in "Mode Flavor" below) gives later
iterations overlaying earlier ones. (Another way to achieve this, perhaps
simpler, is to have the default of "None" here, "Trap Z" or "Sum Z" for "Mode
Flavor" and set the parameter "Trapping Shift (re)" (described below) to a value
of +1 or higher as necessary, using "Funky" for Mode Recipe (described below.)
To match this with a layer other than Distance set Trapping Mode to "None".
Mode Flavor to "Sum Z" and "Trapping Shift (re)" (see below) to 1 (or higher as
needed), using "Straight" as the trapping recipe, and "Funky" as the mode recipe
(see below.) This is also covered in the demo upr at the end of the help file.

MODE FLAVOR
The second part of the above parameter—which helps determine how the trapped
elements are defined. Explore this parameter along with "Trapping Mode" for
different effects—how the elements combine and overlie each other.

ADD PATTERN (FIX SUM Z)
This parameter becomes visible whenever "Sum Z" is chosen in either "Trapping
Mode" or "Mode Flavor" Its effect varies somewhat depending on the combination
of choices in "Trapping Recipe", "Trapping Mode" and "Mode Flavor" but basically
when checked the pixel pattern is added in coloring modes other than Distance.
In Distance mode--using "Sum Z" as outlined in the tip above--having this
parameter checked gives cleaner elements. This parameter also controls whether
Trapping Shift, Strength and Power (see below) do anything using the "Funky"
trapping recipe. Using "Straight" these three parameters have an effect whether
or not this parameter is checked.

TRAPPING FUNCTION
This function has an effect on all coloring modes, though you will find that
many of the function choices have no effect at all. It also depends on the
values chosen for the rest of the parameters in this group.

TRAPPING SHIFT, TRAPPING STRENGTH, TRAPPING POWER
These three parameters control the way the elements overlap when using a
coloring mode other than Distance. They are in the form:
((trapping variable - shift) * strength) ^ power,
so as you can see if either the real component of strength or power is zero the
result is zero and no other value changes will have any effect.

Since these variables are interdependent, and are affected by "Trapping
Function"(and further are dependent on both "Trapping Mode" and "Mode Flavor" it
is quite difficult to write a concise outline of their effects. Essentially they
determine how the elements will be colored at places where they touch or
overlap. I will leave you to explore further.

*NOTE* these particular parameters are grouped here even though they have a
decent effect when Toplight is used, but not in all cases (for example when
using Sum Z in mode flavor with trapping recipe "Straight". When using Sidelight
the "Add Pattern (Fix Sum Z)" parameter must be checked for them to have an
effect, whereas with Toplight they have an effect either way (when they have an
effect at all).

THICKNESS/PATTERN
This parameter changes the thickness of the trapped elements using the "Funky"
trapping recipe—it works best with coloring modes other than Distance. If pixel
pattern is present it affects the thickness and distribution of that pattern in
those coloring modes. Both negative and positive values can be used.

PATTERN DISTRIBUTION
This parameter changes the distribution of the pattern on the trapped elements
using the "Funky" trapping recipe—it works best with coloring modes other than
Distance. Both negative and positive values can be used.

COLOR MODE/TX FN
This function appears only when "Mode Recipe" is set to "Funky" It changes the
way the trapped elements appear when using coloring modes other than Distance.
Might be useful for "artistic" effects. Also changes the way textures are drawn
using any "Trap Z" init variable. Can be very useful when using textures.

COLOR MODE/TX POWER
This exponential parameter also appears only when "Mode Recipe" is set to
"Funky" and is useful in conjunction with "Color Mode/Tx Fn" above. Use it under
the same circumstances to vary the effects of that function.

HEADING: GLOBAL TRAP FUNCTIONS

All of these functions change the appearance of the elements.

HEADING: INDIVIDUAL TRAP PARAMETERS

All of these parameters either change the value of variables in the different
trap shapes or apply a function thereto. They offer yet more ways to change the
shape of the traps. It is especially important when using these that "Mode
Recipe" be set to "Straight" if you want the trapped elements in other coloring
modes to match those of Distance.

HEADING: PROGRESSIVE PARAMETERS

These are an innovation which take existing parameters and increment them
progressively as the iterations increase. Usually they work best in small doses.
Each is opened with a Boolean parameter. Checking and unchecking the Boolean
activates and deactivates the parameter—so that you can check the effect. Try
both positive and negative numbers. The Boolean parameter PROGRESSIVE PARAMETERS
hides the whole group, but does not deactivate any active parameters.

You have another option here, a parameter called PROGRESSION TYPE at the top of
the group. LINEAR means that the amount you enter (in each progressive parameter
under INCREMENT BY) is added to each successive iteration. NON-LINEAR (1-3)
means that the amount added progressively increases by the amount you choose in
the parameter that appears called NON-LINEAR AMOUNT. You should try different
values in this field, both negative and positive. There are three different
type of non-linearity, and they sometimes behave similarly and sometimes behave
quite differently. Some values are unusable with the various choices, giving a
blank screen. You should try different values in this field, both negative and
positive.

HEADING: OTHER OPTIONS
Here is some code I cooked up which varies the effect of the coloring by
changing the definition of #z. Since this happens  at the "top of the stream" it
does not affect the way other parameters and functions work.

HEADING: MORPH PARAMETERS

MORPH PARAMETERS
Another Boolean parameter that hides the whole group to make navigation on the
tab easier and prevent parameter overflow.

MORPH -> MORPH II
Checking the box puts you into Morph II mode, which adds a number of options,
which I will go into a bit later.

MORPH
There are 11 choices of how #z is treated in Morph mode. As you scroll through
these more parameters and functions become visible reflecting the more complex
definition of this root variable and allowing you to alter aspects of it. Each
new #z can consist of up to three old #z's (depending on the "Morph" choice),
and each has some user-settable parameters associated with it. BIAS and TWIST
are complex parameters that vary the value of #z, and MORPH FUNCTION applies a
function to the #z in question. As all Morph choices past one contain either two
or three instances of #z, you will find up to three instances of these three
settings. Morph choices with three variables include an OPERATOR parameter as
well.

"Morph II" is an elaboration of the original Morph code, which adds an extra
instance of some variant of #z to each instance of #z already existing in
whichever Morph choice you set. For example in Morph choice "1" there is a
single #z with two associated parameters and a function. In Morph II choice "1"
there are two #z's each with two parameters and a function. This adds to your
choices. Here are the extra parameters that appear when "Morph II" is chosen:

Z1 MODE: This determines the relationship between the two #z variables. In Morph
"1" there was only one #z variable originally, so you will only see Z1 settings
added. In later Morph choices Z2 and Z3 choices become active and visible. Each
of these determines the kind of arithmetic operator used to define the
relationship between the two variables. Z+Z1 adds them, Z-Z1 subtracts Z1 from
the original #z and Z1-Z subtracts the original #z from Z1. Z2 and Z3 choices
also include Z*Zn, Z/Zn and Zn/Z.

Z1 (or Z2 or Z3) TYPE: This chooses the kind of #z variable which is paired with
the original #z. Choices are |Z|, ATAN2(Z), REAL(Z) and IMAG(Z).

In addition each Zn includes an appropriately labeled BIAS (such as BIAS Z1)
parameter, TWIST "Zn" parameter and MORPH "Zn" FUNCTION.

*NOTE* any given #z block" can be turned off by setting its function to "zero".
In Morph II Z2 and Z3 are off by default (their functions are set to "zero"). To
activate Z2 and Z3 set their associated functions to "Ident" or some other
function choice.

HEADING: TEXTURES

This section contains a number of different textures that can be added to the
image. Choices include RANDOM, FBM, DECIMAL, POPGNARL, GEOMETRIX and ADDITIONAL.
Each has a group of (mostly similar) parameters and functions, headed by a
Boolean parameter to allow you to hide the group in question. These textures are
additive, and one or more can be applied to any given layer.

OVERALL TX PERCENT
This allows you to vary the percentage of all the textures at once. Each texture
also has a parameter for varying only the amount of that texture as well.

RANDOM TX AMOUNT
This is the only control for the "Random" Texture.

As for the rest, I will let you play with them by yourself. There are a number
of hints which should help you understand the meaning and use of the parameters.
The behavior of the different types is fairly consistent, although you will find
that different textures sometimes require different amounts to create a similar
intensity of effect (depending on lots of circumstances). Only "Additional"
texturing varies significantly, and will require a readjustment of the gradient
when applied.

Don't forget that some of the choices under "Advanced Trapping and Texture will
have a global impact on the effect of all these textures in any "Trap Z" mode.
Check there for more.

HAVE FUN! :^})


DEMO TRAPPING UPR

Here is a demo upr which hopefully will graphically illustrate some of the
possibilities of combining various trap layers derived through manipulating some
of the very confusing "Advanced Trapping and Texture" parameters described
above.

Bottom layer (normal): this demonstrates the default arrangement in trap mode.
There is no "Spread", as this would create overlaps at the element edges. The
goal here was to get nice clean, dimensional trap shapes. Try varying Element
Dimension and especially Element Focus to see how this changes the appearance of
the elements.

2nd from bottom layer (ol): this shows how later iterations can be made to
overlay earlier ones. There are two ways to do this, as described in a tip
above. Here I just used the defaults--with one notable exception: "Mode Recipe"
must be set to "Funky." I then increased the "Trapping Shift (re)" until the
elements inside other elements were gone.

3rd from bottom layer (ol+tex): this layer is like the preceding one except that
some "Geometrix" texture has been added. "Mode Flavor" was changed from "Trap Z"
to "Sum Z" for clean textures.

4th from bottom (ol+ghost tex); if you look closely you will see that even
though earlier iterations are hidden, their texture can be seen on the overlying
layer. This was achieved by changing "Trapping Mode" from "None" to "Sum Z" and
"Mode Flavor" from "Sum Z" to "Trap Z".

5th from bottom (straight): Here is a color layer to combine with the monochrome
layers below. This one is created in default mode and fits best with the bottom
layer. Activate the bottom layer with this and notice how the colored iterations
appear to merge where they touch.

6th from bottom (str/funky) activate this with the bottom layer and notice how
the earlier colored iterations now appear to completely lay atop the later ones—
at least the colors take precedence. This is achieved by using the defaults but
setting the "Mode Recipe" to "Funky" instead of "Straight".

7th from bottom (str/funky+3) here is a layer to go with the overlay monochrome
layer. It is exactly as the preceding layer except that "Trapping Shift (re)"
has been increased to +3.

Top layer (funky/funky) here the settings as in "straight" have been kept with
one exception: "Mode Recipe" has been changed to "Funky". The outside edges of
the elements are ringed with color.

Play around with various combinations to see the effect :-)


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================================================================================

HARMONIA, MELODIA, POLYPHONIA, RONDO, SUITES: AEOLIA, DORIA, HYPOAEOLIA,
HYPODORIA, IONIA, LOCRIA, LYDIA, MIXOLYDIA, PHRYGIA

1 APRIL 2004


Astute readers may notice that all have musically-related  titles, with the last
nine representing some of the classical Greek musical modes. In fact, though all
are structurally related, they could be divided into two subgroups comprising
the first four and the last nine. More on this later...

All these offerings are godchildren of Andrea Spinozzi's "Barney-2" formula. In
that one Andrea hit upon or borrowed the idea of creating a number of separate
small loops that each produce a new variable, which are then combined at the
bottom of the loop section. My main contribution has been in recognizing the
value of his idea and systematizing it somewhat, as all these formulae are
variations on the "Barney-2"  structure.

 *Harmonia, Melodia, Polyphonia and Rondo*

Basically what happens is that the first four formulae listed each contain three
"Z sections" and the latter nine each contain four "Z sections". In its simplest
incarnation a "Z section" is a mini-loop that creates a holding variable. For
example, z1 = function(z^param)[select operator]function(c). This is an example
of a simplified "Z1 section". If we set the param =2 and select "+" as the
operator (with the functions left at "ident") we end up with the classic Mandy:
z1 = z^2 + c.

Now imagine that we have three of those sections, and at the end we combine them
like this:
x  =  z1 [select operator] z2
y = z3 [select operator] x
z = y

Here we finally arrive at the new z with which to reiterate the loop, which
combines the products of the three "Z sections" in a number of user-selectable
ways.

That, in essence, is "Melodia". You can get everything from very straight single
spirals and standard-looking Julia thingies through some pretty wild stuff to
total chaos, depending on which functions are set to what in which section, and
how the sections themselves are combined by operators.

In reality the Z sections are more complex than this: There are a number of
user-settable numerical params as well as enumerated params in which the "mode"
of the variables can be selected. For example for "z" you can choose: (plain
vanilla) complex(z), |z|, atan2(z), real(z) or imag(z)--the same goes for the
"c". Also there is an "Initial" section in which the initial definition of z can
be chosen, and functions and params applied to it before it starts its journey
in the loop.

Variations on this theme (in the latter three of this series of formulae)
include one or two of those Z sections being such that the z variable is split
into real and imaginary components and those sub-sections combinable in various
ways applying functions and params. The result then becomes the initializing
variable for the section in question. In all instances where it is applicable
you can choose whether you want the Z section in question to be initialized just
with complex(z) or with that real/imag(z) product.

In addition in these three (Harmonia, Polyphonia and Rondo) a tweakable z is
added to the final section. These three formulae differ mainly by the presence
or absence of feedback loops: In "Harmonia" all three (Z1, Z2 and Z3) sections
start with the chosen initial z value. In "Polyphonia" the "Z1 Section" product
becomes the initializing variable for the Z3 section, and in "Rondo" Z1
becomes the initializing variable for both Z2 and Z3.

********Hmmmm....Did anybody make it this far-- I wouldn't have, I fear. I this
all gets too confusing I recommend just playing around a bit and coming back to
this for reference. In the comments section of each formula I include the
disclaimer that I did not try to simplify these formulae, in fact I aimed for
maximum flexibility. since there are an unconscionable number of numerical
parameters I have included a check box to hide them all. I suggest you do this
while exploring with the switch, then  use the numericals for "fine tuning" in
the Julia.

*The "Suite" series:*

As opposed to the first four formulae, the "Suites" all have four Z sections.
They differ from each other by virtue of the fact that all have fixed Z sections
with a real/imag(z) split, which is combined with a "c" variable in an "Accent"
section with many choices. Some have one re/im(z) section, some have two, and
the products of the sections are routed around to each other differently in the
different formulae. There are also choices for the order of the Z sections.
These differences are explained more fully in the comments at the head of each
formula.

IN GENERAL

I can generalize and say that in my experience the "Suite" formulae, like
Andrea's "Barney-2" tend to be at their best with more simple structures like
single spirals (for which all the defaults are set, anyway). the effects of the
real/imag(z) sections tends to give distorted and twisted elements when using a
trap coloring, but these distortions also tend to introduce "noise" which can
interfere with finer structures like Julia dragons, etc. This may well, however,
reflect my personal bias and I certainly don't discourage you from trying out
wild combos of functions, etc.

The first four formulae tend to be the straight sisters, usually introducing
less distortion than the Suites, but I have found that they are more amenable to
variations. In any case it is necessary to realize that because of the varying
positions and hierarchy of the variables in the loop, setting a certain
combination in one section will usually result in quite a different result than
setting it in another. Try setting the Z function of the various sections to
"sqr" one at a time, for example (using different operators to take it a step
further), and you will see what I mean.

I have tried to include a number of hints for the params and functions, though
not all. There are also a few tips in the comments sections of each.

I personally have despaired of ever getting any kind of an "overview" of how
these formulae behave. There is so much interactivity to the various
user-settable functions and params, and so many choices, that a systematic
exploration would be very difficult. If you are one of those folks who likes to
feel in control these formulae are probably not going to be your cup of tea.
Overall I've had a lot of fun personally exploring these, and my beta testers
(who I would like to thank for all their work) also seemed to enjoy them.
I hope you do too.


DEEP, DARK, FAT, SAVORY, SMOOTH, SPICY, SWEET, THICK BREWS

29 Aug 2002

This is the dreaded collection of 78 formulae formerly contained in the
"tma_pvt" folder. All these formulae are variations branching from one basic
idea, so there is some (and sometimes quite a bit of) similarity in their
behavior and resulting images. But there are also differences significant enough
that I and most of the beta testers thought them worth uploading as a package.
Much depends on the colorings used. Spirals that appear identical using a
general coloring like "Smoothed Iterations" will reveal interesting differences
when rendered with something like "Thingamajigs", but even then the differences
might not appear using certain combinations of functions. Anyway they are all
pretty user-friendly and there is lots to play with in there.

The formulae all use Dave Makin's "extended switch" function, so there are no
commensurate Julia formulae. Please see the explanation in the section below.

The formulae are named and numbered, i.e. Dark Brew 1, Dark Brew 2, etc.
Similarly named formulae bear close familial resemblances to each other, and the
1, 2, 3 etc. sets also have elements in common. Not all the named formulae in
any given set worked well, so there are gaps--some names do not exist in some
numbers.

Basically they work like my other "Brew" formulae--see below for more.

================================================================================

GOBLIN'S, OGRE'S, WIZARD'S, WARLOCK'S, DEMON'S BREWS

29 Aug 2002

These are all developments of "Witch's Brew", and while the results obtained
from them share some of the characteristics of that formula there appears to be
more flexibility in these new ones and a significant number of differences which
(I hope) makes their inclusion worthwhile. Here are a few tips, notes,
observations...

***NEW SWITCH FUNCTION*** With the kind help of Ken Childress and Joe Maddry,
formula wizards par excellence, I have incorporated Dave Makin's ingenious and
elegant "extended switch" function, which obviates the need for a second Julia
formula to activate the Julia switch. In fact you will note that there are no
Julias for these formula, but the switch function works normally. "Velly crever"
as the Japanese would say...The switch also functions from the Julia back to the
Mandy, carrying over all functions and parameters.

FUNCTIONS and PARAMETERS: As with all my formulae, these contain a humongous
number of functions and parameters. You can changes these either in the Mandy
(which will change what you see in the switch window as you explore), or after
you have chosen a Julia set. They work pretty much as they do in all my
formulae.

BAILOUT: I have set a bailout of 10, but it really does behoove you to try
different bailouts. In general the higher the bailout the more completely drawn
are the elements in the image. A low bailout tends to produce truncated elements
with sharp edges, higher bailouts allow the elements to be drawn more
completely, but depending on what you have chosen (in the Julia) a high bailout
can also obliterate the image entirely. Much depends on the Julia seed values
and what functions obtain.

IF YOU GET A LOT OF BLACK SPOTS: Sometimes when you increase the bailout, or use
certain functions in certain colorings, you will notice that the images tend to
get riddled with smaller or larger black spots or patches. One possible fix is
to try to decrease (or sometimes increase) the bailout. If that doesn't work
another remedy that sometimes works is to set the "inside" coloring exactly the
same as the "outside coloring" (including the gradient). This will often patch
up the black areas perfectly, and in fact even has some limited use in texturing
if you change the gradient. Credit to Rui Paraccho for turning me on to this.
Gracias.

OPERATORS: One of the joys (for me) of these particular formulae is changing the
operators. It works like this: since these formulae include equations with a
number of complex-variable expressions I have made it possible to changing the
arithmetic operators to vary the equations themselves. Varying the operators can
produce quite different Msets which give very different results in the Julia
sets, but not all combinations of operators work well together. As a rule of
thumb I have found that addition and subtraction operators tend to give more
simple spirals, whereas multiplication, division and exponential (when it
exists) operators make the spirals more complex.

BLACK M SETS: When setting operators you will sometimes run across black M sets.
It is worthwhile to try changing the "Start/Seed" parameter to 1 or 2 and this
might solve the problem. Or it might not.

GROUPING PARAMETER: These formulae contain a wide variety of different
"groupings" which simply refer to differing parenthization of the formulae.
"None" means no parentheses. "First2" means that the first and second variables
in the formula are enclosed in parentheses. "Second3" means that the second,
third and fourth variables are enclosed in parentheses, etc. Depending on the
operators chosen these will often be redundant, e.g. (a*b)*c is equivalent to
a*b*c, but (a-b)*c is not equivalent to a-b*c. So as you scroll through the
different groupings many will seem to be the same, but which are the same will
depend on the operator settings.

Hope you have fun with these.

================================================================================


SECRET BREW

22 Jun 2002

Here we take the Brews a step farther. This is by far the longest of my
formulae, with 12000 text lines, so please be a bit patient when loading and
using the switch. I call this one "Secret" because certain colorings and
parameters within the colorings seem to reveal complex elements created by all
the variables in this formula. With other colorings (and other parameter
settings within colorings) these elements remain hidden and the formula produces
simple spirals.

Secret Brew introduces a parameter called "Operator 4". It is worth changing
this in the Mandel formula before choosing a Julia with the switch. This
operator set at "^" produces some unusual spirals with discontinuous elements.
This is the default set in the Julia formula.

One thing worth noting is that several of the functions can be changed in the
Julia formula and will alter the spirals in interesting ways. Function 8 is
definitely worth scrolling through as it changes the spirals in fairly subtle
ways with almost all the function choices. Other functions in which at least
some of the choices change but do not destroy the spirals include Function 1 and
Function 3.

As with all my formulae, you can switch back to the Mandel formula from the
Julia without losing the parameter and function values of the latter. This is
very useful if you want to go back in from a Julia and explore further with the
switch, or reload a specific constellation of parameters from a .upr file.

================================================================================

BIG BREW

18 Jun 2002

So 102 wasn't the last brew...

This one seems like a very flexible formula. I had big trouble deciding on a
default parameter set because of the wealth of possibilities. In the end I chose
one which is user-friendly but not necessarily the most interesting. Following
is a listing of interesting sets out of the 625 different possibilities. Each
change of grouping and/or operators creates a new formula. I went through them
all quickly and wrote down the ones in which I found spirals or nice symmetric
elements. For each grouping I have listed sets of three arithmetic operators. In
each set the first operator corresponds to "Operator 1", the second to
"Operator 2" and the third to "Operator 3". Set these values in the respective
parameter fields in the "Formula" tab of "Big Brew", then explore the resulting
set with the switch.

Grouping 1: *+*  *+-  *-*  -*-  -*+  --/  --*  ---  --+  -+*  -+-  +/-  +*-  +-*
+--  +-+  ++-  +++  ++*  +^+  +^-

Grouping 2: -//  -/-  -/+  -*/  -*-  --*  ---  --+  -+*  -+-  -++  +/-  +*/  +*-
+-*  +--  +-^  ++*  ++-  +++

Grouping 3: *--  *+/  *+-  -*-  -*+  -**  -+1  -+*  -+-  -++  -+^  +/-  +*-  +-*
+--  +-^  ++*  ++-  +++  ^*-

Grouping 4: -*+  --+  --*  --^  ---  -+*  -+-  -++  -^+  +-*  ++^

Grouping 5: *--  *-+  *--  *+-  *++  -/-  -*/  -*-  -*+  -+/  -++  -^/  +//  +/-
+/+  +*/  +*-  +*+  +-/  +--  +-+  ++/  ++-  +++

Whew! but that's a lot less than 625...Other sets may also afford interesting
stuff sometimes, but these seemed the most friendly. BEAR IN MIND however, that
I only went through using the default functions. There are eight changeable
function fields with 29 functions so that gives 29^8 possible full sets of 625
equations. Changing the functions definitely changes the character of the forms,
and sometimes gives good results in sets not listed above. So there is plenty to
play with, and that's before you begin to increment the
parameters...

Please read the other "brew" hints and tips below for more info.

================================================================================


BREW 102

13 Jun 2002

Here is the last (I think) of the "Brew" family, named in honor of the late,
great beer that was once brewed in LA (well maybe not great, but definitely
late). Something from my childhood--I still remember the brewery downtown--
passing it on the Golden State Freeway, all factory-looking, with the yellow,
white and black sign proclaiming "Home of Brew 102". Here's to you...

It took me a while to road test this formula--many of the options are not
particularly fruitful--but I did find a very cool default set, and I must say
that this appears to be the most "plastic" of this genre in the way it pulls,
squeezes, stretches and deforms the elements of Dennis Magar's "Thingamajigs",
which is my default test coloring. I think that this may end up being the
"ne plus ultra" of the Brew family if you like weird spirals.

I strongly suggest that you explore changing functions 1-6 while exploring the
original formula with the switch. The functions change the areas where
interesting stuff can be found, but at the same time they also quite radically
change the nature of the spirals. Function 7 changes things rather too
drastically, but I'm leaving it in case someone wants to play with abstracts or
textures. It kills the spirals, but does do some interesting stuff.

Please read the other Brew tips below for hints and suggestions too.

================================================================================


WITCH'S BREW

05 June 2002

Here is another, more complex formula of the "Strange Brew" family. I have set
the defaults to produce rather tame, relatively symmetrical single spirals.
However if you are interested or masochistic you can find some really strange
stuff in here--bent and distorted spirals and all manner or weirdnesses. Try
setting grouping to "1", operator 1 to "*", operator 2 to "+", operator 3 to "^"
and all functions to "ident" and go from there. This was the original\default
setting and can be fun for asymmetry aficionados.

As the defaults stand the formula is very amenable to changes of function. If
you find a nice Julia set try scrolling through the various possibilities in
functions 1-8. Many will produce interesting variants of what you already have.

Other than this I suggest reading the section on "Strange Brew". The switch is
set to retain all parameters and functions when you switch back to "Witch's
Brew" from "Witch's Brew Julia". This is handy if you want to explore other
areas without having to reset all the parameters and functions. As with the
other brews, I've set the default drawing method to "Guessing" to speed the
rendering. I suggest setting this to "Multi-pass Linear" for the final render,
especially if you are using masking in your coloring.

================================================================================


NEW BREW

10 May 2002

As the name indicates another cousin of "Strange Brew". Nothing new here--read
"Cool Brew" below to get an idea. I've set the default drawing method to
"Guessing" to speed render times. It is a good idea to set this to "Multi-pass
Linear" for the final render.

================================================================================

COOL BREW

08 May 2002

This formula is a close relative of "Strange Brew" and behaves rather similarly.
It is not as user-friendly as many of my other formulae--the areas where spirals
appear using the Julia switch are limited and when the spirals do appear they
are quite small. Don't let that fool you. Of all of mine this is the one I would
choose if I were stranded on a desert island and could choose only one (along
with "Thingamajigs" of course). It produces a multitude of complex spiral forms
(in my searches at least) and often a multitude of similar spirals of various
sizes with varying degrees of distortion and stretching. I know that all the
functions and parameters are daunting but it is well worth changing functions
and operator and grouping parameters while exploring with the switch.

The switch is written so that you can switch back from the Julia to the Mandel
formula and retain all functions and parameters. This is helpful if you want to
re-explore a certain set.

You will find that some combinations of operators yield a black M set(especially
operator 1 set to "divide", and some combinations with operator 2 set to
"multiply", "divide" or "exponent"). This also depends somewhat on the
"grouping" function chosen (which changes the exponentiation within the
formula). If you encounter this try setting the "init z" value higher. This can
help. Even a black M set, however, can yield interesting results using the
switch.

Bailout values are especially critical with this formula--I've seen an
interesting spiral at bailout 4 disappear completely at bailout 5--but this
depends on large part on the colorings and functions chosen. By all means
experiment, both in the Mandel and Julia formulae. Any value between 2 and 200
might give good results.

This formula is not easy, but it seems veeery interesting. Good luck and happy
hunting.

================================================================================

SOMETHING ELSE

04 May 2002

Some info and helpful hints

"Something Else" is a tweak of a formula by Jos Leys called "Something New 4" I
was playing with it and exponentiated one expression and some interesting things
happened. I added a parameter and a couple of functions but otherwise it is
unchanged. I thank Jos for his kind permission allowing me to publish it.

Spirals (actually helices) seem to be the forte of this formula. In its default
state using the Julia switch it produces a continuum of lovely, self-contained
spiral forms, somewhat in the manner of the "Mighty Atom" formula. Spiral
aficionados should enjoy it.

An interesting characteristic is the grid of spirals produced in the Mset. You
need to set function 1 to default and zoom in but you should find them fairly
easily. Those points also produce spirals when used as Julia seeds.

As with my other formulae you can set the plethora of functions in the Mset and
then use the switch to find something you like in the Julia formula. Function 1
I would call the "main" function since it changes the behavior and
characteristics of the Mset and switch most radically. Functions 2 and 3 behave
similarly, and I would have left one off except that the effects are cumulative.
Functions 4,5 and 6 are also similar (if not identical—I’m not sure) in effect.
They seem to generally change the Mset the least, and can be scrolled through
after you have found something nice in the Julia formula to alter the
spirals/helices in interesting ways. The parameters 1-4 are as Jos wrote them,
and all seem to produce significant effects in the ways the spirals render. They
can effectively be altered after you have chosen a Julia set with the switch.

It is definitely worthwhile to alter the bailout—there is a wide variance in
effective values depending on the functions and colorings chosen.

Hope you have fun with this one.

================================================================================

MANDALA SERIES

07 Feb 2002

Here is a series of formulae similar in structure but with small changes I
believe affect  their individual behavior enough to warrant including all. In
any case they are not monsters with a lot of enumerated parameters so they do
not suffer from the inertia of my other formulae. I have given them Greek names
because it looks cool'n stuff...or something...

Basically each has a number of functions and parameters which affect the final
form of the fractal. I suggest you start with the various non-Julia formulae and
use the Julia switch to see what you can find. Change the various functions
before or during use of the switch. You can also try switching functions in the
Julia formulae, although many times you will lose your figure because the change
is too radical. But not always. Parameters can be incremented at any stage, but
are especially useful for fine-tuning in the Julia formula after you have chosen
something to work on.

The single most important parameter is "exponent1". Changing the real component
of  this parameter determines the number of times the Julia figure is repeated
in a circular fashion. You can also enter non-integers and you will get partial
"arms".

I have purposely left the number of iterations low on most of the formulae to
keep rendering times low. If you get black holes in the centers of spirals
increase this number to fill them. Often when you find small black holes
spread throughout the figure a higher number of iterations will get rid of them.

It is also worthwhile to try different bailouts, which will increase the
complexity of the figures. You might have to commensurately increase the maximum
iterations if the figure starts to break up.

Have fun and happy fractaling.
================================================================================
STRANGE BREW

14 Jan 2002

This formula is similar in structure to the Atom series below, although the
equation is different and yields quite different results. Please read the Atom
"suggestions and hints" for general guidelines.

Bailout note: You will find that different bailouts yield quite different
"flavors". Low bailouts (in the neighborhood of 4) give very clean-looking
results but depending on the coloring chosen can result in very truncated
elements in the spirals and Julia figures. As the bailouts increase the elements
are drawn more completely, often at the expense of the inclusion of lots of
"dust" and debris. Different functions affect this dramatically, so I suggest
that you check different bailouts to find what looks best to you. If you find
that your images tend to "break up" and get riddled with black holes as you
increase the bailout then setting the max. iterations higher might solve the
problem. This goes for the Atom formulae as well.

The parameters "eye of newt", "frog's breath", "bat wings" and "henbane" are
similar to "budge", "nudge" and "kludge" in the Atom series, affecting the
form, iterations and relationships of elements of the spirals (if spirals they
be).

Many thanks to Jos Leys and Rui Parracho whose knowledge and patience helped get
this formula in shape.

================================================================================

THE ATOM SERIES: SUGGESTIONS AND HINTS

Revised: 23 Dec 2001 (rev.1)

There have been some changes with the latest upload (24 Dec 2001) of Atom
formulae. They are as follows:

1--All present formula with the exception of the original "Mighty Atom"
have been given extra parameters--exponents 1, 2, 3 and sometimes 4.
They operate similarly to the "power" parameter but offer somewhat
different effects and finer control of how the spirals (or whatever  forms)
are affected. These changes in the formulae are backwards  compatible and SHOULD
NOT affect the rendering of your present uprs.

2--"Mighty Atom Mk II" has a new enumeration in the "grouping" parameter:
"middle".

3--A new set of formulae have been added, identical in name to the
first series but with a "z" tag at the end of the name (i.e. Grand Atom
01z, Mighty Atom MkIIz). The equations are the same as for the original
formulae with the exception that the "z" term is exponentiated to the
power of itself which changes their behavior quite dramatically. It
adds quite a bit of complexity to the Julia sets produced--something
like the present "Dragon" formulae. I have done some playing with them
and think the results justify their addition to the folder. You will
probably not find them generally as friendly as the non-z formulae, but
there is still plenty in there. Functionally they are identical to
their non-z counterparts with the exception of an additional function
which modifies the extra term. This becomes fn 6 or fn 7 depending on
the formula.

All these changes bring good news and bad news. The bad news is that
because if the sheer size of the tma folder (it now runs to some 37000
text lines) it takes some time to load, especially if you are running
an old box like I am. The good news is that there is no effect on the
rendering time of the fractals themselves, and now you have lots of new
toys to play with ;~)


I hope you enjoy them.
================================================================================
DISCLAIMER:
Mathematicians and fractalists are doubtless going to cringe at my meager
understanding of the subjects at hand. The following explanations may
well be inaccurate and will certainly display the gaps in my knowledge
about fractals and how they are created. People should extract whatever
information from the following will help them in the practical
application of the concerned formulae and ignore the rest.

I am always happy for any enlightenment that anyone might care to offer
and welcome comments or correspondence to: kymarto@yhc.att.ne.jp

Thanks,

Toby


INTRO:
This is intended to help people find their way through the various
functions and parameters of the series of formulae that I have called
the "Atom" series. The name derives from the form of the first fractals
I made with the core formula in another fractal generator, which
resembled the classic image of electrons spinning around a nucleus. The
formulae I have written for UF are widely expanded in functionality and
flexibility from that first one, but the images derived from them tend
to retain a characteristic look, so I have kept the name.

Spirals seem to be the forte of these formulae, and variations thereon.
My main testing has been done with Dennis Magar's "Thingamajigs", but
in brief exploration the Atoms seem quite friendly with many different
colorings. They are also quite amenable to formula tweaks, and I have
ended up incorporating numerous parameters and functions in each
formula to fully take advantage of this.

There are three (more or less) arbitrarily named classes of this
formula. "Mighty" is the simplest. "Grand" contains one or more extra
terms and seem to generally produce more complex spirals. "Dragon"
contains an added "z" term, which generally changes the form from
single spirals to more complex arrangements. Some of these resemble the
"dragon" from the classic Julia set, hence the name, but the variations
are manifold.

Also a new set of formulae with a "z" tag at the end of
the names have been added. They all behave somewhat like the "Dragon"
class because of the presence of an added "z" exponentiation in the
equations.

The added parameters and the various functions included in each formula
create many variations within each "class" of formula. Sometimes the
Dragons produce simple images, for instance, and depending on the
coloring chosen the spirals of one class sometimes closely resemble
those of another class. My suggestion is just to play and see what you
come up with.

**The original "Mighty Atom" formula has been retained for backwards
compatibility. It has been fixed so that the Julia switch works
properly. This should not affect existing uprs! If you do have a
problem please contact me at the above address and I will send you the
old formula. "Mighty Atom Mk II" is an improved version with a fix in
the power parameter and a few added parameters and a new function. I
suggest you use this from now on instead of the original formula.

HOW TO USE:

These formulae are all designed to be used with the Mandel-to-Julia
switch function. Opening any Atom (non-Julia form) will give you an
M-set that can be explored with the switch to find interesting forms
(usually spirals). Changing parameters and/or functions will change the
corresponding M set to a lesser or greater extent, and change the
location and characteristics of the Julia forms that you will find with
the switch.

After a bit you will learn to recognize where the Julia "forms"
generally occur in the M sets. Sometimes you will need to zoom out to
explore more territory, and then zoom in to the "rich" areas for finer
control of the switch.

FUNCTIONS AND PARAMETERS:

Functions and parameters tend to be interactive, so you can spend many
pleasant hours trying out various combinations both in the M set or
after you have chosen a Julia set ;~)

Following is a list of parameters and functions that will affect the
images and a rough explanation of what they do.

Functions 1-7: These change the functions (tan, sin, sqr, etc.)
modifying various terms of the formulae. Changing them often radically
affects the M set and the resulting Julia sets. Because they involve
major changes they are best tried in the M set before you have found a
Julia you like.  But it is worthwhile experimenting with them when you
have found a nice Julia as well. (I have found this especially true in
the Julia sets with the last two functions in any given equation
(either f4 and f5 or f5 and f6--fn7 exists only in the "z" formulas)).
Many of them will give you blank results but occasionally you will run
across something very nice. And since there are a discrete number of
possibilities it is easy to return to your starting point.(One useful
technique is to highlight the function and run through them quickly using
the keyboard arrow keys or mouse wheel).


Bailout: As with any formula this determines at what point the program
stops drawing the image. Too low a value results in truncated forms.
Too high a value can result in a lot of "clutter" or black spots or
holes in the figures. Often (but not always) inputting a lower bailout
will get rid of those. Changing the functions often changes the ideal
bailout, so it is worth experimenting. (n.b. Only the "real" value of
the bailout has an effect).

Seed: Determines the point on the two axes of the M set from which the
Julia set is derived. Changing it changes the characteristics of the
resulting Julia figure (i.e. tighter or looser spirals, different
shapes, etc.) or if you go too far obliterates it altogether. I have
set defaults for the Julia formulae that already form some sort of spirals.

Operators: These affect the mathematical relationships between various
terms (a-b, a*b, a^b, etc.) in the formulae. Often they have a major
effect on the map and are most effectively used in choosing an M set
map to explore with the switch. A few combinations of operators give a
blank M set, and some are more "fruitful" than others. It is generally
quite easy to see what effect any particular combination has by using
the switch. I have chosen defaults more or less at random, and it
behooves you to change them and play with different M sets.

Grouping: Affects the parenthezation (is that a word--) of the formula.
These Atom formulae have either no groupings or one or two different
groupings, which again work interactively. These usually have a major
effect on the map, though depending on combinations it is sometimes
minor. Occasionally you will see no change when changing between
grouping options (depending on the operators chosen). This is not as
difficult as it sounds: just change whatever you wish and use the
switch to see what you find. You can try changing operators again after
you have found a nice Julia. If it disappears just switch back...


Power, Exponents 1-4, Budge, Nudge, Kludge: These are values that
affect various terms in the equation. Changing them modifies the
spirals in various interesting ways: changing the number of iterations,
the form of the spiral, the relation of the spiral elements to each other, etc.
While these can also be changed in your initial exploration
of the M set, the fact that they can be incremented in small amounts
make them very useful (along with the seed parameter) in tweaking and
tailoring a basic Julia form. Try them and see. Each does something
slightly different, and the effect varies from formula to formula and
even within formulae depending on the options chosen.

A FEW MORE HINTS:

Since the default is set at 100 iterations, more tightly coiled spirals
often have a black hole at the center. Set the number of iterations
higher to cure this (at the expense of somewhat slower rendering).

Sometimes I have noticed that masked figures are not drawn
symmetrically, or seem to lack small parts. If you notice this, change
the "Drawing Method" from "Guessing" to "Multipass Linear". This
sometimes slows the rendering time noticeably, so you might consider
only doing this when rendering the final image.


Interesting figures are often bordered by areas past the escape value
(at least that's what I think they are) which appear as discrete holes.
A higher bailout, if not otherwise detrimental, will sometimes serve to
shrink (but never eliminate) these. While it is not possible to fill
them, they can effectively be masked out by certain colorings with a
masking function.

Good luck and happy fractaling. I wish you as much fun with these as I
am having.

My deep thanks to Ken Childress, Erik Nathan Reckase, Mark Townsend,
Frederik Slijkerman, Jos Leys and the hardy beta testers for their
time, help and suggestions.