Attractor
Traps Tutorial for UltraFractal - Page 1
Introduction
This tutorial is
intended to provide examples and explanations to aid the user in the creation of
fractal art using the formula Attractor Traps. This is a general coloring formula that uses strange
attractors as the trap functions. There are over 200 trap functions
available and 9 trap variants for each. Some of the same attractor traps
are used in the formulas described in the Enhanced
Traps Formulas Tutorial, but the implementation of the traps in those
formulas is different.
In the expert mode there are an almost unlimited
combination of options for modifying how a given trap functions and the
image appears. It assumes that the user is
familiar with the Ultra Fractal program and has some understanding of fractal
theory.
The Attractor Trap formula include ideas from Damien Jones, Mark
Townsend, Toby Marshall and Michèle Dessureault. Three of the attractor types,
Sprott3D, Sprott3D_ODE and SprottQuad, are polynomial attractors based upon the
work of Julian Sprott. A publicly available BASIC formula of Julian Sprott was
used to generate the coefficients for the polynomials. Attractor Traps was designed to provide multiple texture, coloring, and shape options for the user.
Only the parameters which are visible when the expert
mode box is unchecked are listed below.
- Expert mode - Because of the large number of
parameters available, the more advance parameters are hidden unless the
Expert mode box is checked.
The following parameters are in the General Parameters section:
- Trap mode: These are the modes for trapping
an orbit during iteration of the fractal formula. They are:
- Closest - The trapped value is the orbit
value closest to the trap function and is less than the trap width.
- First - The trapped
value is the first one that has a trap distance less than the trap width. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Last - The trapped
value is the last orbit value that has a trap distance that is less than
the trap width. If no value of the orbit is less than the trap width,
the pixel is colored with a solid color.
- Smallest - The
trapped value the orbit value with the smallest absolute value.
- Average - A running
average of the distance between the orbit value and the trap function is
calculated. The trapped value is the minimum average value.
- Farthest - The trapped value is the
greatest distance that is less than the trap width. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Sum - This trap mode is like First,
except that the trapped distance values are summed. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum - This trap mode is like First,
except that the trapped distance values are a weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum 2 - This trap mode is like First,
except that the trapped distance values are a second variant of a
weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum 3 - This trap mode is like First,
except that the trapped distance values are a third variant of a
weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Sum 2 - This trap mode is like Last,
except that the trapped distance values are summed. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum 4 - This trap mode is like Last,
except that the trapped distance values are a weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum 5 - This trap mode is like Last,
except that the trapped distance values are a second variant of a
weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Weighted Sum 6 - This trap mode is like First,
except that the trapped distance values are a third variant of a
weighted sum. If
no value of the orbit is less than the trap width, the pixel is colored
with a solid color.
- Coloring Mode
- The coloring mode determines how the color is calculated.
- Distance: The color
index is proportional to the distance between the orbit value and the
trap function
- Iteration: The color
index is proportional to the number of iterations to reach the trapped
orbit. This has two options.
- Normal - This is the
"classical" iteration mode used by most trap formulas
- Discrete colors - This is a special
mode that takes the modulus of the iteration value and colors
accordingly. The modulus is an additional integer parameter, called #
of iteration colors.
- Magnitude: The color
index is proportional to the absolute magnitude of the trapped orbit.
- Real: The color
index is proportional to the absolute magnitude of the real component of
the trapped orbit.
- Imaginary: The color
index is proportional to the absolute magnitude of the imaginary
component of the trapped orbit.
- Angle: The color
index is proportional to the angle of the trapped orbit.
- Trap Magnitude: The
color index is proportional to the absolute magnitude of the trap
function for the trapped orbit.
- Trap Real: The color
index is proportional to the absolute magnitude of the real component of
trap function for the trapped orbit.
- Trap Imaginary: The
color index is proportional to the absolute magnitude of the imaginary
component of trap function for the trapped orbit.
- Trap Angle: The
color index is proportional to the angle of trap function for the
trapped orbit.
- Exp Iter: The color
index is proportional to the product of the trap magnitude and the
exponentially smoothed value of the trap function.
- Exp Iter 2: The
color index is proportional to the exponentially smoothed value of the
trap function.
- Trap variants - This parameter has a pull
down list with 9 options which change how the final distance is calculated.
- Trap modifier - Modifies the imaginary
component of z before any trap calculations
- Trap width - This parameter sets the distance between the orbit and the trap function that is used for
calculating which pixel should be colored with a solid color or the trap
color. This is used automatically for trap modes for all modes except Closest
and Smallest.
- Distance modifier - Modifies the distance
calculated from the trap.
- Solid background - This is a check box which
is displayed for all modes except Closest and Smallest.
- Trapping mask - For trap modes Closest
and
Smallest this parameter is visible. It has a pull down list with
three options: None, Normal and
Reverse for mask usage
The next set of parameters are in the Trap Parameters section.
- Trap Type - There are 15 trap types.
- Chip - The Chip attractor is used as the trapping function.
There are three Chip parameters.
- Chip parameter #1 - Affects the shape
- Chip parameter #2 - Affects the shape
- Chip parameter #3 - Affects the shape
- CosMartin - The CosMartin attractor is used as the trapping function.
There is one CosMartin parameter.
- CosMartin parameter - Affects the
shape
- Henon - The Henon attractor is used as the trapping function.
There is one Henon parameter.
- Henon parameter - Affects the shape
- Hopalong - The Hopalong attractor is used as the trapping function.
There are three Hopalong parameters.
- Hopalong parameter #1 - Affects the
shape
- Hopalong parameter #2 - Affects the
shape
- Hopalong parameter #3 - Affects the
shape
- Gingerbread - The Gingerbread attractor is used as the trapping
function.
- Latoocarfian - The Latoocarfian attractor is used as the trapping
function. There are four Latoocarfian parameters.
- Latoocarfian parameter #1 - Affects
the shape
- Latoocarfian parameter #2 - Affects
the shape
- Latoocarfian parameter #3 - Affects
the shape
- Latoocarfian parameter #4 - Affects
the shape
- Liar - The Liar attractor is used as the trapping function.
There are four Liar parameters.
- Liar Type - There are three Liar
types in a drop down list - real, imag and abs.
- Liar parameter #1 - Affects the shape
- Liar parameter #2 - Affects the shape
- Liar parameter #3 - Affects the shape
- Pickover - The Pickover attractor, which is a 3D attractor, is used as
the trapping function. There are 12 Pickover parameters.
- Initialize Height - Initializes the
starting value for the attractor 3rd dimension.
- Height weight - Weighting function
for the attractor 3rd dimension value applied to #z. It is applied
after the attractor iterations.
- Pickover #1 - Affects the shape
- Pickover #2 - Affects the shape
- Pickover #3 - Affects the shape
- Pickover #4 - Affects the shape
- Pickover #5 - Affects the shape
- Pickover Fn #1 - Affects the shape
- Pickover Fn #2 - Affects the shape
- Pickover Fn #3 - Affects the shape
- Pickover Fn #4 - Affects the shape
- Pickover Fn #5 - Affects the shape
- Quadruptwo - The Quadruptwo attractor is used as the trapping
function.
- Quadruptwo parameter #1 - Affects the shape
- Quadruptwo parameter #2 - Affects the shape
- Quadruptwo parameter #3 - Affects the shape
- Sierpinski - The Sierpinski attractor is used as the trapping
function.
- Sierpinski parameter #1 - Affects
the shape
- Sierpinski parameter #2 - Affects
the shape
- Sierpinski parameter #3 - Affects
the shape
- Sierpinski parameter #4 - Affects
the shape
- Sprott3D - The Sprott3D attractor, which is a 3D quadratic attractor
with 50 choices for the attractor, is used as the trapping
function.
- Sprott selector - Selects a Sprott
attractor
- Initialize Height - Initializes the
starting value for the attractor 3rd dimension.
- Height weight - Weighting function
for the attractor 3rd dimension value applied to #z. It is applied
after the attractor iterations.
- Sprott3D_ODE - The Sprott3D_ODE attractor, which is an attractor
resulting from the solution of 3D quadratic differential equations and
has 50 choices for the attractor, is used as the trapping
function.
- Sprott selector - Selects a Sprott
attractor
- Initialize Height - Initializes the
starting value for the attractor 3rd dimension.
- Height Weight - Weighting function
for the attractor 3rd dimension value applied to #z. It is applied
after the attractor iterations.
- SprottQuad - The SprottQuad attractor, which is a 2D quadratic attractor
with 100 choices for the attractor, is used as the trapping function.
- Sprott selector - Selects a Sprott
attractor
- Threeply - The Threeply attractor is used as the trapping function.
- Threeply parameter #1 - Affects the shape
- Threeply parameter #2 - Affects the shape
- Threeply parameter #3 - Affects the shape
- Zito - The Zito attractor is used as the trapping
function.
- Zito parameter - Affects the shape
- Distance scale - Modifies the trap distance
calculation
- Attractor scale - Modifies the magnitude of
the attractor value
- Attractor iterations - Number of attractor
iterations before the trap calculations
- Attractor pre-func - Applied to #z before the
attractor iterations
- Attractor post-func - Applied to #z
after the attractor iterations
- Trap rotation - Rotation of the trap in
degrees.
- Trap skew - Skew of the trap in degrees.
- Start Offset - Sets the offset for #z
- Trap Offset - Sets the
offset for the trap function calculation. For
other values, start small and increase the value to see the effect.
- Move Trap Offset - This moves the trap offset
relative to the pixel value being calculated. If this is checked, a new
parameter becomes visible, called Move Amount,
which determines the relative move.