 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.