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Fractal/!Fractal/Help/!Functions/Readme
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________________________________________________________________________
/ \
| !Fractal : General Fractal Plotter |
| |
| Fractal Function Help |
| |
| March 1992 |
\________________________________________________________________________/
Introduction
============
This help file covers the various fractal and 3d transformers available
within !Fractal and their associated parameters and menu options. Note: the
3d_Plane, 3d_View, Riemannn and Render options are not fractal functions,
but take a fractal image and perform a 3d transformation. They are described
after the fractal functions.
The algorithms where shown usually give only the core maths, omitting things
like the scaling needed to handle zooming.
Most functions have a set of variables which are accessed from the
Image->Data menu option or the Numbers tool icon. Normally the first four
variables are for controlling the zooming and image positioning.
--------------------------------------------------------------------------
Bifurcate
---------
!Fractal would not be complete without bifurcation diagrams, one of the
original manifestations of chaos. They were originally discovered from a
very simple equation to describe animal populations:
New Population = Growth Rate * Old Population * ( 1 - Old Population)
where population is a value between 0 and 1. With growth rates < 200% a
single line results, over 200% it splits (bifurcates) into 2, then 4, then
becomes chaotic.
In the plot the x-axis is the growth rate, the y-axis is the population.
Where multiple points are plotted at the same pixel, the colour is
decremented. The data values are:
Growth Rate: value at x0.
Pop. Size : value at y0.
Rate Range : add to x0 to give growth rate at far right.
Pop. Range : add to y0 to give population at top of screen.
Init Popn. : The inital value of the population used in the equation.
Max Gen. : Number of generations to calculate for each growth rate.
Min Gen. : Number of generations before plotting.
The Min Gen value is used as a low filter to let the population stabilise
before plotting, whilst the Max Gen is the total number of generations to
calculate before moving onto the next growth rate.
You can zoom in to examine the chaotic patterns, which often reveal detail.
Increase Max Gen as you zoom in to see the range of population values.
!Fractal switches from fast 32 bit to full floating point calculations as
you zoom in.
There are many formulae that lead to bifurcation diagrams - some of these
will be included at a later date. Mitchel Feigenbaum discovered that the
ratios of lengths of adjacent areas of bifurcation were always 4.6692..., a
constant that is now named afer him.
--------------------------------------------------------------------------
Fault
-----
Generates a landscape by a very simple simulation of geological faulting.
Each time a fault occurs, a part of the landscape is moved up (or down). By
repeating the process hundreds or thousands of times at random locations
across the image, an uneven terrain is created. The portion of the terrain
faulted is determined by the fault type. These are based on various shapes
as selected by Fault->Type.
The data parameters are:
Count: the number of faults to be run before stopping. Higher numbers
produce more realistic and uneven shapes.
Shape Size: The size of the fault shape in OS units. Larger sizes tend to
produce more generalised faults but may lead to obvious boundary lines which
can look unrealistic. Smaller shapes give more uneven, bumpy landscapes.
Sizes of 1/5th the total image give a good balance between smoothness and
variety. Larger shapes will take longer to plot.
Step: The maximum height increment for each fault. The maximum height that
can result is 255 unless Fault->Wrap is turned on in which case heights wrap
around to 0. For 3d shapes such as Cone, the Step represents the maximum
height (usually the centre). Not used for Sprites.
Seed: The random number used as a seed to generate the landscape. To
recreate a landscape, set Fault->Random off and set the seed value. This
allows the effects of different shapes, sizes and step values to be
evaluated.
From the Fault menu the following options are selectable:
Type: leads to a submenu showing the type of fault to use. The first set are
shapes, the size of which is set from the data value Shape Size, and maximum
height by Step.
Circle : a circle of radius Shape Size/2.
Cone : a conic rising evenly in height (max Step) to the centre.
Hexagon : a hexagon of radius Size/2.
Octagon : an octagon of radius Size/2.
Octagon 3d: a 3d octagon rising in height to the centre.
Sprites : leads to a panel listing a sprite shape to be used as the
fault pattern. Click on Sprite to use this feature, and click
on the name of the sprite to use.
Random: when on generates a random landscape. Turn off to recreate a
landscape using the Seed value.
Superimpose: when on causes the faults to be applied to the existing image.
This allows different shapes, sizes and step values to be cumulatively
applied. When off, the image will be cleared on redraw.
Wrap: when on allows height values to wrap around from 255 to 0, otherwise
values peak at 255.
Using Sprite Fault Shapes
-------------------------
Any shape can be used as the fault pattern. The colour values of the sprite
are used as the fault height, whilst the overall size of the sprite is set
by Shape Size as for other shapes. The supplied sprites are in the file
FaultSpr.
You can experiment with your own shapes by dragging a sprite file onto any
of !Fractal's windows whilst Fault is selected. The sprites must be defined
in a 256 colour mode - 13 and 21 are best since the pixels are square. Each
sprite should be square in terms of OS units, but the pattern inside can be
any shape.
It is best to draw the pattern in distinct colours since the low colour
values are difficult to see. When you are happy with the shape, use !Paint's
colour replace option (the bucket!) to set the colours to the height values
(low colour numbers). Typically these would be in the range 0-20. Circular
patterns tend to give the most realistic landscape effect, but feel free to
experiment since strange patterns can be produced with linear shapes. You
can think of Fault as a special type of Tiling mechanism.
Hints And Tips
--------------
The resulting image looks best with the Landscape display palette. If the
image is too low you'll just see blue, so use the Palette adjust facility
(striped Tool icon) to alter the mapping until you get a mixture of sea and
land. Alternatively load your own customised palette made with !FracPal. For
instance, a palette of various shades of white, grey and light blue give a
very realistic cloud effect.
To view the landscape in 3d try 3d_View or Render using either a Log or
Linear 3d mapping (in the 3d panel). Use the 3d Scalar value to accentuate
or smooth the image. Again, select the Landscape palette. Try the Riemann
plot to create your own globe.
When using the Superimpose facility it is best to start with a large shape
size, running for several hundred faults, to generate the basic rolling
terrain. Then use a smaller size, increase the number of faults and step
size to give more localised variation and peakiness. The best shapes are the
Cone and Octagon 3d - for these you'll need a relatively larger size than
Circle because of the gradual rise in height. Try mixing shapes using
Superimpose to create different effects. Using large shape sizes can also
produce interesting patterns.
Algorithm
---------
Faulting by shape simply involves selecting a point at random and then
plotting the shape centred on that spot. The value of each shape pixel is
added to the image pixel value - if Fault->Wrap is off, then the maximum
value of 255 is set, otherwise the value wraps around.
Watch This Space: A lot more functionality will be added, such as linear
faulting and fault programming!.
--------------------------------------------------------------------------
H�non Maps : An implementation of H�non's strange attractor.
----------
Two sub-functions are provided. The Original function shows a simple orbit,
whilst the SIN/COS function is a fuller implementation. They both relate to
orbits of asteroids and so on. The menu options are greyed when not
applicable to a particular function. The menu options are:
Original: selects the basic function. This function is infinite so you will
need to manually stop it. See Chaos by J.Gleick p.141 for full details.
SIN(a)/COS(a): selects the full function. See J.Dewdney, Sc.American, July
1987.
Random a: The value of a is chosen at random by the program when ticked.
Random b: The value of a is chosen at random by the program when ticked.
Auto Step: When on, the number of points calculated will be increased as you
zoom in, to give greater resolution, though this will take longer.
Random Colour: when on the colour will be changed randomly every 100 points.
Original Algorithm:
------------------
The default value for a is 1.4 and b is 0.3. The best results come from
slightly altering these values rather than choosing random figures.
px=1
py=1
do
z=px
px=py+1-a*z*z
py=b*z
plot px,py
until (x>1.0e10 or y>1.0e10)
SIN(a)/COS(a) Algorithm:
-----------------------
Here the main control is the data variable a. The step rate is decreased
when zooming in. Thanks to Joyce Haslam for passing on this algorithm.
for i=-0.8 to 0.8 step 0.05
for j=-0.8 to 0.8 step 0.05
c=0
x=i
y=j
repeat
c+=1
xx=x*COSa-(y-x*x)*SINa
y=x*SINa+(y-x*x)*COSa
x=xx
plot x,y
until (i>100 or x>1000 or y>1000)
next
next
--------------------------------------------------------------------------
IFS 2d: An implementation of Michael Barnsley's IFS system in 2 dimensions.
------
Up to 40 IFS transformations are stored internally, one of which is the
working IFS. From the IFS menu the options are:
Data: The current IFS data except for the zooming co-ordinates which are
accessed via the standard Image�Data� menu. Click Input to enter new values,
or Re-draw to enter and re-draw the image. Click Store to store the data in
the internal file, or Restore to revert to the version held internally.
Click Copy to take a copy of the current data and create a new IFS function,
or click Delete to delete the current entry.
The data on display consists of:
Name: used to identify the IFS function, any 1-11 characters.
Number: location within the internal IFS file. Click on the arrows to access
prior or next entries.
Transformations: number of affine transforms to use (1-9). If the sum of the
probabilities is less than 1, this number is automatically increased by 1
and a null transformation is added.
Iterations: The number of iterations before stopping image generation.
The affine data is shown according to its place in the affine function, not
as a,b,c... as shown in some programs, so be careful when porting values.
The sum of the probabilties must be >=1. If not, an extra null
transformation is added. To move around the grid use !Fractal's standard
editing keys. Numbers may be entered in scientific format. Remember to click
Input or Re-draw afterwards to get the new values processed.
List: shows a panel listing all the internal IFS functions available. Click
over a name to make it the current one. Most of these came from Fractint, so
see there for credits.
Save: allows the internal IFS file to be saved as a text file. Saved files
can be re-loaded by dragging onto the !Fractal icon or window. Please send
in exciting new IFS functions to AFG so that we may all see them.
Algorithm
---------
The algorithm to generate each point is essential as shown on the input panel,
ie:
r=random number from 0 to 1
select affine transformation whose probability sum is just less than r
x=a*x+b*y+c
y=d*x+e*y+f
plot x,y
where a..f refer to the columns in the data input panel left to right.
--------------------------------------------------------------------------
Julia Set
---------
Plots the Julia set corresponding to the equivalent Mandelbrot function.
The initial zoom uses integer maths, then reverts to to a rather slower
floating point. From the menu you can switch to the equivalent Mandelbrot -
the current Real and Imaginary values are used to set the Mandelbrot x & y
value for the centre of the image. Other options are as described under
Mandelbrot.
It is usually best to select a Julia from the equivalent Mandelbrot display.
Trying to set cReal and cImaginary otherwise can be a slow process to get
reasonable results. When switching to the Quaternion set, Quaternion q0 is
set to cReal & q2 to cImaginary.
When using functions other than Square, floating point is used much sooner
so as to maintain accuracy.
Algorithm: the Julia algorithm is essentially the same as for Mandelbrot,
but with a different start point, namely:
u=x
v=y
x=cReal
y=cImaginary
--------------------------------------------------------------------------
L-System : Turtle graphics as used for drawing plants etc.
--------
The L-system uses a LOGO like turtle language to describe shapes as a series
of movements, and was introduced by A. Lindenmayer to help describe plants.
Refer to the book The Algorithmic Beauty of Plants for full details of the
work by Prusinkiewicz and Lindenmayer. Up to 80 definitions are stored
internally which can be saved as a text file, or a new file may be loaded.
Each shape definition is made up of the following components:
Name: 1-19 character name for identification.
Iterations: the number of times the algorithm is recursively expanded.
Angles: Rather than describing the amount of each turn as so many degrees,
each turn is expressed as a fraction of a circle. Thus 6=60� (360�/6). The
number should be even if you plan to use the '|' command to allow turns of
180�.
Axiom: a string of 1-31 characters incorporating turtle commands and rules.
Rules: a single letter identifier and 0-79 characters consisting of further
rules or turtle commands. Note that the case of letters is significant.
The routine starts by taking the axiom string and then replaces each rule
character in it by the equivalent rule string. This process is repeated
for the number of iterations specified.
Thus if the axiom is F-F++F-F, after the first expansion we get :
F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F
The string is in then interpreted as a series of turtle commands, any rule
characters being ignored.
The key to the routine is this recursive expansion, and often the use of
F as a rule and turtle command (which is somewhat confusing I admit, but
very important).
Drawing
-------
Because the resulting size of the image is unknown at the outset, the plot
has to be repeated, with the image being correctly scaled to fit on the
screen on the second pass. However you can use zooming, in which case only a
single plot is performed.
Use Adjust to stop a plot. Note that currently it is not possible to restart
a plot after Adjust is pressed. You are advised to start at low iterations
and work up.
Basic Turtle Commands
---------------------
The characters used are those in standard use. Note that 'F' is different
from 'f'. (Fractint uses 'G' for 'f' since it is case insensitive).
F : draw a line forwards in the current direction.
D : same as F, ie. draw a line forwards.
f : move forwards without drawing a line.
- : turn anti-clockwise 1 angle (see definition of angle above).
+ : turn clockwise 1 angle.
| : turn 180�. Note - angle must be even for this to work correctly.
Advanced Turtle Commands
------------------------
To get more complex displays requires the use of one or more these commands.
[ : stack the current position, angle, line length, width and colour.
] : unstack the turtle details.
! : reverse meaning of '+' and '-'.
>n : increment colour by n where n is one or more digits.
<n : decrement colour by n.
@n : multiply the line length by n. The initial length is 1. 'n' is 1 or
more decimal digits and may contain a decimal place.
To divide precede the number by I (for inverse). eg. @I3 divides by 3,
or use @0.333.
Precede the number by 'Q' for the square root, eg. @Q2 = length * 1.414
#n : Multiply the line width by n - the initial width is a single pixel.
Numbers may be specified as for line length.
{sprite} : Plots a sprite of this name at the current position and rotation.
The sprite is plotted the same size as the current line size.
See below for full details.
{sprite,n} : As above but plots at the scale specified as 'n'.
Colour
------
The initial colour is 255 (white). Colour numbers are made up of the bits
BBGGRRTT where B=Blue, G=Green, R=Red, and T=Tint (amount of white). Thus
adding 64 each time increments the blue, 16 the green, and 4 the amount of
red. This colour scheme is the same as BASIC's COLOUR*4+TINT.
If the colour exceeds 255 it wraps around. Thus putting >16 in the axiom
sets the colour to maximum red (255+16-256=15).
Menu Options
------------
There are 4 menu options, providing data entry, selection of the
L-system to use, saving the internal file of data, and opening of the
current L-System data directory.
Data Entry Panel
----------------
Use this panel to update the L-system data, or select a new L-system. To
move the caret around the icons use !Fractal's standard editing keys. The
various icons are:
Input: You must click on this for your changes to take effect.
Re-draw: This is the same as Input followed by a re-draw request.
Store: This takes the data and stores it in the internal file. If you move
to another L-system without using this option your updates will be lost!
Restore: Refreshes the data on display from the copy in the internal file.
This lets you back out changes without having to re-load the file.
Copy: Takes a copy of the current L-system so that you can edit it without
modifying the original. You should give the copy a new name to avoid
confusion.
Delete: Deletes the current L-system from the internal file.
Name: the 1-19 character identifier for this L-system.
Item: click on the left or right arrows to move to a different L-system.
Remember to use Store if you want to keep any changes to the current
L-system.
Iterations: Use the arrows to change or click on the number to key in a new
value. Higher iterations give more complex patterns but will take much
longer to draw.
Angle: determines the amount of turn for '+' and '-'.
Axiom: the 1-31 character string that is the initial rule.
Rules: Each rule must have a unique character for identification. Up to 9
rules may be use, each of 0-79 characters. A null rule can be used to strip
out turtle commands during expansion (see Cesaro for example). Remember to
click Input or Redraw to enter your values.
L-System List Panel
-------------------
This panel lists all the L-systems that are held internally, showing the
first 11 characters of their name with current name being highlighted. Click
on a name to select one.
Save
----
The internal file of L-system definitions can be saved as a text file by
dragging the text icon to a filer window. The associated sprites will be
saved in the same directory if not already present. Saved definitions can be
reloaded at any time - !Fractal will automatically select L-system if it is
not the current fractal. Two files of definitions are supplied - click on
L-System->Directory to open a Filer window showing the files available.
You can edit the text file, but be careful to keep to the format which is:
L-System
sprite filename
l-system name } for each
default iterations,angles,number of rules } L-System
axiom }
ruleid=rule text 1 for each rule }
The supplied definitions come from the Science of Fractals, Algorithmic
Beauty of Plants, Fractint, and John Greening. Please send in more functions
to AFG so we can all see them. I have added colour and sprites to some of
the shapes.
Hints And Tips On Creating L-Systems
------------------------------------
To get an idea how L-systems work, look at the shapes prefixed 'Sier' and
start at low iteration values. Note that shapes grow because a rule string
contains more than one rule itself.
A good way to trace how a shape is built up is to add a colour change at the
start or end of a rule, ie. add a >4 or >1. The different colours in the
result show where a new copy of the rule was expanded. Try the colour
cycling on this as well!
Shapes that are made up of lots of smaller copies of themselves are created
by making the rules recursive, that is the rule string contains the same
letter as the rule name (eg. F=FF) or a subsequent rule refers back to the
first one (eg. F=AB, A=FB).
To set the initial direction, width or colour, place the required turtle
commands in the Axiom. The initial direction is to the right so use '-' to
make a shape start plotting upwards. That way they are only executed once -
see Bush3 for an example.
The stack is a very powerful feature and worth experimenting with - it is
the key to shapes that feature points, such as the plant examples. It also
allows you to create a sub-shape with its own colour, size and width which
may be placed anywhere within the main pattern. See Tree3 which uses 'L' for
the Leaf rule. Typically these sub-shapes are placed at the end of a stacked
sequence - PlantLeaf uses [--L] to place a leaf at the tip at particular
points.
Two draw commands (F and D) are provided so that one may be expanded (by
making the command a rule) and the other unexpanded. Tree3 uses this to
allow the leaves to be drawn the same size. If the L rule used 'F' for
drawing, the 'F' rule would expand making the leaf larger depending on where
it was drawn - try it and see.
The length command (@) is generally used to make successive parts of the
shape relatively smaller. See Tree1 for example which uses it to make the
branches smaller and smaller. A similar effect can be achieve by using a
rule of the form F=FF - at higher iterations each F character is expanded
into more F draw commands.
Use the width command (#) to give body to plant and tree shapes for extra
reality.
Sprites
-------
The sprite plotting feature is a powerful extension to the normal command
syntax. Instead of using F to draw a line specify the name of a sprite, which
will be plotted at the same length as the line and at the same rotation. To
avoid long rule lines it can be useful to use a rule for a sprite eg.
P=+T-TT++T
T={trunk}
Plotting with sprites will take longer, and it is often better when creating
a new shape to start off with the outline, then substitute the sprites at a
later time.
Creating Sprites
----------------
The following guidelines must be followed for plotting to work correctly.
The sprites are contained in a sprite file in the same directory as the
L-system definitions. The name of the sprite file is specified in the second
line of the L-system definition.
The supplied sprite file is called "L-Sprites" - click on
L-System->Directory to locate. Look at the example sprites provided to get
an idea of the requirements. Each sprite must be:
1) The sprite must be square and should be a multiple of 4 pixels in size.
2) A 256 colour must be chosen that has square pixels, such as mode 21 (or
mode 13).
3) The sprite should be created with a mask. All pixels that are not to be
plotted should be set to the mask colour so these pixels become transparent.
4) When creating the shape bear in mind that top is to the right since that
is the initial direction of plotting. You may find it easier to create the
shape and rotate it 90� clockwise.
5) Since the sprite gets rotated, pixels in the corners will tend to get
lost at angles other than 0,90,180, & 270 degrees. To avoid black space
however, you will normally need to set pixels up to the edge of the sprite.
The above rules are necessary because of the scaling and rotation that is
performed. If you get strange results, check that you have followed these
guidelines. The main limitation on the size of sprites is the amount of
memory in your computer. Large spries will soon consume all that is
available. Sizes from 16-64 pixels should be adequate for most purposes.
Larger sprites will tend to look better when scaled and rotated - small
sprites will look more distorted. To maintain clarity try to avoid too much
detail in the sprite.
To reload a sprite file after editing, drag the corresponding text file onto a !Fractal window or its icon.
Using Sprites
-------------
The sprite name is specified by {spritename} in a rule definition. If the
sprite cannot be found, it is ignored. The sprites are displayed at the
current line scale no matter how large the sprite is. To plot the sprite at
a different relative size use {spritename,9.9} where '9.9' is the scaling
factor.
The size of sprites is also controlled by the line length specifier '@' -
this allows sprites to be plotted at increasingly smaller (or larger) sizes
which is useful for plant like shapes.
Sprites can be used with the all the other plotting commands, though the
colour and width commands do not affect the sprite. Interesting effects can
be achieved by taking the shapes provided and replacing the 'F' commands by
a sprite. To do this it will be necessary to preserve the recursive nature.
Thus if the original rule was F=F+F-, then the replacement rule would need
to be S=S{sprite}+S{sprite}-.
To avoid out of memory conditions it is best to use a number of L-system
definition files with their own sprite files. To create a new file delete
the entries not required, and save the remaining definitions under a new
filename. Then edit the text file to specify a new sprite file name.
--------------------------------------------------------------------------
Lorenz : The Edward Lorenz Strange Attractor.
------
This was probably the first fractal to be recognised as such, and started
the ball rolling for chaos investigation. The shape is three dimensional and
you can use !Fractal's rotation and elevation icons to control the view
position. There are three variables which may be set or randomly chosen, and
there are three colour schemes. Refer to Chaos by J.Gleick for a description
of the story behind its discovery.
The plot is infinite. You can control the speed by setting the Step Rate
factor - smaller values will slow the plot so you can trace its shape.
Notice that the plot revolves around one centre and then suddenly jumps
across to the other. Try altering the default values of a/b/c by small
amounts and see the effect this has on the shape and when the jump occurs.
The menu options are:
Random a/b/c: Click to set/unset random selection of the variable when a
plot is started. For each variable there is an associated maximum, which can
be changed from the data panel. New variables are not chosen when zooming.
Once you have found an interesting shape, turn off random variables, and use
the 3d options to explore the shape.
Random colour: when set a random colour is chosen after every 100 plots.
Step Colour: the colour value is increased after every 100 plots.
If neither Random or Step colour is set, then the plot is made in white
using exclusive or'ing to trace the current position.
You can change the viewpoint whilst plotting by pressing the cursor keys,
and then pressing the space bar to restart. See the 3d section in the
!MainHelp file for more details. Try Step Rate=2.0e-2 for the best use of
this function - note that bigger values destroy the shape.
Algorithm
---------
x=1; y=1; z=1;
do
dx=a*(py-px)
dy=px*(b-pz)-py
dz=px*py-c*pz
px+=dx*step_rate
py+=dy*step_rate
pz+=dz*step_rate
draw px,pz where px/pz are 3d transforms of x/y/z
forever
--------------------------------------------------------------------------
Mandelbrot
----------
The classic fractal, supplied here in 8 different methods. Thanks to Joyce
Haslam for providing the enhanced versions.
The initial zoom uses integer maths, then reverts to to a rather slower
floating point. The data values provided are the co-ordinates and size of
the image, and the maximal and minimal iteration count. High Max Iter values
will slow down plotting. Setting Min Iter allows exclusion of low iteration
colour values. Set Min Iter=Max Iter to only plot the interior. From the
menu the options are:
Julia Set: switches to the equivalent Julia set. Moving the mouse changes
the Real (x) and Imaginary (y) values as shown on top of the screen. Press
select to generate the fractal. The best places to choose are along the
Mandelbrot boundary. After seeing the Julia image you can jump back to the
Mandelbrot by using Display->Previous if you have only drawn the Julia once.
Quarternion: switches to the equivalent Quaternion map in a similar way to
above. The selected x values becomes q0 and y -> q2 in the Quaternion set.
Function: 8 different Mandelbrot functions: square, cubic, quartic,
Bruce Ikenaga's function, a function discovered by Ushiki called Phoenix,
and Inverse square, cubic and quartic.
Infill: provides 5 different colouring options for the interior. Black uses
no colour whilst Max. Iter uses the value of Max Iter as the colour. The
other 3 generate a colour based on the last iterated value of z.
Mandelbrots and Julias can be drawn directly in 3d by turning on the 3d X/Y Plot option. See the 3d section of !MainHelp for details.
Algorithms
----------
For each pixel we calculate the x & y value and then iterate. The colour is
the iteration number unless uu+vv>4.0, when we set the colour to 0 (or as
set by the Interior menu option).
Square: z=z*z+c
iter=0
u=0; v=0;
repeat
uu=u*u;
vv=v*v;
m=uu-vv+x
v=u*(v+v)+y
u=m
iter+=1
until (iter>maxiter or uu+vv>4.0)
Cubic: z=z*z*z+c
iter=0
u=0; v=0;
repeat
uu=u*u;
vv=v*v;
u=uu*u-3*u*vv+x
v=3*v*uu-vv*v+y
iter+=1
until (iter>maxiter or uu+vv>4.0)
Quartic: z=z*z*z*z+c
iter=0
u=0; v=0;
repeat
uu=u*u;
vv=v*v;
m=uu*uu+6*uu*vv+vv*vv+x
v=4*u*v*(uu-vv)+y
u=m
iter+=1
until (iter>maxiter or uu+vv>4.0)
Ikenaga: iter=0
u=0; v=0;
repeat
uu=u*u;
vv=v*v;
m=u*(uu-3*vv+x-1)-v*y-x
v=v*(3*uu-vv+x-1)+u*y-y
u=m
iter+=1
until (iter>maxiter or uu+vv>4.0)
Phoenix: iter=0
u=0; v=0; ux=0; uy=0
repeat
uu=u*u;
vv=v*v;
m=2*u*v+y*vx
n=uu-vv+x+y*ux
ux=u
vx=v
u=m
v=n
iter+=1
until (iter>maxiter or uu+vv>4.0)
Inverse: r=x*x+y*y
x=x/r
y=y/r
then as per Square/Cubic/Quartic
--------------------------------------------------------------------------
Martin Map: uses the algorithm of Dr. Martin.
----------
Change the values of a, b or c to get different patterns.
The colours are plotted at random. The zoom option does not give more
detail, but allows the size and position of the image to be changed. The
routine is infinite and so you will need to select Stop to end it.
From the menu you can select whether a, b or c are set randomly. Due to the
way the algorithm works it is not possible to guess the size or position of
the resulting image. New variables are not chosen when zooming.
For each variable there is an associated maximum for the random number,
which can be changed from the data panel.
--------------------------------------------------------------------------
Midpoint
--------
The midpoint displacement algorithm is one of the simplest used to generate
fractal landscapes. Start with a square and randomly choose the heights of
each corner. Divide this square into smaller ones and set the height of each
unplotted corner to the average height of the surrounding points, +/- a
random displacement. This process is repeated until we reach a unit
square. As we proceed the maximum displacement is reduced proportionately.
Starting with just 4 random points produces a smooth rolling landscape. To
get a more varied landscape more of the squares are plotted with random
heights prior to performing the midpoint displacement. For more information
see The Science Of Fractals, especially p96.
Menu Options
------------
Random: when on a different landscape is plotted. Set to off to allow
re-generation of a landscape with changes in the data variables.
There are 5 data variables that can be used to control the process.
Roughness: Increase to generate more random seed points, giving more hills
and depressions. A value of 0 plots just the initial outer 4 points (1
square), 1 plots 4 squares, 2 plots 16 squares and so on.
Height: Specifies the initial maximum displacement as a power of 2. Thus 8
(the default) means 256. Increase this value to give steeper gradients.
Seed: The initial random value use to generate the landscape. A new seed is
chosen every time unless Random is set off.
Sea Level: An height less than the sea level will be plotted in blue. This
is useful when creating the source images for a Riemann globe.
Sea Colour: The physical colour number used for the sea. Blues are 128-131
and 136-139, but any colour may be chosen.
Notes On Usage
--------------
This algorithm works best in modes with square pixels, ie. modes 21, 13 etc.
Mode 15 will tend to produce more oblong shapes.
Try colour cycling for a psychedelic effect. The landscape palettes give
more realistic results - use Palette shifting to get a sea/land balance.
To generate a landscape use one of the 3d viewing functions within !Fractal.
The 3d palette should be set to Linear for the best effects, though
experiment. Try a Riemann sphere for a planet effect - this works best when
the sea level is set above 0 or when using a landscape palette.
Midpoint does not multi-task since it is pretty fast.
--------------------------------------------------------------------------
Newton
------
This function calculates the cube root of -1 ie. z*z*z-1=0, using the
Newton-Raphson method. There are three possible roots which are: z=1,
z=-0.5+0.8666i, z=-0.5-0.866i. Floating point arithmetic is required so
don't expect it to zip along. Initially 32 bit accuracy is used which is
reasonably fast, then it switches to 64 bit. The menu options are:
Plot Root: in this mode, the colour is chosen depending on which root we are
converging to. This is the quickest option, but not very spectacular.
Plot Iters: in this mode the colour is set from the number of iterations
required to reach the root. This produces Mandelbrot type displays of
infinite complexity, but it can take a lot of time.
There are 2 data variables:
Max Iter : iteration limit, mainly of use when using Plot Iters and zooming
in.
e : The accuracy check value. The iteration is stopped when a root is found
that solves the equation to within the 'e' accuracy value. Higher values of
'e' force greater accuracy and more iterations to be used, useful when
zooming in.
A 3d version can be drawn directly by turning on the 3d X/Y Plot option.
See the 3d section of !MainHelp for details.
--------------------------------------------------------------------------
Pickover - A Strange Attractor developed by Clifford Pickover
--------
This is a 3 dimensional shape controlled by 4 variables. To get a feeling of
the shape you really need to try the 3d viewing options of !Fractal - use
the Rotate & Elevate icons to set your viewing position or better still use
the cursor keys whilst plotting. The plot is infinite and is made with
points rather than lines to avoid too much clutter.
The menu options are:
Random a/b/c/d: Click to set/unset random selection of the variable when a
plot is started. For each variable there is an associated maximum, which can
be changed from the data panel. New variables are not chosen when zooming.
Once you have found an interesting shape, turn off random variables, and use
the 3d options to explore the shape from different view points. The default
variables were obtained in this way.
Random colour: when set a random colour is chosen after every 100 plots.
Step Colour: the colour value is increased after every 100 plots.
If neither Random or Step colour is set, then the plot is made in white
using exclusive or'ing to trace the current position.
You can change the viewpoint whilst plotting by pressing the cursor keys,
and then pressing the space bar to restart. See the 3d section in the
!MainHelp file for more details.
Algorithm
---------
x=1; y=1; z=1;
do
xnew = sin(a*y) - z*cos(b*x)
ynew = z*sin(c*x) - cos(d*y)
znew = sin(x)
plot px,pz where px/pz are 3d transform of x/y/z
forever
--------------------------------------------------------------------------
Popcorn
-------
An algorithm developed by Clifford Pickover featured in Scientific American
July 1989, then in Fractint (where the name Popcorn was given), and
contributed here by Joyce Haslam. There are 9 different equations used with
an option to plot the pattern in a wave like form or a pixel by pixel form
(Julia).
The equation is selected from the Popcorn->Types menu. Click on
Popcorn->Julia to activate the pixel mode plot. The Julia is slow due to the
repeated use of SIN and TAN, so use small values for limit and threshold to
speed things up.
The data items after the standard x/y width/height are:
Limit: for Julia the iteration limit, otherwise the iteration limit sets the
length of each arm plotted.
Step Rate: controls the effect each iteration has on previous values. Higher
values produce more exaggerated plots.
Detail: for the non-Julia plots - 0 gives maximum detail, higher values
give rougher images which allows quicker examination.
Threshold: for Julia, the escape value. Higher values take longer to
calculate but produce more detail.
n: The multiplier in the SIN and TAN equations. Try different values to see
the effect.
A 3d version of the Julia image can be drawn directly by turning on the
3d X/Y Plot option. See the 3d section of !MainHelp for details.
Algorithms
----------
The nine equations are as shown in the Popcorn->Types menu. They are
implemented in code as shown in the 2 examples below:
Julia: for each pixel set the x and y value then :
repeat
sqsum=x*x+y*y
if sqsum < threshold
xx=x-dt*sin(y+tan(n*y)) where dt=step rate
y=y-dt*sin(x+tan(n*x))
x=xx
iter=iter+1
until sqsum>=threshold or iter=limit
colour=iter
non-Julia: for each nth x/y pixel where the value of n is increased with the
value of "detail".
increment colour
repeat
xx=x-dt*sin(y+tan(n*y)) where dt=step rate
y=y-dt*sin(x+tan(n*x))
x=xx
iter=iter+1
plot x*scalar,y*scalar
until iter=limit
--------------------------------------------------------------------------
Quaternion : Invented by William Hamilton, 1843
----------
A Quaternion is a 4d equivalent of a complex number, written as:
Q = a+bi+cj+dk
where i, j and k are imaginary numbers. They are used here in a similar way
to Julia sets with Q -> Q�+q where q is a quaternion constant, made up of
q0, q1, q2 and q3 in this routine. The routine maps out a 2-dimensional
slice using x=a and y=c, with the values of b and d being set to choose
which slice of the 4d plane we are drawing.
The initial values chosen for q0 and q2 are the same as those for the Julia
Set, and you will note the image is superficially very similar. Changing q1,
q3, b or d however allows different planes to be examined, giving a great
area to investigate.
As with the Julia set, the best way to choose values for q0 and q2 is from
the Mandelbrot set - use the Quaternion menu option to pick a point from
near the boundary.
The menu options operate in the same way as for the Julia set.
Algorithm
---------
A 32 bit and floating point version are provided, but both will be slower
than Julia's due to the extra multiplications.
For each pixel we calculate the a (x) & c (y) value and then iterate. The
colour is the iteration number unless Q�>4.0, when we set the colour to 0
(or as set by the Interior menu option).
iter=0;
repeat
iter + 1
if (a�+b�+c�+d�>4) then escape
new_a=a�-b�-c�-d�+q0
b=2*a*b+q1
c=2*a*c+q2
d=2*a*d+q3
a=new_a
until iter>=max_iter
Notes: It is possible to use other functions such as Q->Q�+q as done for
Mandelbrot's. Also it is possible to produce a 3d mapping, though a lot of
the swirling detail apparently is lost. Iterating the function Mandelbrot
style gives a variation on the Mandelbrot shape.
--------------------------------------------------------------------------
Rossler : The Otto Rossler Strange Attractor.
-------
This strange attractor is similar to the Lorenz with a very distinctive
shape. The shape is three dimensional which looks like a treble cleff from
some angles. You can use !Fractal's Rotation and Elevation icons to control
the view position to explore the shape. There are three variables which may
be set or randomly chosen, and there are three colour schemes.
The plot is infinite. You can control the speed by setting the Step Rate
factor - smaller values will slow the plot so you can trace its shape.
Notice that the plot revolves around a point then leaps off in a strange
loop. Try altering the default values of a/b/c by small amounts and see the
effect this has on the shape and when the jump occurs. Alternatively try
using one or more random variables to see what shapes you can arrive at.
The menu options are:
Random a/b/c: Click to set/unset random selection of the variable when a
plot is started. For each variable there is an associated maximum, which can
be changed from the data panel. New variables are not chosen when zooming.
Once you have found an interesting shape, turn off random variables, and use
the 3d options to explore the shape from different view points.
Random colour: when set a random colour is chosen after every 100 plots.
Step Colour: the colour value is increased after every 100 plots.
If neither Random or Step colour is set, then the plot is made in white
using exclusive or'ing to trace the current position.
You can change the viewpoint whilst plotting by pressing the cursor keys,
and then pressing the space bar to restart. See the 3d section in the
!MainHelp file.
Algorithm
---------
x=1; y=1; z=1;
do
new x=x-y*dt-z*dt dt=step rate
new y=y+x*dt+a*y*dt
new z=z+b*dt+x*z*dt-c*z*dt
draw px,pz px/pz are 3d transform of x/y/z
forever
--------------------------------------------------------------------------
Unity
=====
This transformation was developed by Mark Peterson of Fractint and is a
simple "Newton" style algorithm, which is:
One=x*x+y*y; y=(2-One)*x; x=(2-One)*y;
which is iterated for each screen pixel until 1-One=0 within the accuracy of
plotting. The colour is the number of iterations (+32 to avoid dark
colours). The plot looks very similar to a Henon map, with thin lines that
get thinner when you zoom in.
Unity uses integer arithmetic and thus soon hits a zoom limit.
--------------------------------------------------------------------------
3d Transforms
=============
These take a fractal image and transform it. The best images are those of
Mandelbrots using a standard or inverted palette (but feel free to
experiment). Except for Riemann, they do not multitask since they are quite
quick. They offer varying degrees of control.
There is a common menu option which allows you to re-display the source
image, and then store it again as the 3d source. This allows you to do three
things:
a) change the colours of the original image, by using the Effects->Palette
menu. For Render and 3d views this will change the displayed height.
b) Rotate the source image via the tools rotate icon.
c) Use a 3d image as the source of another (or same) 3d transform.
3d View and Riemann provide rotation directly - for the others use the
Rotate rotate tool icon.
Return to function: This menu option takes you directly back to the original
function, displaying the source image to allow zooming to re-commence.
Height Mapping
--------------
The 3d_Plane, 3d_View and Render functions use the pixel colour to determine
the height. Use the Effects->3d Details panel to control this mapping
process, described in the !MainHelp file. When these functions are selected
the min and max pixel colours are automatically calculated. Override these
values to smooth out peaks or clip the image. To reset the values use the
Initial button on the Image->Data panel.
3d Plane
--------
Presents an oblique view of an image, scaled to fit onto the screen. The
height is based on the source colour (0=low, 255=high). The data variables
are:
x Scalar: a value of 1 plots the x axis full size. Use smaller values to
accomodate the shift incurred by the viewing angle.
y Scalar: a value of 1 plots the y axis full size, but 0.6 provides a more
realistic image.
The rotation angle is used as the angle of the y axis, with 90� being the
far right and -90� the far left. Larger angles will tend to lower the
viewpoint and see more to the left or right. You can use the Rotation tool
icon but note that the viewpoint angle is only an indication of the
apparent direction.
3d View
-------
This offers a more realistic 3d image than 3d Plane, and provides two
viewing options - either full rotation or a front end view with perspective
control. The data parameters are:
Distance: when Rotation=0 this parameter controls the vanishing point. A
value of 1 gives the maximum distortion, whilst larger values give more
subtle effects.
From the 3D_View menu you can set Smooth: when off contour lines are
accentuated, which works well with Mandelbrot images. When on contour lines
run into each other, which works best with landscapes such as Midpoint and
Fault functions.
Render
------
Draws a 3d representation as seen full on, using the colour as the height. A
light source is assumed from the bottom left corner and is used to add a
shadow effect.
From Render's menu you can select the elevation viewpoint, or you can use
the Elevation tool icon to control this. Note that only 4 angles are
available.
Riemann
-------
Takes the image and maps it onto a sphere, using a Riemann transformation.
Basically the source image is mapped onto a sphere at the south pole. A
value can only be mapped onto the north pole if it is at infinity. See the
article in Fractal Report 16 by Roger Castle-Smith for a full explanation.
The data values are :
Image Scale: the source image size in relation to the sphere size. Larger
values will mean that the source image is stretched up towards the north
pole.
Globe Scale: the globe size as a fraction of the total image size (0 - 1.0).
Thus 0.5 produces a globe half the total image size. Smaller values help
reduce the distortion introduced by the mapping process.
Fill Colour: The colour used to fill in black pixels. The colour number is
in BBGGRRTT format where B=Blue (x64), G=Green (x16), R=Red (x4) & T=Tint
bits.
----------------------------------------------------------------
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*
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*
00002100 2d 2d 0a 0a 48 e9 6e 6f 6e 20 4d 61 70 73 20 3a |--..H.non Maps :|
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*
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00002780 20 20 20 20 72 65 70 65 61 74 0a 20 20 20 20 20 | repeat. |
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000027a0 20 20 20 20 20 20 20 20 20 20 78 78 3d 78 2a 43 | xx=x*C|
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000027e0 4f 53 61 0a 20 20 20 20 20 20 20 20 20 20 20 20 |OSa. |
000027f0 20 78 3d 78 78 0a 20 20 20 20 20 20 20 20 20 20 | x=xx. |
00002800 20 20 20 70 6c 6f 74 20 78 2c 79 0a 20 20 20 20 | plot x,y. |
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00002830 20 79 3e 31 30 30 30 29 0a 20 20 20 20 20 20 20 | y>1000). |
00002840 20 20 6e 65 78 74 0a 20 20 20 20 20 20 20 6e 65 | next. ne|
00002850 78 74 0a 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |xt..------------|
00002860 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
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00003160 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
000031a0 2d 2d 2d 2d 2d 0a 0a 4a 75 6c 69 61 20 53 65 74 |-----..Julia Set|
000031b0 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 50 6c 6f 74 73 |.---------.Plots|
000031c0 20 74 68 65 20 4a 75 6c 69 61 20 73 65 74 20 63 | the Julia set c|
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00003210 7a 6f 6f 6d 20 75 73 65 73 20 69 6e 74 65 67 65 |zoom uses intege|
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00003550 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
00003590 2d 0a 0a 4c 2d 53 79 73 74 65 6d 20 3a 20 54 75 |-..L-System : Tu|
000035a0 72 74 6c 65 20 67 72 61 70 68 69 63 73 20 61 73 |rtle graphics as|
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*
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000078b0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
000078c0 75 3d 30 3b 20 76 3d 30 3b 0a 20 20 20 20 20 20 |u=0; v=0;. |
000078d0 20 20 20 20 20 20 20 20 20 20 20 20 72 65 70 65 | repe|
000078e0 61 74 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |at. |
000078f0 20 20 20 20 20 20 20 75 75 3d 75 2a 75 3b 0a 20 | uu=u*u;. |
00007900 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007910 20 20 20 76 76 3d 76 2a 76 3b 0a 20 20 20 20 20 | vv=v*v;. |
00007920 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 75 | u|
00007930 3d 75 75 2a 75 2d 33 2a 75 2a 76 76 2b 78 0a 20 |=uu*u-3*u*vv+x. |
00007940 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007950 20 20 20 76 3d 33 2a 76 2a 75 75 2d 76 76 2a 76 | v=3*v*uu-vv*v|
00007960 2b 79 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |+y. |
00007970 20 20 20 20 20 20 20 69 74 65 72 2b 3d 31 0a 20 | iter+=1. |
00007980 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007990 20 75 6e 74 69 6c 20 28 69 74 65 72 3e 6d 61 78 | until (iter>max|
000079a0 69 74 65 72 20 6f 72 20 75 75 2b 76 76 3e 34 2e |iter or uu+vv>4.|
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000079c0 2a 7a 2a 7a 2a 7a 2b 63 0a 20 20 20 20 20 20 20 |*z*z*z+c. |
000079d0 20 20 20 20 20 20 20 20 20 20 20 69 74 65 72 3d | iter=|
000079e0 30 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |0. |
000079f0 20 20 20 20 75 3d 30 3b 20 76 3d 30 3b 0a 20 20 | u=0; v=0;. |
00007a00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007a10 72 65 70 65 61 74 0a 20 20 20 20 20 20 20 20 20 |repeat. |
00007a20 20 20 20 20 20 20 20 20 20 20 20 75 75 3d 75 2a | uu=u*|
00007a30 75 3b 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |u;. |
00007a40 20 20 20 20 20 20 20 76 76 3d 76 2a 76 3b 0a 20 | vv=v*v;. |
00007a50 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007a60 20 20 20 6d 3d 75 75 2a 75 75 2b 36 2a 75 75 2a | m=uu*uu+6*uu*|
00007a70 76 76 2b 76 76 2a 76 76 2b 78 0a 20 20 20 20 20 |vv+vv*vv+x. |
00007a80 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 76 | v|
00007a90 3d 34 2a 75 2a 76 2a 28 75 75 2d 76 76 29 2b 79 |=4*u*v*(uu-vv)+y|
00007aa0 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |. |
00007ab0 20 20 20 20 20 75 3d 6d 0a 20 20 20 20 20 20 20 | u=m. |
00007ac0 20 20 20 20 20 20 20 20 20 20 20 20 20 69 74 65 | ite|
00007ad0 72 2b 3d 31 0a 20 20 20 20 20 20 20 20 20 20 20 |r+=1. |
00007ae0 20 20 20 20 20 20 20 75 6e 74 69 6c 20 28 69 74 | until (it|
00007af0 65 72 3e 6d 61 78 69 74 65 72 20 6f 72 20 75 75 |er>maxiter or uu|
00007b00 2b 76 76 3e 34 2e 30 29 0a 0a 49 6b 65 6e 61 67 |+vv>4.0)..Ikenag|
00007b10 61 3a 20 20 20 20 20 20 20 20 20 20 69 74 65 72 |a: iter|
00007b20 3d 30 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |=0. |
00007b30 20 20 20 20 20 75 3d 30 3b 20 76 3d 30 3b 0a 20 | u=0; v=0;. |
00007b40 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007b50 20 72 65 70 65 61 74 0a 20 20 20 20 20 20 20 20 | repeat. |
00007b60 20 20 20 20 20 20 20 20 20 20 20 20 75 75 3d 75 | uu=u|
00007b70 2a 75 3b 0a 20 20 20 20 20 20 20 20 20 20 20 20 |*u;. |
00007b80 20 20 20 20 20 20 20 20 76 76 3d 76 2a 76 3b 0a | vv=v*v;.|
00007b90 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007ba0 20 20 20 20 6d 3d 75 2a 28 75 75 2d 33 2a 76 76 | m=u*(uu-3*vv|
00007bb0 2b 78 2d 31 29 2d 76 2a 79 2d 78 0a 20 20 20 20 |+x-1)-v*y-x. |
00007bc0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007bd0 76 3d 76 2a 28 33 2a 75 75 2d 76 76 2b 78 2d 31 |v=v*(3*uu-vv+x-1|
00007be0 29 2b 75 2a 79 2d 79 0a 20 20 20 20 20 20 20 20 |)+u*y-y. |
00007bf0 20 20 20 20 20 20 20 20 20 20 20 20 75 3d 6d 0a | u=m.|
00007c00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007c10 20 20 20 20 69 74 65 72 2b 3d 31 0a 20 20 20 20 | iter+=1. |
00007c20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 75 6e | un|
00007c30 74 69 6c 20 28 69 74 65 72 3e 6d 61 78 69 74 65 |til (iter>maxite|
00007c40 72 20 6f 72 20 75 75 2b 76 76 3e 34 2e 30 29 0a |r or uu+vv>4.0).|
00007c50 0a 50 68 6f 65 6e 69 78 3a 20 20 20 20 20 20 20 |.Phoenix: |
00007c60 20 20 20 69 74 65 72 3d 30 0a 20 20 20 20 20 20 | iter=0. |
00007c70 20 20 20 20 20 20 20 20 20 20 20 20 75 3d 30 3b | u=0;|
00007c80 20 76 3d 30 3b 20 75 78 3d 30 3b 20 75 79 3d 30 | v=0; ux=0; uy=0|
00007c90 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |. |
00007ca0 20 20 20 72 65 70 65 61 74 0a 20 20 20 20 20 20 | repeat. |
00007cb0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 75 75 | uu|
00007cc0 3d 75 2a 75 3b 0a 20 20 20 20 20 20 20 20 20 20 |=u*u;. |
00007cd0 20 20 20 20 20 20 20 20 20 20 76 76 3d 76 2a 76 | vv=v*v|
00007ce0 3b 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |;. |
00007cf0 20 20 20 20 20 20 6d 3d 32 2a 75 2a 76 2b 79 2a | m=2*u*v+y*|
00007d00 76 78 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |vx. |
00007d10 20 20 20 20 20 20 20 6e 3d 75 75 2d 76 76 2b 78 | n=uu-vv+x|
00007d20 2b 79 2a 75 78 0a 20 20 20 20 20 20 20 20 20 20 |+y*ux. |
00007d30 20 20 20 20 20 20 20 20 20 20 75 78 3d 75 0a 20 | ux=u. |
00007d40 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007d50 20 20 20 76 78 3d 76 0a 20 20 20 20 20 20 20 20 | vx=v. |
00007d60 20 20 20 20 20 20 20 20 20 20 20 20 75 3d 6d 0a | u=m.|
00007d70 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00007d80 20 20 20 20 76 3d 6e 0a 20 20 20 20 20 20 20 20 | v=n. |
00007d90 20 20 20 20 20 20 20 20 20 20 20 20 69 74 65 72 | iter|
00007da0 2b 3d 31 0a 20 20 20 20 20 20 20 20 20 20 20 20 |+=1. |
00007db0 20 20 20 20 20 20 75 6e 74 69 6c 20 28 69 74 65 | until (ite|
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00007dd0 76 76 3e 34 2e 30 29 0a 0a 49 6e 76 65 72 73 65 |vv>4.0)..Inverse|
00007de0 3a 20 20 20 20 20 20 20 20 20 20 72 3d 78 2a 78 |: r=x*x|
00007df0 2b 79 2a 79 0a 20 20 20 20 20 20 20 20 20 20 20 |+y*y. |
00007e00 20 20 20 20 20 20 20 78 3d 78 2f 72 0a 20 20 20 | x=x/r. |
00007e10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 79 | y|
00007e20 3d 79 2f 72 0a 20 20 20 20 20 20 20 20 20 20 20 |=y/r. |
00007e30 20 20 20 20 20 20 20 74 68 65 6e 20 61 73 20 70 | then as p|
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00007e50 51 75 61 72 74 69 63 0a 0a 2d 2d 2d 2d 2d 2d 2d |Quartic..-------|
00007e60 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
00007ea0 2d 2d 2d 0a 0a 4d 61 72 74 69 6e 20 4d 61 70 3a |---..Martin Map:|
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00007f40 6f 6f 6d 20 6f 70 74 69 6f 6e 20 64 6f 65 73 20 |oom option does |
00007f50 6e 6f 74 20 67 69 76 65 20 6d 6f 72 65 0a 64 65 |not give more.de|
00007f60 74 61 69 6c 2c 20 62 75 74 20 61 6c 6c 6f 77 73 |tail, but allows|
00007f70 20 74 68 65 20 73 69 7a 65 20 61 6e 64 20 70 6f | the size and po|
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00007fc0 6f 20 79 6f 75 20 77 69 6c 6c 20 6e 65 65 64 20 |o you will need |
00007fd0 74 6f 20 73 65 6c 65 63 74 20 53 74 6f 70 20 74 |to select Stop t|
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00008000 20 73 65 6c 65 63 74 20 77 68 65 74 68 65 72 20 | select whether |
00008010 61 2c 20 62 20 6f 72 20 63 20 61 72 65 20 73 65 |a, b or c are se|
00008020 74 20 72 61 6e 64 6f 6d 6c 79 2e 20 44 75 65 20 |t randomly. Due |
00008030 74 6f 20 74 68 65 0a 77 61 79 20 74 68 65 20 61 |to the.way the a|
00008040 6c 67 6f 72 69 74 68 6d 20 77 6f 72 6b 73 20 69 |lgorithm works i|
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00008060 65 20 74 6f 20 67 75 65 73 73 20 74 68 65 20 73 |e to guess the s|
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00008080 6f 66 0a 74 68 65 20 72 65 73 75 6c 74 69 6e 67 |of.the resulting|
00008090 20 69 6d 61 67 65 2e 20 4e 65 77 20 76 61 72 69 | image. New vari|
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000080e0 6e 20 61 73 73 6f 63 69 61 74 65 64 20 6d 61 78 |n associated max|
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00008100 64 6f 6d 20 6e 75 6d 62 65 72 2c 0a 77 68 69 63 |dom number,.whic|
00008110 68 20 63 61 6e 20 62 65 20 63 68 61 6e 67 65 64 |h can be changed|
00008120 20 66 72 6f 6d 20 74 68 65 20 64 61 74 61 20 70 | from the data p|
00008130 61 6e 65 6c 2e 0a 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d |anel...---------|
00008140 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
00008180 2d 0a 4d 69 64 70 6f 69 6e 74 0a 2d 2d 2d 2d 2d |-.Midpoint.-----|
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000081a0 20 64 69 73 70 6c 61 63 65 6d 65 6e 74 20 61 6c | displacement al|
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000081c0 66 20 74 68 65 20 73 69 6d 70 6c 65 73 74 20 75 |f the simplest u|
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000081e0 66 72 61 63 74 61 6c 20 6c 61 6e 64 73 63 61 70 |fractal landscap|
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00008260 20 73 65 74 20 74 68 65 20 68 65 69 67 68 74 20 | set the height |
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00008290 61 76 65 72 61 67 65 20 68 65 69 67 68 74 20 6f |average height o|
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000082c0 72 61 6e 64 6f 6d 20 64 69 73 70 6c 61 63 65 6d |random displacem|
000082d0 65 6e 74 2e 20 54 68 69 73 20 70 72 6f 63 65 73 |ent. This proces|
000082e0 73 20 69 73 20 72 65 70 65 61 74 65 64 20 75 6e |s is repeated un|
000082f0 74 69 6c 20 77 65 20 72 65 61 63 68 20 61 20 75 |til we reach a u|
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00008330 6e 74 20 69 73 20 72 65 64 75 63 65 64 20 70 72 |nt is reduced pr|
00008340 6f 70 6f 72 74 69 6f 6e 61 74 65 6c 79 2e 0a 0a |oportionately...|
00008350 53 74 61 72 74 69 6e 67 20 77 69 74 68 20 6a 75 |Starting with ju|
00008360 73 74 20 34 20 72 61 6e 64 6f 6d 20 70 6f 69 6e |st 4 random poin|
00008370 74 73 20 70 72 6f 64 75 63 65 73 20 61 20 73 6d |ts produces a sm|
00008380 6f 6f 74 68 20 72 6f 6c 6c 69 6e 67 20 6c 61 6e |ooth rolling lan|
00008390 64 73 63 61 70 65 2e 20 54 6f 0a 67 65 74 20 61 |dscape. To.get a|
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000083b0 64 73 63 61 70 65 20 6d 6f 72 65 20 6f 66 20 74 |dscape more of t|
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000083e0 6f 6d 0a 68 65 69 67 68 74 73 20 70 72 69 6f 72 |om.heights prior|
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00008400 68 65 20 6d 69 64 70 6f 69 6e 74 20 64 69 73 70 |he midpoint disp|
00008410 6c 61 63 65 6d 65 6e 74 2e 20 46 6f 72 20 6d 6f |lacement. For mo|
00008420 72 65 20 69 6e 66 6f 72 6d 61 74 69 6f 6e 0a 73 |re information.s|
00008430 65 65 20 54 68 65 20 53 63 69 65 6e 63 65 20 4f |ee The Science O|
00008440 66 20 46 72 61 63 74 61 6c 73 2c 20 65 73 70 65 |f Fractals, espe|
00008450 63 69 61 6c 6c 79 20 70 39 36 2e 0a 0a 4d 65 6e |cially p96...Men|
00008460 75 20 4f 70 74 69 6f 6e 73 0a 2d 2d 2d 2d 2d 2d |u Options.------|
00008470 2d 2d 2d 2d 2d 2d 0a 52 61 6e 64 6f 6d 3a 20 77 |------.Random: w|
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000084b0 6f 66 66 20 74 6f 20 61 6c 6c 6f 77 0a 72 65 2d |off to allow.re-|
000084c0 67 65 6e 65 72 61 74 69 6f 6e 20 6f 66 20 61 20 |generation of a |
000084d0 6c 61 6e 64 73 63 61 70 65 20 77 69 74 68 20 63 |landscape with c|
000084e0 68 61 6e 67 65 73 20 69 6e 20 74 68 65 20 64 61 |hanges in the da|
000084f0 74 61 20 76 61 72 69 61 62 6c 65 73 2e 0a 0a 54 |ta variables...T|
00008500 68 65 72 65 20 61 72 65 20 35 20 64 61 74 61 20 |here are 5 data |
00008510 76 61 72 69 61 62 6c 65 73 20 74 68 61 74 20 63 |variables that c|
00008520 61 6e 20 62 65 20 75 73 65 64 20 74 6f 20 63 6f |an be used to co|
00008530 6e 74 72 6f 6c 20 74 68 65 20 70 72 6f 63 65 73 |ntrol the proces|
00008540 73 2e 0a 0a 52 6f 75 67 68 6e 65 73 73 3a 20 49 |s...Roughness: I|
00008550 6e 63 72 65 61 73 65 20 74 6f 20 67 65 6e 65 72 |ncrease to gener|
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00008570 73 65 65 64 20 70 6f 69 6e 74 73 2c 20 67 69 76 |seed points, giv|
00008580 69 6e 67 20 6d 6f 72 65 20 68 69 6c 6c 73 0a 61 |ing more hills.a|
00008590 6e 64 20 64 65 70 72 65 73 73 69 6f 6e 73 2e 20 |nd depressions. |
000085a0 41 20 76 61 6c 75 65 20 6f 66 20 30 20 70 6c 6f |A value of 0 plo|
000085b0 74 73 20 6a 75 73 74 20 74 68 65 20 69 6e 69 74 |ts just the init|
000085c0 69 61 6c 20 6f 75 74 65 72 20 34 20 70 6f 69 6e |ial outer 4 poin|
000085d0 74 73 20 28 31 0a 73 71 75 61 72 65 29 2c 20 31 |ts (1.square), 1|
000085e0 20 70 6c 6f 74 73 20 34 20 73 71 75 61 72 65 73 | plots 4 squares|
000085f0 2c 20 32 20 70 6c 6f 74 73 20 31 36 20 73 71 75 |, 2 plots 16 squ|
00008600 61 72 65 73 20 61 6e 64 20 73 6f 20 6f 6e 2e 0a |ares and so on..|
00008610 0a 48 65 69 67 68 74 3a 20 53 70 65 63 69 66 69 |.Height: Specifi|
00008620 65 73 20 74 68 65 20 69 6e 69 74 69 61 6c 20 6d |es the initial m|
00008630 61 78 69 6d 75 6d 20 64 69 73 70 6c 61 63 65 6d |aximum displacem|
00008640 65 6e 74 20 61 73 20 61 20 70 6f 77 65 72 20 6f |ent as a power o|
00008650 66 20 32 2e 20 54 68 75 73 20 38 0a 28 74 68 65 |f 2. Thus 8.(the|
00008660 20 64 65 66 61 75 6c 74 29 20 6d 65 61 6e 73 20 | default) means |
00008670 32 35 36 2e 20 49 6e 63 72 65 61 73 65 20 74 68 |256. Increase th|
00008680 69 73 20 76 61 6c 75 65 20 74 6f 20 67 69 76 65 |is value to give|
00008690 20 73 74 65 65 70 65 72 20 67 72 61 64 69 65 6e | steeper gradien|
000086a0 74 73 2e 0a 0a 53 65 65 64 3a 20 54 68 65 20 69 |ts...Seed: The i|
000086b0 6e 69 74 69 61 6c 20 72 61 6e 64 6f 6d 20 76 61 |nitial random va|
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000086d0 61 74 65 20 74 68 65 20 6c 61 6e 64 73 63 61 70 |ate the landscap|
000086e0 65 2e 20 41 20 6e 65 77 20 73 65 65 64 20 69 73 |e. A new seed is|
000086f0 0a 63 68 6f 73 65 6e 20 65 76 65 72 79 20 74 69 |.chosen every ti|
00008700 6d 65 20 75 6e 6c 65 73 73 20 52 61 6e 64 6f 6d |me unless Random|
00008710 20 69 73 20 73 65 74 20 6f 66 66 2e 0a 0a 53 65 | is set off...Se|
00008720 61 20 4c 65 76 65 6c 3a 20 41 6e 20 68 65 69 67 |a Level: An heig|
00008730 68 74 20 6c 65 73 73 20 74 68 61 6e 20 74 68 65 |ht less than the|
00008740 20 73 65 61 20 6c 65 76 65 6c 20 77 69 6c 6c 20 | sea level will |
00008750 62 65 20 70 6c 6f 74 74 65 64 20 69 6e 20 62 6c |be plotted in bl|
00008760 75 65 2e 20 54 68 69 73 0a 69 73 20 75 73 65 66 |ue. This.is usef|
00008770 75 6c 20 77 68 65 6e 20 63 72 65 61 74 69 6e 67 |ul when creating|
00008780 20 74 68 65 20 73 6f 75 72 63 65 20 69 6d 61 67 | the source imag|
00008790 65 73 20 66 6f 72 20 61 20 52 69 65 6d 61 6e 6e |es for a Riemann|
000087a0 20 67 6c 6f 62 65 2e 0a 0a 53 65 61 20 43 6f 6c | globe...Sea Col|
000087b0 6f 75 72 3a 20 54 68 65 20 70 68 79 73 69 63 61 |our: The physica|
000087c0 6c 20 63 6f 6c 6f 75 72 20 6e 75 6d 62 65 72 20 |l colour number |
000087d0 75 73 65 64 20 66 6f 72 20 74 68 65 20 73 65 61 |used for the sea|
000087e0 2e 20 42 6c 75 65 73 20 61 72 65 20 31 32 38 2d |. Blues are 128-|
000087f0 31 33 31 0a 61 6e 64 20 31 33 36 2d 31 33 39 2c |131.and 136-139,|
00008800 20 62 75 74 20 61 6e 79 20 63 6f 6c 6f 75 72 20 | but any colour |
00008810 6d 61 79 20 62 65 20 63 68 6f 73 65 6e 2e 0a 0a |may be chosen...|
00008820 4e 6f 74 65 73 20 4f 6e 20 55 73 61 67 65 0a 2d |Notes On Usage.-|
00008830 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 54 68 |-------------.Th|
00008840 69 73 20 61 6c 67 6f 72 69 74 68 6d 20 77 6f 72 |is algorithm wor|
00008850 6b 73 20 62 65 73 74 20 69 6e 20 6d 6f 64 65 73 |ks best in modes|
00008860 20 77 69 74 68 20 73 71 75 61 72 65 20 70 69 78 | with square pix|
00008870 65 6c 73 2c 20 69 65 2e 20 6d 6f 64 65 73 20 32 |els, ie. modes 2|
00008880 31 2c 20 31 33 20 65 74 63 2e 0a 4d 6f 64 65 20 |1, 13 etc..Mode |
00008890 31 35 20 77 69 6c 6c 20 74 65 6e 64 20 74 6f 20 |15 will tend to |
000088a0 70 72 6f 64 75 63 65 20 6d 6f 72 65 20 6f 62 6c |produce more obl|
000088b0 6f 6e 67 20 73 68 61 70 65 73 2e 0a 0a 54 72 79 |ong shapes...Try|
000088c0 20 63 6f 6c 6f 75 72 20 63 79 63 6c 69 6e 67 20 | colour cycling |
000088d0 66 6f 72 20 61 20 70 73 79 63 68 65 64 65 6c 69 |for a psychedeli|
000088e0 63 20 65 66 66 65 63 74 2e 20 54 68 65 20 6c 61 |c effect. The la|
000088f0 6e 64 73 63 61 70 65 20 70 61 6c 65 74 74 65 73 |ndscape palettes|
00008900 20 67 69 76 65 0a 6d 6f 72 65 20 72 65 61 6c 69 | give.more reali|
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*
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00009fd0 68 65 72 65 20 71 20 69 73 20 61 20 71 75 61 74 |here q is a quat|
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0000a020 68 65 20 72 6f 75 74 69 6e 65 20 6d 61 70 73 20 |he routine maps |
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0000a160 65 76 65 72 20 61 6c 6c 6f 77 73 20 64 69 66 66 |ever allows diff|
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0000a220 6e 69 6f 6e 20 6d 65 6e 75 20 6f 70 74 69 6f 6e |nion menu option|
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0000a2a0 74 68 6d 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 41 20 |thm.---------.A |
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0000a420 20 74 68 65 6e 20 65 73 63 61 70 65 0a 20 20 20 | then escape. |
0000a430 20 20 6e 65 77 5f 61 3d 61 b2 2d 62 b2 2d 63 b2 | new_a=a.-b.-c.|
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0000a580 20 67 69 76 65 73 20 61 20 76 61 72 69 61 74 69 | gives a variati|
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0000a5a0 62 72 6f 74 20 73 68 61 70 65 2e 0a 0a 2d 2d 2d |brot shape...---|
0000a5b0 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------|
*
0000a5f0 2d 2d 2d 2d 2d 2d 2d 0a 52 6f 73 73 6c 65 72 20 |-------.Rossler |
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0000a610 65 72 20 53 74 72 61 6e 67 65 20 41 74 74 72 61 |er Strange Attra|
0000a620 63 74 6f 72 2e 0a 2d 2d 2d 2d 2d 2d 2d 0a 54 68 |ctor..-------.Th|
0000a630 69 73 20 73 74 72 61 6e 67 65 20 61 74 74 72 61 |is strange attra|
0000a640 63 74 6f 72 20 69 73 20 73 69 6d 69 6c 61 72 20 |ctor is similar |
0000a650 74 6f 20 74 68 65 20 4c 6f 72 65 6e 7a 20 77 69 |to the Lorenz wi|
0000a660 74 68 20 61 20 76 65 72 79 20 64 69 73 74 69 6e |th a very distin|
0000a670 63 74 69 76 65 0a 73 68 61 70 65 2e 20 54 68 65 |ctive.shape. The|
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0000a690 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 77 68 69 63 |dimensional whic|
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0000a6b0 72 65 62 6c 65 20 63 6c 65 66 66 20 66 72 6f 6d |reble cleff from|
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0000a6e0 61 6c 27 73 20 52 6f 74 61 74 69 6f 6e 20 61 6e |al's Rotation an|
0000a6f0 64 20 45 6c 65 76 61 74 69 6f 6e 20 69 63 6f 6e |d Elevation icon|
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0000a730 61 70 65 2e 20 54 68 65 72 65 20 61 72 65 20 74 |ape. There are t|
0000a740 68 72 65 65 20 76 61 72 69 61 62 6c 65 73 20 77 |hree variables w|
0000a750 68 69 63 68 20 6d 61 79 0a 62 65 20 73 65 74 20 |hich may.be set |
0000a760 6f 72 20 72 61 6e 64 6f 6d 6c 79 20 63 68 6f 73 |or randomly chos|
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0000a830 74 68 61 74 20 74 68 65 20 70 6c 6f 74 20 72 65 |that the plot re|
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0000a910 20 6f 72 20 6d 6f 72 65 20 72 61 6e 64 6f 6d 20 | or more random |
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0000a930 20 77 68 61 74 20 73 68 61 70 65 73 20 79 6f 75 | what shapes you|
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0000bf68 
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