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*** FracTrip Version 2.0 for Windows ***
1.1 Overview
1.2 About the Mandelbrot Set
1.3 Complex numbers
1.4 Fractal images
2.1 Disclaimer
2.2 Terms of Use
2.3 Suggestions and Bug Report
2.4 Next Releases
2.5 How to contact the Author
1.1 Overview
FracTrip allows you to generate images of the Mandelbrot Set. It lets
you explore this fascinating Set and enter the wonderful world of fractal
geometry in a simple and intuitive way, allowing you to save the
generated images, and to create "film sequences" that can demonstrate
one of the main features of the fractals: the recursivity. You can also
save, load and review the film sequences.
The film sequences are created zooming in or out an image leaving
unchanged the center. With FracTrip you can also zoom inside any
region of a shown image by just selecting an area to zoom into and
issuing the command Zoom in, or even automatically, if the Autostart
and the Autozoom option are set.
You can easily modify the pre-set color palette by mixing together 8
main colors, set the background color, which corresponds to the
Mandelbrot Set, change the color intensity and swap to a B/W palette.
All the parameters that FracTrip uses to generate the images are pre-set
in the best way, however you can change any of them in order to alter
the normal proceeding of generation of an image (E.G. the more you
zoom in, the higher the number of iterations must be, if you want to
leave unchanged the detail of the new image).
At start-up, the program draws the first image automatically, since
the "Autostart" option is set by default. The image generated by
default is the whole Mandelbrot Set.
1.2 About the Mandelbrot Set
"Yes, beautiful, ... but what is it?"
The Mandelbrot set is without doubt the most famous of the fractals. It
is the graphical representation of a function of a variable defined in the
complex plane. The Mandelbrot set consists of all the complex numbers
for which the sequence
Zn+1 = Zn^2 + C (Z0 = C)
does not approach infinity. This sequence has the property that when
the magnitude of Zn exceeds the value 2, the magnitude of Zn+1 will
also exceed this value and become larger and larger within a few
iterations.
Every pixel on the window corresponds to a complex number. FracTrip
calculates the sequence for each of these pixels, for at most a certain
number of iterations (iteration = the value of n). If the magnitude of Zn
does not exceed 2, the point probably belongs to the Mandelbrot set,
otherwise, if the magnitude exceeds 2 before the n iterations are
calculated, the complex number surely lies outside the set (see
menu Options - Iterations).
The dot on the screen is plotted in a color that depends on the number
of iterations FracTrip has to calculate before the sequence diverges. So,
the colored area represents the set of complex numbers for which the
sequence diverges after a certain number of steps.
The beauty of the Mandelbrot set is at its boundary. The sequence
calculated for points with almost identical coordinates may have a totally
different behaviour: in one case it may diverge, in the other not.
Exploring this area may produce magnificent, unexpected images.
1.3 Complex numbers
A complex number is a number in the form
C = X + i * Y,
where X and Y are real numbers, and i is the imaginary unit, defined as
the square root of -1. The algebra of complex numbers follows that of
real numbers, with the particularity of i, so, for instance, the square of a
complex number is
C^2 = (X+iY)^2 = X^2 - Y^2 + 2XY.
A complex number is usually represented as a point on a plane where
the real numbers are represented on the X axis and the imaginary
numbers on the Y axis. It is called the complex plane.
The magnitude of a complex number is defined as the square root
of the sum of the squares of its components:
|C| = sqrt (X^2 + Y^2).
1.4 Fractal images
A fractal image is generally a geometrical shape that can be
subdivided in parts similar to each other, each of which is a lower-scale
copy of the whole. A fractal image reveals more and more details the
"deeper" one goes, and every detail in a smaller scale looks like the
detail itself in a larger scale.
The boundary of the Mandelbrot Set is a fractal curve. It consists of
innumerable parts composed of miniatures of themselves.
2.1 Disclaimer
THE PROGRAM ON THIS DISKETTE IS PROVIDED "AS IS". THE
AUTHOR DISCLAIM ALL WARRANTIES, EITHER EXPRESS OR IMPLIED,
AS TO THE PROGRAM OR ITS PERFORMANCE OR QUALITY,
INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. IN
NO EVENT WILL THE AUTHOR BE LIABLE FOR ANY DAMAGES,
INCLUDING WITHOUT LIMITATION DIRECT, INDIRECT, INCIDENTAL,
SPECIAL OR CONSEQUENTIAL DAMAGES, LOST PROFITS OR LOST
DATA, RESULTING FROM THE USE OF OR INABILITY TO USE THE
PROGRAM.
IF THESE TERMS ARE NOT ACCEPTABLE TO YOU, THEN PLEASE
DELETE ALL THE FILES FROM YOUR DISKS IMMEDIATELY.
2.2 Terms of Use
FracTrip is distributed as a shareware program. It is NOT a public
domain program. However, you are encouraged to copy it for trial
purposes. The program and all associated files can be freely copied and
shared to allow others to try FracTrip. You may upload this program and
all associated files to any bulletin board system (BBS) or on-line computer
service. You may not give FracTrip away with commercial products
without an explicit authorization from the Author.
If you try FracTtrip and decide to use it, you must register your copy. If
you do not register your copy, you are not authorized to use the
program beyond an initial evaluation period of thirty (30) days.
By registering, the Author grant you a license to use the copyrighted
computer program FracTrip on a single computer at a time, subject
to the terms and conditions of this license. You agree not to:
a) modify, disassemble, or decompile the program,
b) use this program on more than one terminal of a network, on a multi-
user computer, on a time-sharing system, on a service bureau, or on any
other system on which the program could be used (other than for trial
purposes) by more than one person at a time.
The registration fee for FracTrip is US$30. When you register, you will
receive the registration code and will receive information on future
upgrades to the program.
To register for FracTrip, send the registration fee to the Author.
2.3 Suggestions and Bug Report
If you spot bugs, or have comments or suggestions for how to improve
the program, please write to the he Author.
All your suggestions will be accepted and, if possible, included in the
subsequent release. In any case you will be noticed on the way your
suggestion has been considered.
If you write, please include your Internet e-mail address. It will be
easier and faster to me to answer such mail rather than ordinary mail.
2.4 Next Releases
Soon coming:
More fractals (other than Mandelbrot's),
3D projection,
Cooperative processing on a network for speed improvement.
All your suggestions for other improvements will be welcome.
2.5 How to contact the Author
If you want to contact the author of FracTrip you can write to:
Stefan Forni
Via C. Massini, 43/D
00155 ROMA
ITALY
or, better, send an e-mail message to
pier_dario_diodati@ntt.it on the INTERNET.
REGISTER NOW!