| Type: | Package |
| Title: | Fractal Image Data Generator |
| Version: | 1.3.5 |
| Date: | 2023-08-27 |
| Maintainer: | Mehmet Suzen <mehmet.suzen@physics.org> |
| URL: | https://github.com/msuzen/Julia |
| BugReports: | https://github.com/msuzen/Julia/issues |
| Description: | Generates image data for fractals (Julia and Mandelbrot sets) on the complex plane in the given region and resolution. Benoit B Mandelbrot (1982). |
| License: | GPL-3 |
| NeedsCompilation: | no |
| Packaged: | 2023-08-27 16:04:29 UTC; msuzen |
| Author: | Mehmet Suzen [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2023-08-27 16:20:05 UTC |
Julia Set Generator in a Square Region
Description
'JuliaImage' returns two dimensional array representing escape values from on the square region in complex plane. Escape values (which measures the number of iteration before the lenght of the complex value reaches to 2).
Usage
JuliaImage(imageN, centre, L, C)
Arguments
imageN |
Number of pixels to equally space division of one side if the square region. |
centre |
A complex number that determines the centre of the square region |
L |
A side length of the square region on the complex plane. |
C |
Complex coefficient |
Details
Julia Set is defined as the set of initial complex values where the z = z^2 +C does not diverge to infinity. C is an arbitrary complex constant that does not change during the iteration by definition.
Value
It returns a 2D array of real values from 0 to 1. The array correspods to image on the complex plane.
Note
Post processing to plot/color mapping of the Julia set for visualisation can be done by using the array generated. See examples to get a png output.
Author(s)
Mehmet Suzen <mehmet.suzen@physics.org>
References
Gaston Julia (1918) "Memoire sur l'iteration des fonctions rationnelles," Journal de Mathematiques Pures et Appliquees, vol. 8, pages 47-245.
See Also
Examples
#
# Generating png of the Julia set
# C is 1 minus the golden ratio
#
imageN <- 5; # increase this to see images
centre <- 0.0
L <- 4.0
C <- 1i-1.6180339887;# Golden Ratio
image <- JuliaImage(imageN,centre,L,C);
#library(png)
#file <- "julia1.png"
#writePNG(image,file); # possible visulation
#
# Generating png of the Julia set
# different coefficient.
#
imageN <- 5; # increase this to see images
centre <- 0.0
L <- 4.0
C <- -0.70176-0.3842i
image <- JuliaImage(imageN,centre,L,C);
#library(png)
#file <- "julia2.png"
#writePNG(image,file); # possible visulation
JuliaIterate
Description
'JuliaIterate' returns the number of iteration until a complex value diverges for the Julia map for a give complex number.
Usage
JuliaIterate(z, C)
Arguments
z |
A complex coordinate (initial value for the map). |
C |
A complex constant. |
Details
'JuliaIterate' returns the number of iteration until a complex value diverges for the Julia map for a give complex number.
Value
Number of iterations.
Note
Iterative function.
Author(s)
Mehmet Suzen <mehmet.suzen@physics.org>
References
The Fractal Geometry of Nature, Benoit B. Mandelbrot, W.H.Freeman & Co Ltd (18 Nov 1982)
See Also
JuliaIterate and MandelIterate
Examples
z<-0+0i
C <- 1-1.6180339887;# Golden Ratio
it<- JuliaIterate(z,C)
Mandelbrot Set Generator in a Square Domain
Description
'MandelImage' returns two dimensional array representing escape values from on the square region in complex plane. Escape values (which measures the number of iteration before the lenght of the complex value reaches to 2.)
Usage
MandelImage(imageN, centre, L)
Arguments
imageN |
Number of pixels to equally space division of one side if the square region. |
centre |
A complex number that determines the centre of the square region. |
L |
A side length of the square region on the complex plane. |
Details
Mandelbrot set is defined as the set of initial complex values where the z = z^2 +z_0 does not diverge to infinity. Initial value for the map is taken to be zero and z_0 is the complex coordinate.
Value
Returns a matrix.
Note
Returns a matrix
Author(s)
Mehmet Suzen <mehmet.suzen@physics.org>
References
The Fractal Geometry of Nature, Benoit B. Mandelbrot, W.H.Freeman & Co Ltd (18 Nov 1982)
See Also
Examples
# png image
imageN <- 5; # increase this to see image
centre <- 0.0
L <- 4.0
image<-MandelImage(imageN,centre,L);
#file <- "mandelbrot1.png"
# writePNG(image,file); # possible visualisation
# Closer lookup to set
imageN <- 5;
centre <- -0.5
L <- 2.0
image<-MandelImage(imageN,centre,L);
# file <- "mandelbrot.png"
#writePNG(image,file); # possible visualisation
MandelIterate
Description
'MandelIterate' returns the number of iteration until a complex value diverges for the Mandelbrot map for a give complex number.
Usage
MandelIterate(z_0)
Arguments
z_0 |
A complex coordinate (constant coefficient value for the map) |
Details
Iterate function.
Value
Returns an integer
Note
Iterate function
Author(s)
Mehmet Suzen <mehmet.suzen@physics.org>
References
The Fractal Geometry of Nature, Benoit B. Mandelbrot, W.H.Freeman & Co Ltd (18 Nov 1982)
See Also
JuliaIterate and MandelIterate
Examples
z_0 <- 0-0.5i
it <- MandelIterate(z_0)