Generic Julia Set Plotter (version 8.2)

This is an extension of our previous Julia Set (J-set) plotters. The iteration functions F(z,c) are extended from simple "z^n+c" to polynomial functions, rational functions, transcendental functions, and composition of them. This is a simple J-Set Plotter based on escape-time or jump-iteration-number. The plotter divides the points on the complex plane into two groups, bounded points and escaped points, and thus for polynomial functions, the bounded point set is actually filled-in Julia set. The mathematically strict Julia set is the boundary of the filled-in Julia set and escaped point set.

Although, for some functions used in this plotter, the produced J-set may not meet the classic J-set definition in a strict mathematical sense, they can still produce fancy patterns and images. In these cases, J-set images may be sensitive to the Escape Radius, and the Max Iteration number. This plotter gives more options open to the user and thus makes it a more flexible experimental tool. Because the mathematical strictness are sometimes compromised, we may call this experimental tool as a Generic J-Set Plotter.

For every iteration function F(z,c), and every point on the complex plane c(xc,yc), there is a J-set image. That is, for a selected iteration function F(z,c), the plotter can generate different J-set for different selected point c(xc,yc).

Quickstart:
Click "Run/Re-Run Plotter" button to plot the default J-set image.
Plot a new J-set by right-clicking OR holding the Shift key and left-clicking on the image area, which selects a new c(xc,yc) point.
On a touch-screen device, tap on the image area to select a new c(xc,yc) point.
To zoom in on an area of the image, hold down the left button and drag a selection with your mouse.
Change "F(z,c)" with the select menu to explore 146 different iteration functions or composition functions.
Further explore the image by changing the "Escape Radius", "Max Iteration","Hue(Color)", and "Plot Size" options.

→ CLICK HERE TO [ SHOW ] MORE USAGE INFORMATION

1. Click "Run/Re-Run Plotter" button to start, the plotter will display a J-set for a pre-defined F(z,c) and c(xc,yc) point.

2. Then, you can select a different c(xc,yc) point inside the "Plot Area" using one of the following methods:
use mouse "right" button.
use mouse "left" button + "shift key", if your mouse does not have a right button.
for a "Touch Screen" device, just "touch" at the point you want to selected.
manually input xc and yc values into the text boxes, and then click "Run/Re-Run Plotter" button.

The following operations are similar to that of the M-set plotter.

3. You can use the Mouse Left button(down-move-up) to select a specific portion of the image to zoom-in.
For "Touch Screen Device", J-set plotter currently does NOT provide zoom-in function.

4. The Zoom-In operation can be repeated/iterated to see more details of an image area.
The "Zoom-out/Back" button can be used to reverse the Zoom-In operation step by step.
At any zoom-in stage, the user have options to:
select a larger "Max Iteration" number to increase the image resolution,
select different "Hue(Color)" algorithms to signify different aspects of image details,
select different "Escape Radius" to explore its effect on the image,
change the "Plot Size" to get a bigger size and higher resolution image,
use "Reset/Init" button to jump back by one step to the initial stage and displays the whole image again,
input a number in text box "Set Init/Default Plot Range" and press "Reset/Init" button to increase the plot range, if the image is larger than the canvas.
or, select a different F(z,c) function to start a new plot session.

5. In theory, the J-set image zoom-In levels can be infinite, whereas the actual zoom-in levels of this plotter are limited by its accuracy. 
A fuzzy image, and/or the displayed mouse (x, y) position showing no or little change reached its accuracy limit.
r(x,y) is the distance between current mouse position and the origin point (0,0).

6. J-set sets plotting can sometimes be computation intensive. Combination of a big "Max Iteration" number, a large "Plot Size", and a high order(degree) F(z,c), can trap the plotter in a long time calculating status. 
In this situation, if you do not like waiting, you can click on the "Restart/Reload This Web Page" link to stop the running plotter and start a new session.

More Information for the J-set/Multijulia sets Plotter:

The J-set is generated by iteration of a complex equation:
`z(n+1)=F(z(n),c(xc,yc))`
For J-set plotting, c(xc,yc) is constant for a image plot session.

Using a point z0(x0,y0) on the complex plane as the initial condition for the iteration, with designated iteration parameters of a maximum iteration number, a escape radius, and an iteration condition abs(z(n))<=escape radius:
If the iteration reaches the maximum iteration number, the z0(x0,y0) point is defined as a member of filled-in J-set or bounded set;
If during the iteration, the condition abs(z(n))<=escape radius fails, the iteration is stopped and the point is then defined as a non-J-set point or escaped point.

The escape or jump iteration number, denoted as Jmp for each non-J-set point, represents the property of the point. Coloring each escaped point according to its Jmp value will produce a colorful and beautiful J-set image or pattern.

This plotter provides 24 types of simple coloring/rendering algorithms and they are displayed in the Hue(Color) selection menu.

0. Hue(Color)="((Jmp-Min)*360)/(Max-Min)" 
1. Hue(Color)="((Jmp-Max)*360)/(Min-Max)" 
2. Hue(Color)="(Jmp/IteMax)*360"
3. Hue(Color)= "((IteMax-Jmp)/IteMax)*360"
4. Hue(Color)= "Jmp"
5. Hue(Color)= "IteMax-Jmp"
6. Hue(Color)= "((2*Jmp+10)/IteMax)*360"
7. Hue(Color)= "((Jmp/IteMax)*360)^2%360"
8. Hue(Color)="((Jmp-Min)*120)/(Max-Min)"
9. Hue(Color)= "120+((Jmp-Min)*120)/(Max-Min)"
10. Hue(Color)="240+((Jmp-Min)*120)/(Max-Min)"
11. Hue(Color)= "240+((Jmp-Max)*120)/(Min-Max)"
12. Hue(Color)= "120+((Jmp-Max)*120)/(Min-Max)"
13. Hue(Color)= "((Jmp-Max)*120)/(Min-Max)"

14. "H=Hmin,L=Lmin->Lmax"
Hue(Color)=Hmin
Lightness =lmin+((Jmp-Min)*(lmax-lmin))/(Max-Min); //L=Lmin->Lmax           
15. "H=Hmin,L=Lmax->Lmin" 
Hue(Color)=H=Hmin
Lightness=lmin+((nJmp-Max)*(lmax-lmin))/(Min-Max); //L=Lmax->Lmin
16. "H=Hmin->Hmax,L=Lmin->Lmax" 
Hue(Color)=Hmin+((Jmp-Min)*(Hmax-Hmin))/(Max-Min);//H=Hmin->Hmax
Lightness=lmin+((Jmp-Min)*(lmax-lmin))/(Max-Min); //L=Lmin->Lmax
17. "H=Hmin->Hmax,L=Lmax->Lmin"
Hue(Color)=Hmin+((Jmp-Min)*(Hmax-Hmin))/(Max-Min);//H=Hmin->Hmax
Lightness=lmin+((nJmp-Max)*(lmax-lmin))/(Min-Max); //L=Lmax->Lmin
18. "H=Hmax->Hmin,L=Lmin->Lmax" 
Hue(Color)=Hmin+((nJmp-Max)*(Hmax-Hmin))/(Min-Max);//H=Hmax->Hmin
Lightness=lmin+((Jmp-Min)*(lmax-lmin))/(Max-Min); //L=Lmin->Lmax
19. "H=Hmax->Hmin,L=Lmax->Lmin"
Hue(Color)=Hmin+((nJmp-Max)*(Hmax-Hmin))/(Min-Max);//H=Hmax->Hmin
Lightness=lmin+((nJmp-Max)*(lmax-lmin))/(Min-Max); //L=Lmax->Lmin
20. "H=7th Hue,L=Lmin->Lmax" 
Hue(Color)="((Jmp/IteMax)*360)^2%360"
Lightness=lmin+((Jmp-Min)*(lmax-lmin))/(Max-Min); //L=Lmin->Lmax
21. "H=7th Hue,L=Lmax->Lmin" 
Hue(Color)="((Jmp/IteMax)*360)^2%360"
Lightness=lmin+((nJmp-Max)*(lmax-lmin))/(Min-Max); //L=Lmax->Lmin
22. Hue(Color)="((Jmp/IteMax)*360)^4%360"             
23. Hue(Color)="((Jmp/IteMax)*360)^8%360" 

where, IteMax = "maximum iteration number", 
Max = maximum of Jmp of the non-J-set points or escaped points for the current/plotted image,
Min = minimum of Jmp of the non-J-set points or escaped points for the current/plotted image,
Hmax, Hmin, Lmax, and Lmin are user input color and lightness parameters,
default Lightness=0.5. 

The following wikipedia link provides more Information for the Julia set:

https://en.wikipedia.org/wiki/Julia_set
https://en.wikipedia.org/wiki/Complex_dynamics
https://en.wikipedia.org/wiki/Escaping_set
https://en.wikipedia.org/wiki/Newton_fractal
https://en.wikipedia.org/wiki/Filled_Julia_set