Julia Set and Multijulia Set Plotter (version 6.0)

Julia set(J-set) can be generated by many types of complex functions. 
The complex functions used to generate Mandelbrot set(M-set) can also be used to generate J-set. With a little change, the M-set plotter can be used to plot J-set.

For every iteration equation of a M-set(or Multibrot set), and every point on the complex plane c(xc,yc), there is a J-set(or Multijulia Set) image.
That is, for a selected iteration equation, the plotter can generate different J-set for different selected point c(xc,yc).

Check out cool fractal images created with the Mandelbrot and Julia plotting tools in our Fractal Gallery! You will always find the newest creations on our Facebook page.

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 the "Type of M(J)-set" to explore 29 different Julia and Multijulia sets.
Further explore the image by changing the "Plot Size", "Max Iteration", "Hue(Color)", and "Escape Radius" options.

→ CLICK HERE TO [ SHOW ] MORE USAGE INFORMATION
Click "Run/Re-Run Plotter" button to start, the plotter will display a J-set for a pre-defined c(xc,yc) point.
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.

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.
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,
or, select a different Type of J-set/Multijulia sets to start a new plot session.
In theory, the J-set/Multijulia 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 when the mouse is moving over the image, indicate the plotter has reached its accuracy limit.
r(x,y) is the distance between current mouse position and the origin point (0,0).
J-set/Multijulia sets plotting can sometimes be computation intensive. 
Combination of a big "Max Iteration" number, a large "Plot Size", and a high order(degree) Multijulia type, 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.

Click "Run/Re-Run Plotter" Button to Start

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

The J-set is generated by iteration of a simple complex equation: `z_(n+1)=z_(n)^2 + 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 J-set (strictly speaking, filled-in J-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.

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

This plotter provides 14 types of simple coloring algorithms and they are displayed in the Hue(Color) selection menu.
Hue(Color)=((Jmp-Min)*360)/(Max-Min)
Hue(Color)=((Jmp-Max)*250)/(Min-Max)
Hue(Color)=(Jmp/IteMax)*360
Hue(Color)=((IteMax-Jmp)/IteMax)*250
Hue(Color)=Jmp
Hue(Color)=IteMax-Jmp
Hue(Color)=((2*Jmp+10)/IteMax)*360
Hue(Color)=((Jmp/IteMax)*360)^2%360
Hue(Color)=((Jmp-Min)*120)/(Max-Min)
Hue(Color)=120+((Jmp-Min)*120)/(Max-Min)
Hue(Color)=240+((Jmp-Min)*120)/(Max-Min)
Hue(Color)=240+((Jmp-Max)*120)/(Min-Max)
Hue(Color)=120+((Jmp-Max)*120)/(Min-Max)
Hue(Color)=((Jmp-Max)*120)/(Min-Max)
Where, IteMax = "maximum iteration number".
Max = maximum of Jmp of the non-J-set points for the current/plotted image.
Min = minimum of Jmp of the non-J-set points for the current/plotted image.

Variants of J-set equation can also produce interesting images and they are also included in our plotter. 
The variants are called Multijulia sets and their equations are written as: 
`z_(n+1)=z_(n)^(m)+c(xc,yc)`
where m is: 3,4,5,6,7,8,9,10,,,,
or -2,-3,-4,-5,-6,-7,-8,-9,-10,,,,
and
`z_(n+1)=(z_(n)^(m)+c(xc,yc))^(-1)`
where m is: 2,3,4,5,6,7,8,9,10,,,,

Other 2 complex equations are also included.
"Burning Ship" equation:
`z_(n+1)=(abs(Re(z_(n)))-i*abs(Im(z_(n))))^2+c(xc,yc)`
"Tricorn" equation:
`z_(n+1)=(Re(z_(n))-i*Im(z_(n)))^2+c(xc,yc)`

Base on previous version of the M-set plotter, this upgraded plotter uses the offscreen technique to improve GUI responsiveness, the image process technique is used to reduce image plot time, and the multiple web worker(thread) technique is used to speed numerical computation.

Depending on the device hardware, operation system, and browser, the efficiency of multiple web workers can be limited. The reduction of calculation time may not be proportional to the increase in worker number. You can try different worker numbers to find an optimal one for your browser/device.

The following wikipedia link provides more Information for Julia set:
https://en.wikipedia.org/wiki/Julia_set