# Orbit trapped Julia fractal

**Saturn Elephant**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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Given a complex number \(z_c\), called
*the Julia point*, the corresponding *Julia fractal* is
obtained by iterating
\(z \mapsto z^2 + z_c\) for each
complex number \(z\) until the modulus
of \(z\) exceeds a certain threshold or
the maximal number of iterations is attained. Then a color is assigned
to \(z\).

An *orbit trapped Julia fractal* is obtained in the same way, but
the iteration is stopped whenever
\(z\) is close enough to a given set
such as a square or a circle. In the example shown below, we take the
two axes as this trapping set.

I also add something: instead of looking at the distance between \(z\) and the two axes, I look at the distance between \(z\) and the axes after having rotated \(z\) by an angle \(\alpha\). Then I’ll vary \(\alpha\) to get an animation.

So here is the code of this algorithm; the color assigned to the final \(z\) is defined in function of the value of the trapping function (the distance):

# trapping function: # distance (up to factor 1/0.03) between alpha-rotated z and axes f <- function(z, alpha) { z <- z * exp(1i*alpha) min(abs(Re(z)), abs(Im(z))) / 0.03 } # choose the Julia point juliaPoint <- complex(real = -0.687, imaginary = 0.299015) # main function Julia <- Vectorize(function(x, y, juliaPoint, alpha) { # counter i <- 0L # current point, to be iterated z <- complex(real = x, imaginary = y) # iterations while(i < 100L && Mod(z) < 100 && (i < 2L || f(z, alpha) > 1)) { z <- z^2 + juliaPoint i <- i + 1L } # now assign a color to the resulting z fz <- 2 * f(z, alpha) hsv( # h, s, v must be in (0, 1) h = (Arg(z) + pi) / (2 * pi), s = min(1, fz), v = max(0, min(1, 2 - fz)) ) })

The condition `i < 2L`

ensures that the iteration is not
stopped at the beginning. Let’s plot a first image:

# run the orbit trapping of Julia n <- 2048L x_ <- seq(-2, 2, length.out = n) y_ <- seq(-1.5, 1.5, length.out = n) img <- t(outer(x_, y_, Julia, juliaPoint = juliaPoint, alpha = 0)) # plot opar <- par(mar = c(0, 0, 0, 0), bg = "black") plot(c(-100, 100), c(-100, 100), type = "n", asp = 3/4, xlab = NA, ylab = NA, axes = FALSE, xaxs = "i", yaxs = "i") rasterImage(img, -100, -100, 100, 100) par(opar)

And here is the animation obtained by varying the angle \(\alpha\):

Such a fractal is easily and efficiently rendered as a *shader*.
Click here
to play with the shader, in which the cursor of the mouse is used to
take the Julia point. I also modified the trapping function (\(z^3\)
instead of \(z\)).

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**Saturn Elephant**.

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