Orbit trap

A point-based orbit trap colours a point based upon how close a function's orbit comes to a single point, typically the origin.

A line-based orbit trap colours a point based upon how close a function's orbit comes to one or more lines, typically vertical or horizontal (x=a or y=a lines). Pickover stalks are an example of a line based orbit trap which use two lines.

Orbit traps are typically used with the class of two-dimensional fractals based on an iterative function. A program that creates such a fractal colours each pixel, which represent discrete points in the complex plane, based upon the behaviour of those points when they pass through a function a set number of times.

The best known example of this kind of fractal is the Mandelbrot set, which is based upon the function z_n+1 = z_n² + c. The most common way of colouring Mandelbrot images is by taking the number of iterations required to reach a certain bailout value and then assigning that value a colour. This is called the escape time algorithm.

A program that colours the Mandelbrot set using a point-based orbit trap will assign each pixel with a “distance” variable, that will typically be very high when first assigned:

double distance = 10e5

As the program passes the complex value through the iterative function it will check the distance between each point in the orbit and the trap point. The value of the distance variable will be the shortest distance found during the iteration:

private double getDistance(Complex c,
                           Complex point,
                           int maxIteration)
{        
    double distance = 1e20;
    Complex z = new Complex(0, 0);
        
    for (int i=0; i<maxIteration; i++)
    {
        // Perform Mandelbrot iteration
        z = z.multiply(z);
        z = z.add(c);
               
        // Set new distance dist = min( dist, |z-point| )
        Complex zMinusPoint = new Complex(z);
        zMinusPoint = zMinusPoint.subtract(point);
            
        double zMinusPointModulus = zMinusPoint.magnitude();
        if (zMinusPointModulus < distance)
            distance = zMinusPointModulus;
    }
        
    return distance;
}

• Carlson, Paul W. (1999), "Two artistic orbit trap rendering methods for Newton M-set fractals", Computers & Graphics, 23 (6): 925–931, doi:10.1016/S0097-8493(99)00123-5.
• Lu, Jian; Ye, Zhongxing; Zou, Yuru; Ye, Ruisong (2005), "Orbit trap rendering methods for generating artistic images with crystallographic symmetries", Computers & Graphics, 29 (5): 787–794, doi:10.1016/j.cag.2005.08.008.