Mandelbrot - ObjectVision/GeoDMS GitHub Wiki

Miscellaneous functions Mandelbrot

The Mandelbrot function generates the Mandelbrot set fractal as a raster image.

Mandelbrot(gridDomain: SPoint unit, countRange: UInt32 unit, origin: {Void}->DPoint, increment: {Void}->DPoint) -> gridDomain->UInt32

Computes the Mandelbrot set iteration count for each cell in a grid. The Mandelbrot set is defined by the iteration z(n+1) = z(n)² + c, starting with z(0) = c.

For each grid cell:

1. Map cell coordinates to complex plane position using origin and increment
2. Iterate until |z| > 2 (escaped) or maximum count reached
3. Return the iteration count (higher = closer to Mandelbrot set boundary)

Cells inside the Mandelbrot set return maxCount; cells outside return the escape iteration.

The DPoint arguments use:

• X component → Real part of complex number
• Y component → Imaginary part of complex number

Time complexity: O(n × maxCount) worst case, where n is the number of grid cells.

In practice, many cells escape quickly, so average performance is much better. The computation is embarrassingly parallel and benefits from multiple cores.

• gridDomain must be an SPoint (Int16 point) range
• countRange determines the maximum iteration count (higher = more detail, slower)
• origin and increment must have Void domain (parameter values)

// Create a 800x600 grid centered on the Mandelbrot set
unit<SPoint> MandelbrotGrid := range(SPoint, point_xy(0s, 0s), point_xy(800s, 600s));
unit<UInt32> IterationRange := range(UInt32, 0, 256);  // max 256 iterations

parameter<DPoint> origin    := point_xy(-2.5, -1.2);   // complex plane origin
parameter<DPoint> increment := point_xy(0.004, 0.004); // 0.004 per pixel

attribute<UInt32> fractal (MandelbrotGrid) := Mandelbrot(
    MandelbrotGrid,
    IterationRange,
    origin,
    increment
);

// Visualize with color palette based on iteration count

The resulting iteration counts can be visualized by:

• Mapping counts to a color palette
• Using logarithmic scaling for better contrast
• Applying smooth coloring techniques

Classic Mandelbrot set views:

• Full set: origin=(-2.5, -1.2), range ~3.5
• Seahorse valley: origin=(-0.75, 0.1), range ~0.1
• Spiral: origin=(-0.761574, -0.0847596), range ~0.0001

• Miscellaneous functions

5.0