The Distance color scheme colors a point by the distance it is from another object or set of objects. These schemes are a slight alteration of an infinite number of coloring methods called orbit traps.
This program offers several types of distance schemes as well as the ability to combine any number of them in several ways. The Gaussian Integer schemes color points by how close or far the points in their orbits get to Gaussian Integers (a + bi where both a and b are integers). There are also schemes that track how close the orbit gets to the real axis, imaginary axis, the origin, or the unit circle.
To select a color scheme check the box beside its title. If more then one scheme is checked then the program will combine the selected schemes using the combine method selected. There are three combine methods, minimum, maximum, and average. Minimum will calculate the index of each selected scheme and use the smallest one for selecting the palette color, maximum will use the largest one for selecting the palette color, and average will average the indexes for color selection.
When data on these distances is calculated we must eventually normalize the calculation to the 0-1 scale so that the point can be colored. With the Gaussian Integer schemes the closest a point can get to a Gaussian Integer is when it is a Gaussian Integer and hence the distance is 0. On the other hand, the furthest any point can get from a Gaussian Integer is (2)^(1/2)/2 = 0.7071067811... So all of these distances are divided by 0.7071067811... to normalize to the 0-1 scale.
The other methods are not bounded in this manner. Orbits can get very far from an axis or the origin. These schemes will track the maximum distance of the orbits of each of the points on the image and divide all the data points by this maximum, standard way to normalize data to 0-1 is to divide by the maximum value. This has a tendency to favor the very large values, specifically if one is graphing a fractal on the inverse complex plane where the points in the center are transformed to very large values. To help reduce this wash-out we have two options, Scale by and Overload, that allow the user to override the calculated maximums. The Scale by option allows the user to select the default mode of Data, which uses the calculated maximums, and Constant, which replaces the calculated maximum by the constant value that is supplied. If the constant value that is supplied is smaller than the calculated maximum there will be values (after normalization) that exceed 1. The Overload option allows the user to select how these values are handled. The Trim option will set any value larger than 1 to 1 and the Repeat option will cycle the values around, so a value of 1.234 will be changed to 0.234 and the color that is 23.4% of the way down the palette will be used.
Using the closest point to a Gaussian Integer,
Using the furthest point from all Gaussian Integers,
Using the average distances to the closest Gaussian Integer,
Note: With Gaussian integer methods, repeating the palette helps refine the final image.
Using the distance to the unit circle,
Using the distance to the real axis,
Using the distance to the imaginary axis,
Using the distance to the origin,
If one zooms in and works with the palette, some very interesting images can appear.
Note: These methods produce interesting images for many types of fractal families, not just the polynomial-like fractals. As with curvature, you will sometimes want to use a larger bailout radius, depending on the what system is being graphed. The smaller radius has a tendency to cut off the spikes that are common in this coloring scheme.