The Curvature color scheme tracks the orbit of the point and estimates the curvature of the path using discrete approximation methods. Specifically, we calculate |arg((zn - zn-1)/(zn-1 - zn-2))| for each of the three consecutive points in the orbit. The program tracks the minimum, maximum, and the average of these values, each can be used in separate color schemes. This method does not tend to be very interesting for the inside points so we have included it only in the plotting of outside points.
The program offers two interpolation modes linear interpolation and Catmull-Rom Spline interpolation. These interpolation methods make the transition over iteration levels continuous and produces unbroken streams of color. Linear interpolation combines the last and next to last iteration curvature estimate and the Catmull-Rom Spline interpolation uses a cubic spline through the last four curvature estimates. These interpolation methods assume that the dynamical system is based on a polynomial iterative function, i.e. az^n + bz^k + ... + c where the exponents are real number constants. Thus, when using non-polynomial systems the results are not as spectacular. The Interpolation Degree should be the same as the degree (highest power) of the polynomial-like function. So for the Mandelbrot set the degree should be 2 and for z = z^1.5 + c it should be 1.5.
Note: This coloring scheme does not tend to work very well in a convergence bailout mode. Also, to achieve the smoothness, a fractional escape iteration needs to be estimated. The same estimate is done here as for the polynomial smoothing algorithm in the iteration color scheme, hence the curvature coloring works best for polynomial-like systems. One can apply the method to non-polynomial systems with varying success but you almost always see discontinuous coloring. One final note, you will want to use a large bailout radius with this method, a radius of 1000 or even 10^10 is not uncommon. With a small radius you will see the propagation of errors close to the iteration level changes. Also, the larger the bailout radius is the longer the color streaks tend to be.
Using linear interpolation,
Using Catmull-Rom Spline interpolation,
Although the above images are not that spectacular, zooms using curvature coloring can produce some stunning images.
If one repeats the color palette and has a palette with at least one darker color, a flame affect is easy to create.