The disc variation is based on polar coordinates, where points are specified by a distance ρ and an angle θ. The basic idea is simple: it just switches ρ and θ (after scaling by π to make things work nicely). This converts wedges centered at the origin to half-rings and vice versa as shown below. The images on the left side are before and on the right side after disc is applied.
Escape-time fractals are in a way the opposite of flame fractals. Both use the idea of an orbit: the sequence of points generated by repeatedly iterating fractal formulas. But while flame fractals plot the actual orbit points to create the flame, escape-time fractals use the orbit of a point to determine how to color it, commonly by determining how long it takes the orbit to cross an imaginary circle (or to “escape”; hence the name “escape-time”). The two methods usually generate very different fractals.
The most commonly used coordinate systems are rectangular coordinates and polar coordinate. But these aren’t the only possibilities. One of the less common coordinate systems is elliptic coordinates. Although their mathematical application is specialized, they can be used in variations to make interesting flames.
The most commonly used coordinate systems are rectangular coordinates and polar coordinates (see polar). But these aren’t the only possibilities. One of the less common coordinate systems is bipolar coordinates. Although their mathematical application is specialized, they can be used in variations to make interesting flames. Rectangular coordinates are based on two families of parallel lines (horizontal and vertical) where each line in the first family intersects each line in the other at a right angle.
The text_wf variation incorporates a line of text into a flame. It is a blur variation like circleblur and starblur, meaning it ignores the input and just generates a shape; but instead of the shape being a circle or star, it is a line of text. The brain processes words differently from images, so this can add a new level of depth to fractal flames. Or the text can be a single symbol (not necessarily a letter), using text_wf as a blur with a specialized shape. Text_wf can be quite versatile.