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disc

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.

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fract variations

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.

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Elliptic

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.

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unpolar and foci

The unpolar variation is the opposite of polar2; see the polar description. Whereas polar2 treats the log-polar coordinates of a point as rectangular coordinates, unpolar treats the rectangular coordinates of a point and treats them as log-polar coordinates.

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bipolar

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.