Type 2 rational iteration
Each step computes (z² + c) / (z² − c), then squares it. The numerator is zero when z² = −c; the denominator is zero when z² = c. After squaring, every zero and pole is double, leaving the map with two double zeros and two double poles on the Riemann sphere. Its rational degree is four.
As |z| grows, (z² + c) / (z² − c) approaches 1, so the next value lands near 1 no matter how large z becomes. Only a close pass by one of the two poles produces a large value. FractalPark treats |z| above 16 as escaped, so a coloured “fast escape” region records a close encounter with a pole, not a one-way trip to infinity.
Suppose c = 1. The poles are at z² = 1, i.e. z = 1 and z = −1. Choose z₀ = 0.98, very close to the pole at 1. Then (z₀² + c) / (z₀² − c) ≈ (0.9604 + 1) / (0.9604 − 1) = 1.9604 / (−0.0396) ≈ −49.5, and squaring gives z₁ ≈ 2450. The large value then folds back: on the next step, with |z₁| large, the ratio tends to 1 and z₂ lands near 1. The shader colours the pixel according to how quickly the orbit returns near 1 after its pole encounter.
