IQ Mandelbox (2D edition) — iterated box-fold / ball-fold / scale / translate

Math essence

Per pixel: v = (x, y). For k in 1..iterations: v = box_fold(v, 1) where each component is mirrored into [-1, 1]; v = ball_fold(v, r_min, r_fixed) which multiplies by r_fixed^2/|v|^2 when |v| < r_min; v = scale * v; v = v + c(t). Track escape-or-bailout radius per iteration into m; tonemap m/(1+m); map through a + b*cos(2pi*(c*t + d)).

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/shader_legends_2/IQMandelbox2D/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


def _box_fold(v: np.ndarray, s: float) -> np.ndarray:
    """Mirror v into [-s, s]^d.  This is the IQ Mandelbox 'box fold'
    step: any component greater than +s is reflected back, and any
    component less than -s is reflected back, iteratively until
    |v_i| <= s."""
    sign = np.where(v > 0.0, 1.0, -1.0)
    absv = np.abs(v)
    # m = how many s-boxes have we passed in the + direction
    m = np.floor((absv + s) / (2.0 * s))
    # pos within the (-s, s) interval for that box
    p = absv - 2.0 * m * s
    # alternate: if m is even we're in an inward copy; odd — outward
    p = np.where(np.mod(m, 2.0) < 0.5, p, 2.0 * s - p)
    return sign * p * (s / max(s, 1e-6))


def _ball_fold(v: np.ndarray, r_min: float, r_fixed: float) -> np.ndarray:
    """IQ Mandelbox 'ball fold': keep |v| above r_min.  If
    |v|^2 < r_min^2, scale up by r_fixed^2 / |v|^2.  Otherwise
    pass through.  This is the divisiveness of the Mandelbox —
    points too close to the origin are pushed out, points outside
    the fixed ball are squashed toward r_fixed."""
    r2 = v[..., 0] ** 2 + v[..., 1] ** 2
    factor = np.where(
        r2 < r_min * r_min,
        r_fixed * r_fixed / np.maximum(r2, 1e-6),
        1.0,
    )
    out = np.empty_like(v)
    out[..., 0] = v[..., 0] * factor
    out[..., 1] = v[..., 1] * factor
    return out


def _scale(v: np.ndarray, s: float) -> np.ndarray:
    return v * s


def _translate(v: np.ndarray, c0: float, c1: float) -> np.ndarray:
    out = np.empty_like(v)
    out[..., 0] = v[..., 0] + c0
    out[..., 1] = v[..., 1] + c1
    return out


class IQMandelbox2D:
    """Mandelbox, 2D edition — Inigo Quilez (iquilezles.org, ~2010).

    The full 3D Mandelbox (https://iquilezles.org/articles/mandelbulb/,
    https://www.shadertoy.com/view/Mdf3zX) iterates

        v := scale * box_fold(ball_fold(v)) + c