complex op• Data kinds: none → image2d (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.cplx_escape_time(kind='mandelbrot', param=0j, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, escape_radius=2.0) (to call the implementation directly, import mathops; mathops.cplx_escape_time(kind='mandelbrot', param=0j, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, escape_radius=2.0); from the registry, opsmath.get("cplx_escape_time"))
How many steps of z -> z**2 + c it takes to leave the escape disc.
kind="mandelbrot" varies c over the window from z = 0; kind="julia" fixes c = param and varies the starting z. The result is a float image of iteration counts; points that never escape carry max_iter.
★This one is gated by theorems, not by a reference picture.
• *Escape radius*: once |z| > 2 (with |c| <= 2) the orbit diverges, so escape_radius = 2 is not a tuning knob but the exact threshold.
• *Main cardioid*: c lies in the main cardioid iff the fixed point z* = (1 - sqrt(1 - 4c)) / 2 is attracting, i.e. |2 z*| < 1. Every such c provably never escapes, so those pixels must read exactly max_iter — a closed-form interior test for a set usually drawn by iteration alone. The period-2 bulb |c + 1| < 1/4 is the same kind of statement.
• *Symmetry*: both families are invariant under conjugation, so the image must be exactly mirror-symmetric about the real axis (bit for bit, not to a tolerance) when the window is.
Raises ValueError: unknown kind; non-finite param; max_iter < 1; escape_radius <= 0; degenerate window.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
image2d as input)wave_fringe_period · abel_inverse_image · abel_revolve · hankel_image · dct_transform · dwt_transform · chebyshev_coeffs_nd · geodesic_heat_grid
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.