complex op• Data kinds: cimage → rgbimage
• Call: import fullseye as fs; fs.ledger.cplx_domain_colour(field, gamma=1.0, bands=0.0, saturation=1.0) (to call the implementation directly, import mathops; mathops.cplx_domain_colour(field, gamma=1.0, bands=0.0, saturation=1.0); from the registry, opsmath.get("cplx_domain_colour"))
Domain colouring: turn a complex field into an RGB image you can read.
Hue carries `arg(z)` (one full turn of the colour wheel per turn of the
argument, red at `arg = 0); value carries |z|` through the **strictly
increasing** map `t/(1+t) with t = |z|**gamma`, so a zero is exactly
black, `|z| = 1` is half brightness and a large modulus saturates at full
brightness. The saturation is not dropped far from the origin, so the
hue — and with it the argument — stays readable everywhere; pass
`saturation = 0` for a plain grey ramp of the modulus alone. With
`bands = 0` (the default) the
brightness is monotone in `|z|`, which means the picture is *invertible*:
the argument comes back out of the hue and the modulus out of the value.
★The picture proves a theorem. Walk a small circle around a zero of
order *m* and the hue cycles through the colour wheel exactly *m* times;
around a pole of order *m*, *m* times the other way. That is the argument
principle, read off the image with no numbers — and this family's
`cplx_winding_number` counts the same integer from the field itself, so
the drawing and the arithmetic check each other.
Parameters
----------
field : complex 2-D array (`cimage`)
Typically from :func:cplx_rational_field.
gamma : float > 0
Compresses (`<1) or stretches (>1`) the modulus ramp.
bands : float >= 0
Classic modulus contours: `bands` shading cycles per decade of
`|z|`. Non-zero breaks monotonicity (that is the point — it draws
level lines), so the inverse-mapping guarantee above holds only at 0.
saturation : float in [0, 1]
Raises `ValueError`: not a 2-D complex array; non-finite samples
(a pole sampled exactly — see :func:cplx_rational_field); `gamma <= 0`;
`bands < 0; saturation` outside [0, 1].
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
rgbimage as input)—
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.