complex op• データ種: cpoints → measurement
• 呼び出し: import mathops; mathops.cplx_winding_number(z, w=0.0) (または opsmath.get("cplx_winding_number"))
Winding number of a closed contour around a point (turning number).
How many times the polygon `z` (closing segment implicit) travels
counter-clockwise around *w*: `+1` for a simple positively-oriented loop
containing it, `-1 clockwise, 0 outside, ±k for a k`-fold
loop. Computed as the sum of the principal-value argument increments of
`z_k - w divided by 2*pi` and rounded — for a *polygon* that sum is an
exact multiple of `2*pi`, so the result is an exact integer, not an
estimate (the rounding merely removes ~1e-14 of accumulated float error).
Honest limitation — **the count can alias low, and no local check can stop
it**: this is the winding number of the polygon *through the samples*,
which equals that of the underlying curve only if the sampling resolves it.
A segment that turns the ray to *w* by `>= pi` is ambiguous (which side
did it pass?) and raises. Below that there is no contradiction to detect:
`z**5 sampled on a 4-point circle turns exactly pi/2` per step and
counts 1 instead of 5 (measured). From `pi/2` up, a
`RuntimeWarning says so (WIND_ALIAS_WARN`); the only real remedy is
the classical one — refine until the count stops changing.
Raises `ValueError`: *w* coincides with a vertex or lies on a segment
(the winding number is undefined on the contour), a segment subtends
`>= pi` as seen from *w* (undersampled — refine the contour), fewer than
3 points, degenerate contour, non-finite/masked input.
HALCON: no operator (`test_region_point` answers the related but weaker
inside/outside question for regions).
mathops の全 op は入力を検証してから計算する(黙って通さない):
• **complex 入力は ValueError** — float64 への強制変換は虚部を黙って捨てる(numpy は ComplexWarning だけ出して「もっともらしく間違った」実数を返す)。.real/.imag/abs() を明示するか、複素対応の complexops を使う。
• **masked array(masked 要素あり)は ValueError** — マスクを剥がして下の生値を使う暗黙変換を拒否。埋める/落とすを明示する。
• **NaN/Inf は全入力で ValueError**(件数を明示して拒否 — 結果全体に伝播するため)。
• 形状は厳格: 1-D と 2-D を暗黙昇格・ブロードキャストしない(vector 枠に matrix、matrix 枠に vector は ValueError。reshape を明示する)。
• サイズ上限: 行列を取る op と stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。
• サンプルデータ カタログ(DL URL / ライセンス) — 2-D は skimage.data(BSD/public)+ 合成、3-D は実データ源(Stanford/PDS 等)の DL URL。
• 演算子の来歴・参考文献 — この op 族の元になった研究/手法の出典。
• アルゴリズムの正典(著者・年)と用途は上記ファミリ使い方ガイドに記載。
• math_complex — py -3.11 examples/math_complex.py
measurement を入力に取れる)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski · cplx_mobius
*Provenance: mathops.py — MATH operator registry. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.