complex op• 資料種類:cpoints → measurement
• 呼叫:import mathops; mathops.cplx_winding_number(z, w=0.0)(或 opsmath.get("cplx_winding_number"))
閉合圍道繞某點的卷繞數(迴轉數)。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
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 的每個運算子都先檢驗輸入再計算(不讓任何東西無聲通過):
• **complex 輸入一律 ValueError** —— 強制轉成 float64 會無聲丟掉虛部(numpy 只發一個 ComplexWarning,然後回傳一個「看似合理卻是錯的」實數)。請明確寫出 .real/.imag/abs(),或改用支援複數的 complexops。
• **含被遮罩元素的 masked array 一律 ValueError** —— 拒絕「剝掉遮罩直接使用下面原值」的隱式轉換。請明確選擇填補還是丟棄。
• **所有輸入中的 NaN/Inf 一律 ValueError**(明確給出個數後拒絕 —— 它會汙染整個結果)。
• 形狀嚴格:不對 1-D 與 2-D 做隱式提升或廣播(向量槽位收到矩陣、矩陣槽位收到向量都是 ValueError;請明確 reshape)。
• 尺寸上限:接受矩陣的運算子與 stat_histogram 的 bins,超過 mathops.MAX_ELEMENTS(2^26 ≈ 6700 萬個元素)即 ValueError。
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者・年份)與用途見上面的族使用指南。
• math_complex — py -3.11 examples/math_complex.py
measurement 作為輸入)—
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 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.