circle_packing_apollonian — MATH `construct` op

• 資料種類:無 → table(僅由參數決定的運算子 —— 不接受影像或資料輸入)

• 呼叫: import fullseye as fs; fs.ledger.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0)(要直接呼叫實作,import mathops; mathops.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0);從台帳取用則 opsmath.get("circle_packing_apollonian"))

用法

由笛卡兒四元組生成的阿波羅尼奧斯圓填充 —— 每個圓都背著一條定理。

> 以下的詳細說明為原文 —— 摘要與標題已翻譯。

Four mutually tangent circles satisfy the Descartes circle theorem

(k1 + k2 + k3 + k4)**2 == 2 * (k12 + k22 + k32 + k42)

where `k = 1/r` is the curvature (negative for the enclosing circle). The

theorem is quadratic in `k4`, so a triple of mutually tangent circles has

two solutions and the second is `k4' = 2*(k1+k2+k3) - k4`; recursing on

that reflection fills the gasket. The centres follow the complex form

`k4*z4 = k1*z1 + k2*z2 + k3*z3 +- 2*sqrt(k1*k2*z1*z2 + ...)`, so no

geometry is fitted — every circle is produced by an exact algebraic step.

★Why this earns its place: the drawing carries its own proof. Each circle

can be checked against Descartes to machine precision, tangency is

`|z_i - z_j| == |r_i +- r_j|` exactly, and **an integral quadruple stays

integral for ever** — start from `(-1, 2, 2, 3)` and every curvature in the

infinite packing is an integer (Lagarias-Mallows-Wilks). A drawing routine

that is slightly wrong cannot keep integers integral.

Parameters

----------

curvatures : 4 floats

A Descartes quadruple. The default `(-1, 2, 2, 3)` is the smallest

integral gasket. Must satisfy the theorem to `1e-9` relative.

depth : int >= 0

Reflection levels. Level 0 is the four seed circles; each further level

adds `4 * 3**(level-1), so the total is 2 * 3**depth + 2`.

min_curvature : float

Drop circles smaller than `1/min_curvature` (0 = keep all).

Returns a `table: x, y, radius, curvature, depth`

(the enclosing circle has negative curvature and positive radius).

Raises `ValueError`: not four curvatures; the quadruple does not

satisfy Descartes; every curvature negative or zero (no packing); `depth`

negative or so large the packing exceeds the cap; non-finite input.

HALCON: no operator.

該族通用的輸入契約(fail-closed)

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。

詳細使用指南

• math_metrology 族使用指南

參考(範例資料・文獻)

• 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。

可執行的範例(實際呼叫該運算子並已驗證的樣例)

• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py

型別可銜接的下一個運算子(可接受 `table` 作為輸入)

dynsys_poincare_section

同類別(`construct`)

ford_circles · phyllotaxis_pattern · neighbour_index_gaps · ifs_fractal · ifs_similarity_dimension · space_filling_curve · curve_locality

*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)。*

circle_packing_apollonian — MATH construct op

用法