circle_packing_apollonian — MATH `construct` op

• 数据种类:无 → table(仅由参数决定的算子 —— 不接受图像或数据输入)

• 调用: import fullseye as fs; fs.ledger.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0)(要直接调用实现,import mathops; mathops.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0);从台账取用则 opsmath.get("circle_packing_apollonian"))

用法

由笛卡儿四元组生成的阿波罗尼奥斯圆填充 —— 每个圆都背着一条定理。

> 以下的详细说明为原文 —— 摘要与标题已翻译。

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.

该族通用的输入契约(fail-closed)

mathops 的每个算子都先校验输入再计算(不让任何东西无声通过):

• **complex 输入一律 ValueError** —— 强制转成 float64 会无声丢掉虚部(numpy 只发一个 ComplexWarning,然后返回一个「看着合理却是错的」实数)。请显式写出 .real/.imag/abs(),或改用支持复数的 complexops。

• **含被掩元素的 masked array 一律 ValueError** —— 拒绝「剥掉掩码直接使用下面原值」的隐式转换。请显式选择填充还是丢弃。

• **所有输入中的 NaN/Inf 一律 ValueError**(明确给出个数后拒绝 —— 它会污染整个结果)。

• 形状严格:不对 1-D 与 2-D 做隐式提升或广播(向量槽位收到矩阵、矩阵槽位收到向量都是 ValueError;请显式 reshape)。

• 尺寸上限:接受矩阵的算子与 stat_histogram 的 bins,超过 mathops.MAX_ELEMENTS(2^26 ≈ 6700 万个元素)即 ValueError。

详细使用指南

• math_metrology 族使用指南

参考(示例数据・文献)

• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。

• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。

• 算法的正典(作者・年份)与用途见上面的族使用指南。

可运行的示例(实际调用该算子并已验证的样例)

• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py

类型可衔接的下一个算子(可接受 `table` 作为输入)

dynsys_poincare_section

同类别(`construct`)

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

*Provenance: mathops.py — MATH 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*

circle_packing_apollonian — MATH construct op

用法