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"))
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 * (k1**2 + k2**2 + k3**2 + k4**2)
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.
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.
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。
• 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。
• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py
table 作为输入)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
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)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.