construct opなし → table(引数だけで決まる op —— 画像やデータの入力を取らない)import fullseye as fs; fs.ledger.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0) (実装を直接呼ぶなら import mathops; mathops.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0)、台帳から引くなら opsmath.get("circle_packing_apollonian"))Apollonian gasket from a Descartes quadruple — every circle a theorem.
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 の全 op は入力を検証してから計算する(黙って通さない):
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。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。ValueError(件数を明示して拒否 — 結果全体に伝播するため)。ValueError。reshape を明示する)。stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。py -3.11 examples/poc_theorems_as_pictures.pytable を入力に取れる)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 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.