construct op• Data kinds: none → table (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.ford_circles(max_denominator=12, lo=0, hi=1) (to call the implementation directly, import mathops; mathops.ford_circles(max_denominator=12, lo=0, hi=1); from the registry, opsmath.get("ford_circles"))
Ford circles for the Farey fractions — tangency *is* an integer identity.
For a fraction `p/q` in lowest terms the Ford circle sits at
`(p/q, 1/(2q**2)) with radius 1/(2q**2)`. Two such circles are
tangent if and only if `|p*s - q*r| == 1` — the Farey-neighbour
condition — and otherwise strictly disjoint. They never overlap.
★Why this earns its place: the picture's correctness is an identity
between integers, not a tolerance. `|p*s - q*r|` is computed in exact
integer arithmetic and compared with the *geometric* tangency
`|c_i - c_j| == r_i + r_j` measured from the coordinates; the two must
agree on every pair. The number of fractions is the Farey length
`1 + sum(phi(q) for q in 1..n)`, another exact integer.
Returns a `table: x, y, radius, p, q`.
Raises `ValueError: max_denominator < 1; lo >= hi`; the
interval or denominator would exceed the cap.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py
table as input)construct)circle_packing_apollonian · phyllotaxis_pattern · neighbour_index_gaps · ifs_fractal · ifs_similarity_dimension · space_filling_curve · curve_locality
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.