geometry op• Data kinds: mesh → table
• Call: import fullseye as fs; fs.ledger.angle_defect(vertices, faces=None) -> 'dict' (to call the implementation directly, import mathgeometry; mathgeometry.angle_defect(vertices, faces=None) -> 'dict'; from the registry, opsmath.get("angle_defect"))
Per-vertex angle defect (discrete Gaussian curvature) K_v = 2π − Σ(angles of the triangles at that vertex). For boundary vertices, π − Σ (geodesic curvature).
On a closed mesh Σ K_v = 2πχ holds exactly (Descartes' theorem). With a boundary, Σ(interior K) + Σ(boundary π − Σθ) = 2πχ (discrete Gauss–Bonnet). Returns {"defect", "total", "two_pi_chi", "is_boundary"}.
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.
• geometry_tour — py -3.11 examples/geometry_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
geometry)mesh_euler_characteristic · delaunay_triangulate · geodesic_heat · geodesic_heat_grid · mesh_torus · nurbs_curve · nurbs_circle · nurbs_surface
*Provenance: mathgeometry.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.