complex op• Data kinds: cpoints → cpoints
• Call: import mathops; mathops.cplx_mobius(z, a, b, c, d) (or opsmath.get("cplx_mobius"))
Möbius (linear fractional) map `w = (a z + b) / (c z + d)`.
The automorphisms of the Riemann sphere: every Möbius map is conformal and
sends circles-and-lines to circles-and-lines. Two standard cases the tests
pin: the Cayley transform `(z - i)/(z + i)` maps the real axis onto the
unit circle (`|w| = 1) and i to 0; the inversion 1/z` maps the
unit circle onto itself.
The determinant `a d - b c` must not vanish — that degenerate case is not
a map but a constant (every point collapses to `a/c`), which is refused
rather than returned as a suspiciously uniform answer.
Raises `ValueError: |a d - b c| below 1e-12` of the
coefficient scale (degenerate/constant map), a sample at the pole
`z = -d/c` (the image is the point at infinity, which float64 cannot
represent), an overflowed result (a sample microscopically close to that
pole), plus the usual shape and finiteness contracts.
HALCON: no operator (`projective_trans_point_2d` is the real-plane
projective analogue).
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.
• math_complex — py -3.11 examples/math_complex.py
cpoints as input)cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
*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.