Working definition
Angle normalization maps equivalent directions that differ by integer turns into a chosen half-open interval, commonly [0°, 360°). The modulo operation makes comparison deterministic, while metadata must preserve units, reference axis, orientation, and any unwrapped value needed to distinguish repeated crossings.
Notation
θ̄ = ((θ mod 360°) + 360°) mod 360°θ ∈ ℝ; θ̄ ∈ [0°, 360°)Assumptions
- The angular unit is declared.
- The zero direction and positive orientation are fixed.
- A half-open output interval is chosen.
Invariants
- θ and θ + 360°k represent the same direction.
- Normalization is idempotent.
- The output remains inside the declared interval.
Reproducible procedure
- Convert the input to the canonical angular unit.
- Apply floor-modulo rather than language remainder for negative inputs.
- Store unwrapped angle or crossing index when chronology matters.
Error and boundary controls
- Modulo does not reduce upstream coordinate uncertainty.
- Values near 0° require boundary-aware tolerances.
- Rounded display values must not replace full-precision inputs.
What this does not establish
A normalized longitude is geometry. A sign, mansion, aspect, or interpretive label derived from it remains convention-dependent and does not gain predictive validity from the normalization.
Explicit applications
2 cross-domain bridges
Canonical celestial longitude
Normalize continuous ecliptic longitude while retaining full precision, frame, and unwrapped motion.
Inputs
- raw longitude
- angular unit
- reference frame
Outputs
- canonical longitude
- boundary distance
- unwrapped companion value
Transformation: Apply floor-modulo to the declared interval and retain provenance.
Limit: The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity.
Open connected system →House and aspect boundary classification
Classify continuous angular geometry under a named house or aspect convention.
Inputs
- continuous longitudes
- house system
- orb policy
Outputs
- derived labels
- boundary uncertainty
- convention ID
Transformation: Normalize and classify only after preserving boundary distance.
Limit: Classification is convention-dependent and is not a measured physical property.
Open connected system →Authoritative references
- [1]Standards of Fundamental Astronomy · International Astronomical Union
Establishes: Authoritative algorithms and conventions for astronomical timescales, Earth orientation, reference systems, astrometry, and celestial-coordinate transformations.
Boundary: SOFA standardizes astronomical computation. It supplies no astrological symbols, meanings, auspiciousness judgments, or evidence of predictive validity.
- [2]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.