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.
Direct answer
- 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.
Mechanism and method
- 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.
What is measured
- θ and θ + 360°k represent the same direction.
- Normalization is idempotent.
- The output remains inside the declared interval.
Angle normalisation, executed
- Method: normalize-angle-microdegrees, evaluated in fixed-point microdegrees on signed 64-bit integers. Units: microdegree, shown here in degrees.
- Turn equivalence: 400° → 40°, 760° → 40°, −320° → 40°. Three angles differing by whole turns give one result, which is the page's statement that θ and θ + 360°k are the same direction.
- Idempotence: 400° → 40° → 40°. Applying normalisation to its own output changes nothing.
- The half-open interval: 359.999999° → 359.999999° stays inside, while 360° → 0° rather than 360. That is what makes the upper end open rather than closed.
- Assumptions: Angles are exact in microdegrees; a value that is not a whole number of microdegrees is outside this representation. The kernel avoids floating point entirely, so these results carry no rounding from binary fractions.
- Uncertainty: None from arithmetic: the operation is exact on integers and aborts on overflow rather than wrapping. Any uncertainty belongs to the input measurement, not to this step.
- Executed by kernel sha256:9046cc9127b3992b962e1a1f2be7bcf62c9a8e355bd29a4d58f2fb2a597df7f3. Rebuild the kernel with npm run build:wasm-kernel, which is deterministic, confirm its digest matches kernelSha256, then re-execute each operation on the recorded inputs.
- What the receipt does not establish: That the kernel is correct. A receipt records what it computed, not whether the computation is right. That the operation models anything outside itself. That the page’s claims are true. These cases exhibit stated invariants on specific values, and exhibiting is not proving.
Limitations
- Modulo does not reduce upstream coordinate uncertainty.
- Values near 0° require boundary-aware tolerances.
- Rounded display values must not replace full-precision inputs.
- The angular unit is declared.
- The zero direction and positive orientation are fixed.
- A half-open output interval is chosen.
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.
Bridge: Canonical celestial longitude
- Normalize continuous ecliptic longitude while retaining full precision, frame, and unwrapped motion.
- Input: raw longitude
- Input: angular unit
- Input: reference frame
- Output: canonical longitude
- Output: boundary distance
- Output: unwrapped companion value
- Limit: The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity.
Bridge: House and aspect boundary classification
- Classify continuous angular geometry under a named house or aspect convention.
- Input: continuous longitudes
- Input: house system
- Input: orb policy
- Output: derived labels
- Output: boundary uncertainty
- Output: convention ID
- Limit: Classification is convention-dependent and is not a measured physical property.