Working definition
Spherical coordinates represent a point by radial distance and two angular coordinates. Multiple conventions swap angle names or origins, so a reproducible record must declare axis orientation, longitude range, latitude or colatitude, frame, origin, epoch, and whether the position is geometric, astrometric, or apparent.
Notation
x = r cos φ cos λy = r cos φ sin λz = r sin φAssumptions
- The coordinate origin and frame are fixed.
- Angular conventions are explicit.
- Singularities at the poles are handled.
Invariants
- Rotation preserves radial distance.
- Cartesian conversion preserves the represented point within numerical error.
- Longitude is undefined at a coordinate pole.
Reproducible procedure
- Declare the spherical convention and frame.
- Convert to Cartesian form for robust transformations.
- Convert back with quadrant-aware inverse trigonometric functions.
Error and boundary controls
- Longitude becomes ill-conditioned near the poles.
- Parallax matters when the observer origin changes.
- Rounding near classification boundaries can change labels.
What this does not establish
Spherical coordinates locate a direction. They do not supply physical explanation or astrological interpretation.
Explicit applications
1 cross-domain bridges
Angular separation on the celestial sphere
Compute observer-relative separation between measured or modeled directions.
Inputs
- two direction vectors
- reference frame
- epoch
Outputs
- angular separation
- frame record
- uncertainty estimate
Transformation: Convert to a common Cartesian frame and evaluate the stable great-circle separation.
Limit: Angular proximity does not imply physical proximity.
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]JPL Horizons System · NASA Jet Propulsion Laboratory
Establishes: Ephemeris products and documented observer, target, timescale, coordinate, and output conventions for reproducible Solar System state and observable calculations.
Boundary: Ephemeris agreement validates positions under declared conventions; it does not validate downstream symbolic classifications or interpretations.