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Spheroidal Coordinates

Spheroidal Coordinates. (Geographic) Latitude: j Longitude : l Height from reference ellipsoid: h. Geographic Latitude. Geocentric Latitude: f Geographic (=Geodetic) Latitude: j. Zenith. Pole. P. r. Horizon. f. j. Equator. Geocenter. H. Nadir. Spheroidal Coordinates (contd.).

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Spheroidal Coordinates

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  1. Spheroidal Coordinates • (Geographic) Latitude: j • Longitude : l • Height from reference ellipsoid: h Lecture Notes on Astrometry

  2. Geographic Latitude • Geocentric Latitude: f • Geographic (=Geodetic) Latitude: j Zenith Pole P r Horizon f j Equator Geocenter H Nadir Lecture Notes on Astrometry

  3. Spheroidal Coordinates (contd.) • Curvature Radius along/across Meridian: r • Normal of Meridian: N Lecture Notes on Astrometry

  4. Ellipse • Semi-major axis: a • Semi-minor axis: b Lecture Notes on Astrometry

  5. Flattening Factor • Flattening Factor: f • Eccentricity: e, Co-Eccentricity: e’ Lecture Notes on Astrometry

  6. Spheroidal to Rectangular Lecture Notes on Astrometry

  7. Rectangular to Spheroidal • Not-so-easy • Longitude • Latitude Equation • Eliminating h Lecture Notes on Astrometry

  8. Latitude Equation Lecture Notes on Astrometry

  9. Modified Latitude Eq. • New Variable • New Equation Lecture Notes on Astrometry

  10. Derivation Lecture Notes on Astrometry

  11. Solving Modified Latitude Eq. • Reduction of Argument Domain • Variable Domain • Newton Method • Initial Guess Lecture Notes on Astrometry

  12. Newton Method • Most Effective to Solve Nonlinear Eq. • Linear Approx. • Newton Iteration y y=f(x) x0 x1 x x Lecture Notes on Astrometry

  13. Newton Method (contd.) • Second Order • Doubling Number of Correct Digits • Fast but Unstable • Slow when Multiple Roots • Key Points • Bracketing … Simple Root • Stable Starter Lecture Notes on Astrometry

  14. Good Starter for Newton Method • Bracketed Solution • Assumption 1 • Assumption 2 • Stable Starter = Upper Bound Lecture Notes on Astrometry

  15. Application (Mod. Lat. Eq.) • Preparation Lecture Notes on Astrometry

  16. Application (contd.) • Newton Correction • Stable Starter = Lower Bound Lecture Notes on Astrometry

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