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

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

5. Lecture Notes on Astrometry
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. Lecture Notes on Astrometry
Latitude Equation
Lecture Notes on Astrometry

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

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