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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
Spheroidal Coordinates
  • (Geographic) Latitude: j
  • Longitude : l
  • Height from reference ellipsoid: h

Lecture Notes on Astrometry

geographic latitude
Geographic Latitude
  • Geocentric Latitude: f
  • Geographic (=Geodetic) Latitude: j

Zenith

Pole

P

r

Horizon

f

j

Equator

Geocenter

H

Nadir

Lecture Notes on Astrometry

spheroidal coordinates contd
Spheroidal Coordinates (contd.)
  • Curvature Radius along/across Meridian: r
  • Normal of Meridian: N

Lecture Notes on Astrometry

ellipse
Ellipse
  • Semi-major axis: a
  • Semi-minor axis: b

Lecture Notes on Astrometry

flattening factor
Flattening Factor
  • Flattening Factor: f
  • Eccentricity: e, Co-Eccentricity: e’

Lecture Notes on Astrometry

spheroidal to rectangular
Spheroidal to Rectangular

Lecture Notes on Astrometry

rectangular to spheroidal
Rectangular to Spheroidal
  • Not-so-easy
  • Longitude
  • Latitude Equation
  • Eliminating h

Lecture Notes on Astrometry

latitude equation
Latitude Equation

Lecture Notes on Astrometry

modified latitude eq
Modified Latitude Eq.
  • New Variable
  • New Equation

Lecture Notes on Astrometry

derivation
Derivation

Lecture Notes on Astrometry

solving modified latitude eq
Solving Modified Latitude Eq.
  • Reduction of Argument Domain
  • Variable Domain
  • Newton Method
  • Initial Guess

Lecture Notes on Astrometry

newton method
Newton Method
  • Most Effective to Solve Nonlinear Eq.
  • Linear Approx.
  • Newton Iteration

y

y=f(x)

x0

x1

x

x

Lecture Notes on Astrometry

newton method contd
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

good starter for newton method
Good Starter for Newton Method
  • Bracketed Solution
  • Assumption 1
  • Assumption 2
  • Stable Starter = Upper Bound

Lecture Notes on Astrometry

application mod lat eq
Application (Mod. Lat. Eq.)
  • Preparation

Lecture Notes on Astrometry

application contd
Application (contd.)
  • Newton Correction
  • Stable Starter = Lower Bound

Lecture Notes on Astrometry