Spheroidal Coordinates

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# Spheroidal Coordinates - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Spheroidal Coordinates' - perdy

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

Lecture Notes on Astrometry

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

Zenith

Pole

P

r

Horizon

f

j

Equator

Geocenter

H

Lecture Notes on Astrometry

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

Lecture Notes on Astrometry

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

Lecture Notes on Astrometry

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

Lecture Notes on Astrometry

Spheroidal to Rectangular

Lecture Notes on Astrometry

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

Lecture Notes on Astrometry

Latitude Equation

Lecture Notes on Astrometry

Modified Latitude Eq.
• New Variable
• New Equation

Lecture Notes on Astrometry

Derivation

Lecture Notes on Astrometry

Solving Modified Latitude Eq.
• Reduction of Argument Domain
• Variable Domain
• Newton Method
• Initial Guess

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

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
• Bracketed Solution
• Assumption 1
• Assumption 2
• Stable Starter = Upper Bound

Lecture Notes on Astrometry

Application (Mod. Lat. Eq.)
• Preparation

Lecture Notes on Astrometry

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

Lecture Notes on Astrometry