Geographic Coordinate Systems

Geographic Coordinate Systems
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Overview

Geographic Coordinate System:
- A coordinate system that enables every location on a planet (e.g., the Earth) to be specified
History:
- Is said to date back to Eratosthenes in the third century BC

The Most Common System

Latitude:
- Specifies the north-south position using an angular measure
- Ranges from \(0^\circ\) at the equator to \(90^\circ\) at the North and South poles
- Denoted by \(\phi\)
Longitude:
- Specifies the east-west position using an angular measure
- Ranges from \(0^\circ\) at the prime meridian to \(180^\circ\) at the international date line (with positive value to the east and negative values to the west)
- Denoted by \(\lambda\)

Simplifications and Assumptions

Getting Started:
- Use a spherical coordinate system
A Significant Issue:
- Planets are not spherical so the spherical coordinates must be mapped onto a reference ellipsoid
Other Significant Issues:
- Planets are not perfect ellipsoids
- Surfaces of planets are not smooth
- Points on the surface move relative to each other (e.g., motion of plates, subsidence, tidal movements)

Latitude and Longitude

From Longitude/Latitude to Cartesian Coordinates

Notation

From Longitude/Latitude to Cartesian Coordinates (cont.)

Derivation of the Conversion:
- \(\cos \phi = d / R \Rightarrow d = R \cos \phi\)
- \(\cos \lambda = p_{z}/d \Rightarrow p_{z} = d \cos \lambda = R\cos \phi \cos \lambda\)
- \(\sin \lambda = p_{x}/d \Rightarrow p_{x} = d \sin \lambda = R \cos \phi \sin \lambda\)
- \(p_{y} = R \sin \phi\)
Note:
- This is very similar to the conversion of spherical coordinates to Cartesian coordinates (but it uses the latitude not the colatitude)