coordinate systems you are here – and where is that anyway? henry suters
TRANSCRIPT
Coordinate Systems
You are Here – and Where is that Anyway?
Henry Suters
Location
We often need to identify a particular location
Graphing in math
Locations on a map
Position on a computer screen
etc.
Coordinates
Identifying a location on a 2-D surface takes 2 measurements
This is why we call it 2-D
In math these numbers are listed as an ordered pair
The first number is the x coordinate and the second is the y coordinate
Rectangular Coordinates
Each ordered pair corresponds to a point
A common way to do this is to use a rectangular coordinate system
Two number lines cross at right angles
The lines cross at the 0 point of both lines
The x axis is the horizontal line
The y axis is the vertical line
Rectangular Coordinates(cont.)
Graphing
Ordered pairs are graphed as follows:
Draw a vertical line through the location of the first coordinate on the x axis
Draw a horizontal line through the location of the second coordinate on the y axis
Where the lines cross is the point corresponding to the ordered pair.
Graphing Example(6, -2)
Map Coordinates
Some maps use a similar system to identify particular locations
The x coordinate is now called the Longitude
The y coordinate is now called the Latitude
Another name for the rectangular coordinate system is the Cartesian coordinates system – the same root word as Cartography – map making
Sample Map
Longitude and Latitude
The “x axis” is the Equator.
The “x coordinate” (longitude) varies from -180o (West) to 180o (East)
The “y axis” is the Prime Meridian and passes through the Royal Observatory, Greenwich, England
The “y” coordinate (latitude) varies from -90o (South) to 90o (North)
More about Longitude and Latitude
By tradition (and because of something we will discuss later) Longitude and Latitude are measured as angles
You must go through 360o to travel around a circle and you must travel through 360o of longitude to travel around the world from east to west
Minutes and Seconds
The lines of latitude and longitude are located too far apart for many purposes
Each degree of longitude or latitude is divided into 60 minutes
Each minute is divided into 60 seconds
The observatory is located at:
Latitude 35o 49’ 52” Longitude -84o 37’ 5”
Notation
Sometimes N and S are used instead of + and – for latitude, and W and E are used instead of + and - for longitude
Latitude 35o 49’ 52” Longitude -84o 37’ 5”
Latitude 35o 49’ 52” N Longitude 84o 37’ 5” W
More Notation
Seconds may be indicated as fractional minutes
Latitude 35o 49’ 52” Longitude -84o 37’ 5”
Latitude 35o 49’ 52” N Longitude 84o 37’ 5” W
Latitude 35o 49.8695’ N Longitude 84o 37.0899’ W
Still More Notation
Minutes may be indicated as fractional degrees
Latitude 35o 49’ 52” Longitude -84o 37’ 5”
35o 49’ 52” N 84o 37’ 5” W
35o 49.8695’ N 84o 37.0899’ W
35.83116o N 84.61816o W
Practice IdentifyingLocations on a Map
What are the longitude and latitude of the lower left hand corner of your map?
Notice the larger black tick marks along the edges of the map are located every minute.
What is located at 35o 58’ 55” N 84o 34’ 40” W?
What are the longitude and latitude of the Kingston Steam Plant (on the lower right of the map)?
Mapping a Sphere
The Earth is approximately spherical
How do you map a sphere onto a flat sheet of paper?
Using a flat (planar) map to describe a sphere will introduce significant distortions
Mapping Exercise
Materials
Foam ball
Tissue paper
Twist ties
Markers
Scissors
Mapping Exercise (cont.)
Wrap ball with tissue paper and secure with twist ties, trim excess with scissors
Use markers to draw an equator and a sampling of longitude and latitude lines
Draw continents and oceans
Unwrap ball and notice rectangular grid
Also notice distortions to shapes and sizes of drawn objects
Different Projections
There are many different ways to project portions of a sphere onto a planar map
All methods will distort some feature (maybe more)
Shape
Direction
Distance
Area
Mercator ProjectionPreserves Direction but not Area or Shape
Gall-Peters ProjectionPreserves Direction and Area but not Shape
Mollweide ProjectionPreserves Area but not Direction or Shape
Robinson ProjectionPreserves Nothing
Goode homolosine Projection
Preserves Area but not Direction or Shape
Spherical Coordinates
A rectangular coordinate system is not the easiest way to identify a point on a sphere
It is easier to think about angles from the center of the Earth relative to the Prime Meridian (longitude) and the equator (latitude)
This is why longitude and latitude are measured as angles in degrees, minutes and seconds
Spherical Coordinates (cont.)
Astronomy
Identifying a location in the sky can also be done using spherical coordinates
We think of celestial objects as being embedded in a sphere surrounding the Earth (even though they are not)
Declination
We also imagine projecting from the center of the Earth, through the equator to draw a circle around this Celestial sphere
Declination is the angle of an object (viewed from the center of the Earth) above or below the Celestial Equator (similar to latitude)
Right Ascension
We need a celestial analogue to the Prime Meridian
We pick this line to be the perpendicular to the Equator and to cross it at the same place where the Sun crosses on the Vernal Equinox (first day of Spring)
Right Ascension is the angle of an object (viewed from the center of the Earth) to the right or left of the Vernal Equinox (similar to longitude)
Astronomy