coordinate systems you are here – and where is that anyway? henry suters

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