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 -180 o (West) to 180 o (East) The “y axis” is the Prime Meridian and passes through the Royal Observatory, Greenwich, England The “y” coordinate (latitude) varies from -90 o (South) to 90 o (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 360 o to travel around a circle and you must travel through 360 o 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 35 o 49’ 52” Longitude -84 o 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 35 o 49’ 52” Longitude -84 o 37’ 5” Latitude 35 o 49’ 52” N Longitude 84 o 37’ 5” W

More Notation Seconds may be indicated as fractional minutes Latitude 35 o 49’ 52” Longitude -84 o 37’ 5” Latitude 35 o 49’ 52” N Longitude 84 o 37’ 5” W Latitude 35 o ’ N Longitude 84 o ’ W

Still More Notation Minutes may be indicated as fractional degrees Latitude 35 o 49’ 52” Longitude -84 o 37’ 5” 35 o 49’ 52” N 84 o 37’ 5” W 35 o ’ N 84 o ’ W o N o W

Practice Identifying Locations 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 35 o 58’ 55” N 84 o 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 MarkersScissors

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) ShapeDirectionDistanceArea

Mercator Projection Preserves Direction but not Area or Shape

Gall-Peters Projection Preserves Direction and Area but not Shape

Mollweide Projection Preserves Area but not Direction or Shape

Robinson Projection Preserves 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