Mercator Projection

In my last post we saw how plotting round lon,lat coordinates on a flat X,Y map results in the map scale getting stretched in the east-west direction (while staying constant in the north-south direction) the farther we go away from the equator.  This makes it quite difficult to plot a course for navigation since the angles between lines will vary also.

Gerardus Mercator, born March 5, 1512, realized that he could maintain the angles (direction) by stretching the map in the north-south direction the same amount as it was being stretched in the east-west direction.  The scale is proportional to the secant of the latitude, but since there were no computers back then, and a table of secants had not yet been developed, Mercator likely determined the spacing graphically. 

John Snyder’s classic USGS Professional Paper 1395 describes the mathematics behind map projections and includes a section (pages 38-47) on the Mercator Projection.  Here’s a link to the USGS website which includes a link to a digital version :

http://pubs.er.usgs.gov/publication/pp1395

So problem solved - by stretching the map north-south and east-west by the same amount, sailing routes between two points will be shown as a straight line.

But what about Greenland and South America?  While the Mercator Projection has benefits for use in navigation, all the stretching that is done to preserve direction has a down side – area becomes greatly exaggerated.   We saw this in my previous post.  

A hui ho!



Baseball at the North Pole

In my last post we saw how Greenland, which has 1/8^th the land area of South America,  looks just as large or even larger on a flat map.  How can this be?  Well, ask yourself this. 



How can you take a spherical earth and represent it on a flat piece of paper?

There’s no simple way to do it.  Something has to be changed, stretched or broken to change spherical (round) coordinates of longitude and latitude (lon,lat) into planar (flat) X,Y coordinates.  One way to do this is to take the lon,lat values and plot them directly on an X,Y graph.  It looks something like this.

This seems pretty reasonable and simple.  Let’s look a little closer.  If you had a pair of magic shoes that allowed you to walk on water as well as over land, ice and snow, how far would you have to go to walk around the world following the equator? 

Well, it turns out to be about 40,075 kilometers (24,901 miles).  Now what if you followed the 30˚ north latitude line around the world?  You’d have to walk less far – about 34,706 km (21,565 mi).  And at 60˚ north latitude only 17,353 km (10,783 mi).  And at the North Pole?  Well, if you could set up a baseball diamond with the pitcher’s mound on the North Pole, you could hit a home run and literally run around the world in just 360 feet!

In fact, the crew of the nuclear submarine USS Seadragon did this on March 25, 1960.  I got this photo from a GIS user who lives on the Big Island of Hawaii.  Her father took the photo.  You can find a copy in the National Archives also.

http://www.archives.gov/publications/prologue/2006/spring/baseball.html

So what does this mean about the scale of our map?  In order to plot round lon,lat on flat X,Y, we had to keep stretching out our map as we moved to the north (and to the south).  The map scale stays constant at different longitudes, but as we have seen, can vary dramatically at different latitudes.  This makes it very difficult to plot a course for navigation purposes.  Gerardus Mercator (500^th birthday was on March 5, 2012) figured out a way to solve this problem.  I’ll explain his solution in my next post – hint: it’s the Mercator Projection.  For a preview, here’s a link to University of Colorado Peter H Dana’s website on map projections.

http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html

A hui ho!

Greenland vs. South America

Greenland vs. South America.  Which has the larger land area of the two?  Do you remember the map of the world up on the wall when you were in school?  It probably looked something like this:

Greenland looks even bigger than South America so the answer seems pretty clear. 

But wait… let’s look at them on a globe.  If you have a Hugg-A-Planet, go get it now and look.  You’ll see that Greenland is way smaller than South America.  If you don’t have a globe handy, here’s a two dimensional example:

If you Google it, you’ll find that Greenland has a land area of a little more than 2 million square kilometers (836,109 square miles), while South America has a land area of more than 17 million square kilometers (6,890,000 square miles).  So what is going on here?  South America is more than 8 times larger than Greenland.  Why does Greenland look larger on the flat map?  We have Gerardus Mercator (500^th birthday was on March 5, 2012) to thank for this curious phenomenon.  I’ll explain why this happens in my next post.

A hui ho!

The Countdown Begins...

Gerard de Kremer, aka Gerardus Mercator, the man who gave the world the Mercator projection - an amazing feat of mathematics, was born over 500 years ago.  His name is perpetuated in the variations of his original projection, including the Universal Transverse Mercator (UTM) projection and the World Mercator Auxiliary Sphere projection used by web mapping systems today.


Yes, there are issues with using the Mercator projection, and I’ll explore those in future posts, but for now, here are a few links to get you started learning about the man and his projection.


The man: http://en.wikipedia.org/wiki/Gerardus_Mercator


The projection with math: http://en.wikipedia.org/wiki/Mercator_projection


I'd love to hear from anyone interested in Mercator and helping celebrate his birthday.


A hui ho!