The Fly on the Wall

Map coordinate systems sound like pretty geeky stuff but the concept is quite simple.  They define the location of a point on a surface.  We easily think of this as being the distance from X and Y axes, but that is a relatively recent idea.  Columbus would not have known about it, nor Mercator.  Although artists of the Renaissance did use the grid system to create enlargements and reductions, the connection with cartography seems to have been illusive.  Perhaps because mapping is essentially the reduction of a curved surface onto a flat one.

Just getting a handle on the curved surface was hard enough.  Although ancient people knew the Earth was a sphere, measuring the position of various locations on its surface was an enduring challenge for many centuries.  That story quite logically begins with angles since they could be easily observed in the sky overhead.

The Greek astronomer Hipparchus (190–120 BC) was fascinated with arcs and angles and created a table of chords which expressed the relationship between angle and distance.  This became the foundation for trigonometry.  Others, including Menelaus (70-140 AD) and Ptolemy (100-170 AD) contributed important advancements in applying plane geometry to curved surfaces.  By the 8th century, Islamic scholars had perfected methods for calculating the Qibla which is the direction to Mecca from any location on Earth.  This work relied on spherical geometry, shared knowledge and careful astronomical observation.

True north, and the angle of one's position between the poles (latitude) is rather easily determined from the height of the sun at midday.  The angle of one's position around the polar axis from some reference (longitude) is much more difficult.  The ancients knew that time and longitude were related and, under ideal conditions, they could accurately gauge the passage of time over short intervals with hourglasses or water clocks.  This enabled them to record the occurrence of important astronomical events with respect to local solar time.

A pattern of such occurrences enabled prediction, and thus a reference for when could used as a reference for where.  An observer who witnessed the occultation of a particular star by the moon at 10¼ hours past midday could consult an ephemeris to learn when that exact event was predicted to occur in Alexandria.  If it had been expected at 8 hours past midday, the later observation must have occurred 33.75º to the west (perhaps in Grenada).

The challenge for map makers of the day was to flatten all this spherical stuff out onto a plane surface.  This involves a graticule which is the graphical representation of standard lines of latitude and longitude in 2 dimensions.  This concept was known to Ptolemy but the first example does not appear until a 13th century Byzantine map which also showed north at the top.  Plotting the graticule, and the position of known locations upon it, was entirely a matter of estimation.

This was the state of the art for geometry, geography and cartography for four centuries.  Angles and distances are the basis for what is now known as a polar coordinate system.  No other kind even existed.

Then the French philosopher and mathematician René Descartes (1596-1650) became famously inspired by a fly on his bedroom ceiling.  As the fly wandered about, Descartes realized that its exact position could always be described as its distance from 2 adjacent walls.  The cornices of those walls became the X and Y axes.  The distances to them from any muscid position were its coordinates.

Descartes was also famous for staying in bed most of the day, but when he finally got up he proceeded to develop the formulas necessary to generate squares, circles and ellipses using this coordinate system.  He established the convention of noting the horizontal distance first as X and then the vertical distance as Y enclosed in parenthesis and separated with a comma.  We also have Descartes to thank for exponential notation and a great many other innovations and insights which he published in 1637 as Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences.

The idea definitely caught on.  The English savant Sir Isaac Newton (1642-1727) made significant advancements in coordinate mathematics which subsequently lead to his invention of calculus.  Many others made important contributions in the 18th century and the Swiss mathematician Leonhard Euler (1707-1783) is responsible for developing the basic conversion formulae between polar and Cartesian coordinates.  This is hugely significant because that relationship is the foundation of GIS.  The translation of a latitude and longitude (polar coordinate) into a 2-dimensional location (Cartesian coordinate) is what map projections are all about.

The term "geographic projection" is therefore misleading because no coordinate conversion occurs at all.  The angular values for longitude and latitude, which are called geographic coordinates, are simply treated as though they were linear units and used as Cartesian X and Y values.  The result is not particularly useful since shape, size, distance and bearing are all distorted.  This is because a degree is not a unit of distance.

The geographic projection in which a degree is treated as a unit of distance. This makes the graticule appear to be a grid of perfect squares.

All other map projections are derived by calculation and express their dimensions in true linear units (usually meters).  There are literally thousands of modern map projections, each with its own strengths and inherent distortions, but all of them are drawn with Cartesian coordinates.  Longitude and latitude are mathematically converted to X and Y values that represent a distance from the edge or center of the projected plane.  This distance is expressed in real units as though the surface of the world had actually been stretched and flattened accordingly.  These values are called projected coordinates.

A map, of course, cannot possibly be that big.  The scale of the map is therefore the ratio of its physical dimensions to its projected dimensions.  If the map scale is 1:640000 then 1 inch on the map equals 640,000 inches on the projection (approximately 10 miles).  Since the projection itself is a distortion, this may not necessarily be equivalent to 10 miles on the surface of the Earth -- but it might be close.  Scale dimensions are called map coordinates.

For all practical purposes, there are really only 2 kinds of coordinate systems: polar and Cartesian.  What are often called "projected coordinate systems" are really just the formula used to make the conversion.  We have Newton and Euler and many others to thank for those formulas.  We have Descartes to thank for the realization that such formulas were even possible.

Of course, every feature on any given a map must be rendered with the same formula and with the same units.  If they are not, some things just won't line up like they should.  It's as though Descartes' fly wandered off in the wrong the direction.  That is called a bug.


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Posted in Geography, History on Nov 7, 2016