An Overview of Geodesy and Geographic Referencing Systems

The magic of geographic information systems is that they bring together and associate representations from diverse sources and infer relationships based on spatial references. This ability depends on our data sources using well defined coordinate referencing systems. This is not to say that the coordinate systems need to be the same for each data source, only that the relationship between the coordinate references with some shared conception of the surface of the earth needs to be well described. Indeed, there are thousands of perfectly legitimate coordinate systems in active use. The notion of spatial referenicing systems is one of the most fundamental and interesting ideas that all users of GIS should understand. This document provides an overview of the basic ideas.

## A Brief History and Theory of Latitude and Longitude

Almost everybody knows that Latitude and Longitude provide a framework for referencing places on the earth. It is interesting that an understanding that is thousands of years old has always been near the cutting edge of our cybernetic culture.

To tie this lecture together with the previous lecture on the context of geographic information systems, we will return to our framework for Modeling What's Important.

### Big Ideas Covered in this Lecture

• geospatial Datasets employ Coordinate Referencing Systems (CRS) to characterize the locations and shapes of things at or near the surfcae of the earth.
• Topographic maps are assumed to represent planar geometry consistent with cartesian coordinates.
• Latitude and Longitude are a way to distinctly reference any location on the planet. But Latitude and longitude are angular measures. They may be portrayed as if they were cartesian, but in most cases, this will distort the geography in disturbing ways.
• Creating reasonable portrayals of places requires us to understand how latitude and longitude reference a places on the planet, AND how lat and lon are transformed into cartesian coordinate systems that uphold the assumption that a map reflects consistent scale, direction, and shape proportionality corresponding with the things and relationships being represented.

### References

In order to establish a system for referencing places on the earth someone first had to establish a means of measuring the earth so that the measurements could be subdivided into a referencing system that could be used my many people to tag observations in the field. Eratosthenes of Cyrene (Egypt) Measured the Earth around 200 BC. He also figured out that a leap day would be necessary every 4 years in order to keep calendars consistent.

Around 50 years later, a Turk, Hipparchus of Rhodes invented a world-wide referencing system of meridians and paralells that we use to describe earth locations: latitude and longitude.

It is actually very simple to figure out your latitude and logitude (geographic coordinates) from first principles, using a simple protractor and a clock. Here's How! (page from PBS)

## Earth Models

Latitude and Longitude are angles, not, in themselves measures of distance. In order to use these angular displacements to understand distance relationships on the surface of the earth, you need to know the radius of the earth at the points being referenced. This would be simple if the earth was spherical; but it turns out that if you use this assumption, the distances that you calculate will have substantial errors. Since about 1700 it was postulated by Isaac Newton that the earth is not spherical and a few years layer this was confirmed by actual measurements of a degree of latitude in different parts of the world. (see This great essay by Isaac Asimov or this Article from Scientific American. Therefore observations of latitude and longitude made on the surface of the earth must be qualified with a radius which is derived from one of many possible models of the Size and shape of the earth (or Earth Model.)

See page 4 of the ESRI manual on Map Projections. for a discussion of the evolution of our concepts of the shape of the earth. When you gather together data for north america, you will find data registerd to the North American Datum of 1983, The World Geodetic Spheroid of 1994, and occaisionally the North American Datum of 1927. Data collected in other parts of the world likely use other local datums. Mis-specifying the earth model for a dataset can lead to displacements in your ground measurements of as much as 20 meters!

### The Graphical Problem of Latitude and Longitude

One fundamental assumptions about topographic maps of places include the followiong:

• the map has a scale that can be used to measure distances all over the map,
• that the shapes that are represented on the map are a reasonable approximation of the shapes
• the cardinal directions are at right angles to one-another.

These assumptions are generally termed the assumption of Planimetric Scale. They are also the same assumptions of Cartesian coordinate systems that allow us to use plane geometry to infer distances. Furthermore, they seem to be rather instinctive in human beings, in as much as if you show a person a map that violates these assumptions, you are in effect tricking that person into faulty conclusions about your subject.

A very common instance of this sort of trap arises from the fact that datasets we are given often reference locations of points and vertices with decimal degrees of latitude and longitude. And mapping software will graphically portray the geometry so described as if latitude and longitude were coordinates in a cartesian sense. Yet, latitude and longitude are polar coordinates. A degree of Longitude does not have a constant scale in terms of distances on the earth. At the equator, this distance is approximately 100 km. Near the poles, an entire degree could be spanned by your little toe!

 The map of Massachusetts in unprojected Decimal Degrees of Longitude and Latitude Note the proportion of the lines and the circle. The circle gives the impression that Worcester is approximately the same distance from Boston as New Hampshire, and that the state is nearly five times longer, east to west, as it is north-to-south. This misrepresentation of proportion has the same effect no matter how close you are zoomed in. Compare with the same data layers transformed according to the Massachusetts State Plane coordinate system system This shows an image that is much truer relative to distances you would actually experience on the ground. Worcester is actually much closer than it appeards before. And the state is only 3 times longer than it is tall. The grid on each map is one degree on a side. Using degrees of longitude and latitude as a cartesian coordinate system makes the grids appear square. But ion at the latitude of Massachusetts, the race of one degrees of longitude is much shorter than a degree of latitude.

## Transformation of Geographical Coordinates to Cartesian Coordinate Systems

While the system of latitude and longitude provides a consistent referencing system for anywhere on the earth, and it is therefore used in geographic databases that are not specific to a particular place. However, in order to portray our information on maps or for making calculations, we need to transform these angular measures to cartesian coordinates. These transformations amount to a mapping of geometric relationships expessed on the shell of a globe to a flattenable surface -- a mathematical problem that is figurastively refered to as Projection.

It turns out that any way you try to do this, you must unavoidably incorporate some distortion into the picture. Any projection has its area of least distortion. Projections can be shifted around in order to put this area of least distortion over the topographer's area of interest. Thus any projection can have an unlimited number of variations or cases that determined by standard paralells or meridians that adjust the location of the high-accuracy part of the projection.

### Projection Cases

In the case of the orthographic projection above, the area of least distortion is occurs where the figurative projection plane touches the model of the earth. To create a projection that works well for a particular area, we can create a case of the orthographic projection that has its point of tangency wherever we want:

Other projection methods are based on more complicated flattenable projection surfaces, and instead of points of tangency, spacial cases of these projections can be made by adjusting their Standard Parallels or Central Meridians

These images are by Erwin Raisz, who worked at the Harvard Institute for Geographical Exploration between 1931 and 1951 and wrote many fine books on cartography and topography.

### The Mercator Projection

Presentation of uniform scale is not always the goal when using maps as graphical calculators. A case in point is the Mercator's Cylyndrical Projection, which is unique among projections in terms of portraying lines of constant compass direction (rhumb lines) as straight lines on the map, as shown on this site by Carlos A Furuti. In a zone along its standard paralell, the Mercator projection has good scale and shape and direction presentation along with its property that makes a a terriffic tool for aiming missles and artillery! This is why the Transverse Case of the Mercator projection was invented and is in such common use in broad-scale series of national mapping projects.

### On-the-Fly Projection

Creating map projections used to be a VERY HARD THING TO DO! even just 30 years ago. And now we can project and unproject massive quantities of coordinates, transforming them backward and forward from Latitude and Longitude (assuming this or that earth model) to overlay precisely with data that are stored in some other coordinate space. It is truly amazing that humans have perfected a rich library of open-source software that can Forward Project geographic coordinates (latitude and longitude, + earth model) to any projected system; and also backward project) from any well described projected coordinates back to geographic coordinates -- all in the wink of an eye. We can be thankful for that. But there are still some details that we have to understand.

Automatic transformation of coordinate systems requires that datasets include machine-readable metadata. In about 2002, the makers of ArcMap added one more file to the schema of a shape file. The .prj file contains the description of the projection of a shape file, and if it exists, it is always copied with the shape file or dlements that are exported from it. This is the machine-readable metadata that allows ArcMap to know how to handle the dataset if any transformation (reprojection) is required. There are plenty of datasets that do not include such machine readable metadata. This includes data that are not created with ArcMap since 2002 and even some that are. So we should get used to understanding map projections and their properties. If you need to learn to set the coordinate system for a dataset, use ArcCatalog - as explained in the The ArcMap Projections Tutorial.

### Metadata Issues with Map Coordinate Systems

Given that geographic datasets are primarly concerned with spatial references, we really must have metadata (data about the data) describing the coordinate system that is embedded in the files. Here is an example of the FGDC Content Standards for Geospatial Metadata regarding Coordinate Systems But in actuality, because most datasets make use of special cases of projections, much of this metadata can be generated automatically if we know the following facts about the coordinate system of a dataset:

• The Earth Model -- a key parameter for projecting and unprojecting the coordinates.
• If the coordinates are unprojected latitude and longitude, the coordinate system is known as Geographic Coordinates (GCS).
• If the geometry in the datsaset is expressed in Projected Coordinates then the metadata should specify the following:
1. Projection Method
2. Projection Case
3. Coordinate Units (usually either Feet, or Meters)

### Standard Projection Systems

Even though the projection method is almost always some sort of very long system of equations, for which the cases plug in several parameters, there are conventions for projections and cases that have handy names, like the State Plane Coordinate System used by most state and local government agencies. Other countries have their National Grid projections. ANd almost always, systems of quadrangle maps use some variation of the Universal Transverse Mercator system of projectons and cases system that divides the world into 60 longitudinal zones. see map by Alan Morton.

### Parting Thoughts

Aside from being a very important and interesting and continuing thread in the history of humanity's struggle to understand their surroundings, the evolving understanding and application of Geodesy and Topography offer a very important lesson for anyone who would attempt to understand the world through representations or models. Understading the logic of one's referencing systems is crucial to understanding the utility of your conclusions. Choosing a projection means choosing one sort of error over another. This sort of choice comes into play with almost every decision that an educated person makes when creating and evaluating maps and GIS.